COMPLEXIFICATION OF VIRO THEOREM AND TOPOLOGY OF REAL

COMBINATORIAL HYPERSURFACES

Ilia Itenberg IRMAR, Université de Rennes 1 Campus de Beaulieu F-35042 Rennes cedex France

Eugenii Shustin^{∗} Tel Aviv University School of Mathematical Sciences Ramat Aviv, 69978 Tel Aviv Israel

Abstract

We generalize the Viro construction of real algebraic hypersurfaces, based on the use of convex subdivisions of Newton polyhedra, to arbitrary subdivisions and show that the combinatorial hypersurfaces appearing from arbitrary subdivisions satisfy the same topological restrictions (congruences, inequalities etc.) as algebraic varieties. Our construction can be seen as a complex version of the Viro gluing theorem.

Introduction

The Viro method of gluing of polynomials appeared to be the most powerful construction of real algebraic varieties with prescribed topology [32, 33, 36] (see also [13], 11.5, [20, 23]). It provides a nice interaction of real algebraic geometry, toric geometry and combinatorics, and gives rise to various generalizations and applications [4, 6, 19, 28, 29, 30, 31].

∗A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161. The work was completed during the stay of both authors at Max-Planck Institut für Mathematik. The authors thank these funds and institutions for hospitality and financial support.

Consider the simplest example of Viro’s construction. Let T_{d} ⊂ ℝ^{2}, d ∈ ℕ, be the triangle with vertices (0, 0), (0, d), (d, 0),

τ : T_{d} = Δ_{1} ∪ ... ∪ Δ_{N}

be a triangulation with the set of vertices V ⊂ ℤ^{2}, and σ : V → {±1} be any function. Out of this combinatorial data we construct piecewise-linear plane curves. Denote by T_{d}^{(1)}, T_{d}^{(2)} and T_{d}^{(3)} the copies of T_{d} under reflections with respect to the coordinate axes and the origin. Take in T_{d}^{(1)}, T_{d}^{(2)} and T_{d}^{(3)} triangulations symmetric to τ, and define σ at the vertices of new triangulations by

σ(ε_{1}i, ε_{2}j) = ε_{1}^{i}ε_{2}^{j}σ(i, j), (i, j) ∈ V, ε_{1}, ε_{2} = ±1.

Now in any triangle of the triangulation of T_{d}∪T_{d}^{(1)}∪T_{d}^{(2)}∪T_{d}^{(3)}, having vertices with different values of σ, we draw the mean line separating the vertices with different signs. The union C(τ, σ) of all these mean lines is a broken line homeomorphic to a disjoint union of circles and segments. Introduce natural maps:

Φ : T_{d} ∪ T_{d}^{(1)} ∪ T_{d}^{(2)} ∪ T_{d}^{(3)} → ℝP^{2}, Ψ : Int(T_{d} ∪ T_{d}^{(1)} ∪ T_{d}^{(2)} ∪ T_{d}^{(3)}) → ℝ^{2},

where Φ is continuous onto, identifying antipodal points on ∂(T_{d} ∪T_{d}^{(1)} ∪T_{d}^{(2)} ∪T_{d}^{(3)}), and Ψ is a homeomorphism. We call the curves Φ(C(τ, σ)) ⊂ ℝP^{2}, Ψ(C(τ, σ)) ⊂ ℝ^{2} projective and affine T-curves of degree d.

The Viro theorem states that a projective (resp., affine) T-curve of degree d is isotopic in ℝP^{2} (resp., in ℝ^{2}) to a nonsingular algebraic projective (resp., affine) curve of degree d, providing that the triangulation τ is convex, i.e., there exists a convex piecewise-linear function ν : T_{d} → ℝ, whose linearity domains are Δ_{1}, …, Δ_{N} (sometimes such triangulations are called coherent; see [13]). The Viro theorem, in fact, endows the combinatorial broken line C(τ, σ) with a rich structure, which implies a number of restrictions on the topology of C(τ, σ) (see an account of known results in [25, 34, 35, 38]).

On the other hand, there are non-convex triangulations (see the simplest classical example in Figure 1 [7]). There are examples of T-curves beyond the range of known algebraic curves [26], and there is some similarity between T-curves and algebraic curves: up to degree 6 all projective T-curves are isotopic to algebraic ones of the same degree [9], T-curves satisfy some consequences of the Bezout theorem [9] and the Harnack inequality [18, 15]. The natural question arising is if any T -curve is isotopic to an algebraic curve of the same degree, and if not, how far T-curves may differ from algebraic curves.

In the present paper, we propose a complex version of the general Viro construction. This version uses not necessarily convex subdivisions of Newton polyhedra. The construction produces PL-hypersurfaces. We call them C-hypersurfaces. They share a lot of topological properties with algebraic hypersurfaces. In particular, for any T-curve, we construct its complexification, i.e., an equivariant surface in ℂP^{2} or ℂ^{2}, whose real part coincides with the given T-curve, and show that an equivariant smoothing of this complexification is a flexible curve in the sense of [35]. As a

[figure]

Figure 1: A non-convex triangulation

corollary, we obtain that arbitrary T-curves satisfy all topological restrictions known for real algebraic curves.

In section 1 we describe the construction of complex and real C-hypersurfaces. In section 2 we study the topological properties of complex C-hypersurfaces of degree d in ℂP^{n} (i.e., constructed using a subdivision of the simplex in ℝ^{n} with vertices (0, 0, ..., 0), (d, 0, ..., 0), ..., (0, ..., 0, d)) and of double coverings of ℂP^{n} ramified along complex C-hypersurfaces. We prove the following statements.

• A C-hypersurface of degree d in ℂP^{n} is an orientable manifold, homologous to an algebraic hypersurface of degree d in ℂP^{n}.

• Any C-hypersurface M in ℂP^{n} is isotopic to a close smooth manifold M_{sm} of codimension 2 in ℂP^{n}. If M is real, i.e. invariant under the complex conjugation in ℂP^{n}, the isotopy can be made equivariant.

• Given a (real) C-hypersurface M, the tangent bundle to its smoothing M_{sm} is (equivariantly) isotopic to a (equivariant) bundle of complex hyperplanes. In particular, M_{sm} possesses an (equivariant) almost complex structure.

• A C-hypersurface in ℂP^{n} is simply connected if n > 2.

• For a C-hypersurface of degree d in ℂP^{n} one has π_{1}(ℂP^{n}\M) = ℤ/dℤ if n ≥ 2.

• Let M ⊂ ℂP^{n} be a C-hypersurface and M′ ⊂ ℂP^{n} a nonsingular algebraic hypersurface, both of degree d. Then H_{∗}(M) and H_{∗}(M′) are isomorphic as graded groups.

• Let n be a positive odd number, M a C-hypersurface of degree d in ℂP^{n} and M′ a nonsingular algebraic hypersurface of degree d in ℂP^{n}. Then the lattices (H_{n−1}(M), B_{M}) and (H_{n−1}(M′), B_{M′}) (where B_{M} : H_{n−1}(M)×H_{n−1}(M) → ℤ and B_{M′} : H_{n−1}(M′)×H_{n−1}(M′) → ℤ are the intersection forms of M and M′, respectively) are isomorphic.

• A C-surface of degree d in ℂP^{3} is homeomorphic to a nonsingular algebraic surface of degree d in ℂP^{3}.

• Any C-curve in ℂP^{2} is isotopic to an algebraic curve of the same degree.

Section 3 is devoted to topology of real C-hypersurfaces in ℂP^{n}. We prove for them the generalized Harnack inequality and the Gudkov-Rokhlin and Gudkov-Krahnov-Kharlamov congruences. For real C-surfaces in ℂP^{3} we also prove inequalities similar to the Comessatti ones.

1 Construction of complex and real Chypersurfaces

1.1 Notations and definitions

Further on the term polyhedron (polygon) means a convex polyhedron (polygon) in ℝ^{n}, n ≥ 2, whose vertices have nonnegative integral coordinates.

Given a polynomial

F = ∑_{i_{1},...,i_{n}} A_{i_{1}...i_{n}}z_{1}^{i_{1}} · ... · z_{n}^{i_{n}} ,

by Δ(F) we denote its Newton polyhedron, i.e., the convex hall of the set

{(i_{1}, ..., i_{n}) ∈ ℝ^{n} : A_{i_{1}...i_{n}} ≠ 0} .

The truncation of F on a face δ of Δ(F) is the polynomial

F^{δ} = ∑_{(i_{1},...,i_{n})∈δ} A_{i_{1}...i_{n}}z_{1}^{i_{1}} · ... · z_{n}^{i_{n}} .

A polynomial F ∈ ℂ[z_{1}, ..., z_{n}] is called nondegenerate, if F and any truncation F^{δ} on a proper face δ of Δ(F) have a nonsingular zero set in (ℂ^{∗})^{n} (cf. [32]).

1.2 Complexification of the momentum map

Let Δ ⊂ ℝ^{n} be a polyhedron, and let μ_{Δ} : (ℝ^{∗})^{n} → I(Δ) be the momentum map (see [1, 2, 12], [13], 6.1), where I(Δ) is the complement in Δ of the union of all its proper faces,

μ_{Δ}(x_{1}, ..., x_{n}) = \frac{∑_{(i_{1},...,i_{n})∈Δ} x_{1}^{i_{1}}...x_{n}^{i_{n}} · (i_{1}, ..., i_{n})}{∑_{(i_{1},...,i_{n})∈Δ} x_{1}^{i_{1}}...x_{n}^{i_{n}}} .  (1)

Split the complex torus in the product (ℂ^{∗})^{n} = (ℝ_{+}^{∗})^{n} × (S^{1})^{n}:

(z_{1}, ..., z_{n}) ∈ (ℂ^{∗})^{n} ↦ (|z_{1}|, ..., |z_{n}|) ∈ (ℝ_{+}^{∗})^{n}, (\frac{z_{1}}{|z_{1}|}, ..., \frac{z_{n}}{|z_{n}|}) ∈ (S^{1})^{n} .

Put ℂ(I(Δ)) = I(Δ) × (S^{1})^{n} ⊂ (ℂ^{∗})^{n}, and define the complexification ℂΔ of Δ to be the closure of ℂI(Δ) in ℂ^{n}.

Proposition 1.1 The complexification ℂΔ of Δ is a singular PL-manifold with boundary. The singular set of ℂΔ is the union of ℂδ over all faces δ ⊂ Δ, which are intersections of Δ with coordinate planes of dimension > n − dim Δ + dim δ. The real part ℝΔ of ℂΔ is the union of Δ with all its symmetric copies with respect to the coordinate hyperplanes.

Proof. Straightforward. □

Define the complex momentum map ℂμ_{Δ} : (ℂ^{∗})^{n} → ℂ(I(Δ)) by

ℂμ_{Δ}(x_{1}v_{1}, ..., x_{n}v_{n}) = (μ_{Δ}(x_{1}, ..., x_{n}), (v_{1}, ..., v_{n})) ∈ I(Δ) × (S^{1})^{n} = ℂ(I(Δ)), (2)
(x_{1}, ..., x_{n}) ∈ (ℝ_{+}^{∗})^{n}, (v_{1}, ..., v_{n}) ∈ (S^{1})^{n} .

As an easy consequence of classical results we obtain the following statement.

Proposition 1.2 The map ℂμ_{Δ} is surjective and commutes with the complex conjugation Conj. It is a diffeomorphism when dim Δ = n. The real part of ℂ(I(Δ)) is the image of (ℝ^{∗})^{n}.

Below we use the following interaction of the complex momentum map with the action of ℤ^{n} in ℝ^{n}

\overline{a} = (a_{1}, ..., a_{n}) ∈ ℤ^{n}, \overline{t} = (t_{1}, ..., t_{n}) ∈ ℝ^{n} ↦ \overline{a} + \overline{t} = (t_{1} + a_{1}, ..., t_{n} + a_{n}) ∈ ℝ^{n} ,

and the actions of SL(ℤ, n) in ℝ^{n} and (ℝ^{∗})^{n}

A ∈ SL(ℤ, n), \overline{x} ∈ ℝ^{n} ↦ A\overline{x} ∈ ℝ^{n} ,

A ∈ SL(ℤ, n), \overline{x} = (x_{1}, ..., x_{n}) ∈ (ℝ^{∗})^{n} ↦ \overline{x}^{A} = (∏_{i=1}^{n} x_{i}^{a_{i1}}, ..., ∏_{i=1}^{n} x_{i}^{a_{in}}) .

Proposition 1.3 Let Δ ⊂ ℝ^{n} be a polygon. If \overline{a} = (a_{1}, ..., a_{n}) ∈ ℤ^{n}, A ∈ SL(ℤ, n), and \overline{a} + AΔ lies in the nonnegative orthant ℝ_{+}^{n}, then

ℂμ_{\overline{a}+AΔ}(\overline{z}) = (\overline{a} + Aμ_{Δ}(\overline{x}^{A}), \overline{v}) ,  (3)

where \overline{z} = (\overline{x}, \overline{v}) ∈ (ℝ_{+}^{∗})^{n} × (S^{1})^{n}.

Proof. The result follows from the definition

ℂμ_{\overline{a}+AΔ}(\overline{z}) = (μ_{\overline{a}+AΔ}(\overline{x}), \overline{v}) ,

and the classically known relation

μ_{\overline{a}+AΔ}(\overline{x}) = \overline{a} + Aμ_{Δ}(\overline{x}^{A}) □

1.3 Real and complex chart of a polynomial

Let F ∈ ℂ[z_{1}, ..., z_{n}] be a nondegenerate polynomial with Newton polyhedron Δ(F) = Δ. The closure ℂCh(F) ⊂ ℂΔ of the set ℂμ_{Δ}({F = 0} ∩ (ℂ^{∗})^{n}) is called the (complex) chart of the polynomial F. If F is real then ℝCh(F) = ℂCh(F)∩ℝΔ is called the real chart of F.

This definition is a modification of the classical one [23, 32] (cf. [27]).

Proposition 1.4 Suppose that Δ belongs to the open positive orthant (ℝ_{+}^{∗})^{n} = {(x_{1}, ..., x_{n}) ∈ ℝ^{n} : x_{1} > 0, …, x_{n} > 0}. Then the set ℂCh(F) is a PL-submanifold in ℂΔ of codimension 2 with boundary ∂ℂCh(F) = ℂCh(F) ∩ ∂ℂΔ. It is smooth in ℂI(Δ), and, for any proper face δ of Δ of positive dimension,

ℂCh(F) ∩ ℂδ = ℂCh(F^{δ}) .  (4)

If F is real then ℂCh(F) is invariant with respect to Conj, and ℝCh(F) is a PLsubmanifold in ℝΔ of codimension 1 with boundary ∂ℝCh(F) = ℝCh(F) ∩ ∂ℝΔ.

Remark 1.5 If Δ intersects with coordinate hyperplanes, then the statement of Proposition 1.4 holds true when substituting ℂΔ\Sing(ℂΔ) and ℝΔ\Sing(ℂΔ) for ℂΔ and ℝΔ, respectively, but we will not use this below.

Proof of Proposition 1.4.

(i) If dim Δ = n then by Proposition 1.2, ℂCh(F)∩ℂ(I(Δ)) is a smooth manifold of codimension 2.

So, we have to consider only the behavior of ℂCh(F) on ∂ℂΔ.

(ii) Assume that dim Δ < n and A ∈ SL(ℤ, n) is such that AΔ ⊂ (ℝ_{+}^{∗})^{n} is parallel to the coordinate hyperplane i_{n} = 0. By (3) there is a diffeomorphism which takes the pair (ℂΔ, ℂCh(F (\overline{z}))) onto the pair (ℂ(AΔ), ℂCh(F (\overline{z}^{A}))). Now, if i_{n}|_{Δ} = a, then Δ′ = Δ − (0, ..., 0, a) ⊂ \{i_{n} = 0\}, F (\overline{z}) = z_{n}^{a}F′(\overline{z}), and by formula (3) the pair (ℂΔ, ℂCh(F)) naturally splits into the product (CΔ′, ℂCh(F′)) × S^{1}, thereby one reduces the dimension of the ambient space.

(iii) To prove (4) (for dim Δ = n), we note that any point \overline{z}^{0} ∈ ℂCh(F) ∩ ∂ℂΔ is a limit of ℂμ_{Δ}(γ(t)) as t > 0, t → 0, where

γ(t) = (λ_{1}t^{k_{1}} + O(t^{k_{1}+1}), ..., λ_{n}t^{k_{n}} + O(t^{k_{n}+1}))

is a curve lying in {F = 0} ∩ (ℂ^{∗})^{n} with λ_{1}...λ_{n} ≠ 0, (k_{1}, ..., k_{n}) ∈ ℤ^{n}\{0}. Let δ be the maximal (proper) face of Δ, where the function (i_{1}, ..., i_{n}) ∈ Δ ↦ i_{1}k_{1} +...+i_{n}k_{n} achieves its minimum (equal to l). Then

0 = F (γ(t)) = F^{δ}(λ_{1}, ..., λ_{n})t^{l} + O(t^{l+1}) ⟹ F^{δ}(λ_{1}, ..., λ_{n}) = 0 ,

γ(t) = (\overline{x}, \overline{v}) ∈ (ℝ_{+}^{∗})^{n} × (S^{1})^{n}, where

\overline{x} = (|λ_{1}|t^{k_{1}} + O(t^{k_{1}+1}), ..., |λ_{n}|t^{k_{n}} + O(t^{k_{n}+1})), \overline{v} = (\frac{λ_{1}}{|λ_{1}|} + O(t), ..., \frac{λ_{n}}{|λ_{n}|} + O(t)) .

So, one derives

ℂμ_{Δ}(γ(t)) = (μ_{Δ}(\overline{x}), \overline{v}) = (\frac{∑_{(i_{1},...,i_{n})∈Δ}(i_{1}, ..., i_{n})x_{1}^{i_{1}}...x_{n}^{i_{n}}}{∑_{(i_{1},...,i_{n})∈Δ} x_{1}^{i_{1}}...x_{n}^{i_{n}}} , \overline{v})

= (\frac{t^{l} · ∑_{(i_{1},...,i_{n})∈δ}(i_{1}, ..., i_{n})|λ_{1}|^{i_{1}}...|λ_{n}|^{i_{n}} + O(t^{l+1})}{t^{l} · ∑_{(i_{1},...,i_{n})∈δ} |λ_{1}|^{i_{1}}...|λ_{n}|^{i_{n}} + O(t^{l+1})}, \overline{v}) ⟶^{t→0} ℂμ_{δ}(λ_{1}, ..., λ_{n}).

Hence ℂCh(F) ∩ ℂδ ⊂ ℂCh(F^{δ}).

Now let (λ_{1}, ..., λ_{n}) ∈ (ℂ^{∗})^{n}, F^{δ}(λ_{1}, ..., λ_{n}) = 0, where δ is a proper face of Δ, and let (k_{1}, ..., k_{n}) ∈ K(δ), where K(δ) ⊂ ℤ^{n}\{0} consists of vectors (k_{1}, ..., k_{n}) such that δ is the maximal face of Δ on which the functional (i_{1}, ..., i_{n}) ∈ Δ ↦ i_{1}k_{1} + ... + i_{n}k_{n} achieves its minimum. Then F^{δ}(λ_{1}t^{k_{1}}, ..., λ_{n}t^{k_{n}}) = t^{l}F^{δ}(λ_{1}, ..., λ_{n}) = 0, l = i_{1}k_{1} + ... + i_{n}k_{n}, (i_{1}, ..., i_{n}) ∈ δ, and there exist smooth functions φ_{1}(t) = λ_{1} + O(t), ..., φ_{n} = λ_{n} + O(t) such that the curve γ(t) = (t^{k_{1}}φ_{1}(t), ..., t^{k_{n}}φ_{n}(t)) lies on {F = 0}. Indeed,

F (γ(t)) = t^{λ}F^{δ}(φ_{1}, ..., φ_{n}) + t^{λ+1}G(t, φ_{1}, ..., φ_{2}) = 0
⟺ F^{δ}(φ_{1}, ..., φ_{n}) + tG(t, φ_{1}, ..., φ_{2}) = 0 ,

hence the existence of φ_{1}, ..., φ_{n} defined for small t > 0 follows from the implicit function theorem and the fact that F^{δ} = 0 is nonsingular in (ℂ^{∗})^{n}. The above argument shows that

ℂμ_{δ}(λ_{1}, ..., λ_{n}) = lim_{t→0} ℂμ_{Δ}(γ(t)) .

(iv) Assume that dim Δ = n. Let δ ⊂ Δ be a proper face, dim δ = s > 0, and \overline{w} = (y_{1}v_{1}^{0}, ..., y_{n}v_{n}^{0}) ∈ ℂI(δ)∩ℂCh(F), where (y_{1}, ..., y_{n}) ∈ I(δ), (v_{1}^{0}, ..., v_{n}^{0}) ∈ (S^{1})^{n}. We will describe ℂCh(F) in a neighborhood U\overline{w} of the point \overline{w} in ℂΔ.

Represent ℝ^{n} as ℝ^{s} × ℝ^{n−s} and denote by pr_{s} : ℝ^{n} → ℝ^{s}, pr_{n−s} : ℝ^{n} → ℝ^{n−s} the natural projections. According to Proposition 1.3, we can assume that pr_{n−s}(δ) is a point (p_{s+1}, ..., p_{n}) with positive integers p_{s+1}, ..., p_{n}, so δ̃ = pr_{s}(δ) = δ − (0, ..., 0, p_{s+1}, ..., p_{n}), and Δ̃\δ̃ lies in the positive orthant. Then F (z_{1}, ..., z_{n}) = z_{s+1}^{p_{s+1}}...z_{n}^{p_{n}} F̃(z_{1}, ..., z_{n}), where

F̃(z_{1}, ..., z_{n}) = F̃ δ̃(z_{1}, ..., z_{s}) + ∑_{(i_{1},...,i_{n})∈Λ} A_{i_{1}...i_{n}}z_{1}^{i_{1}}...z_{s}^{i_{s}} · M_{i_{s+1}...i_{n}} ,  (5)

Λ = {(i_{1}, ...i_{n}) ∈ Δ̃ : i_{s+1}, ..., i_{n} > 0} ∩ ℤ^{n}, M_{i_{s+1}...i_{n}} = z_{s+1}^{i_{s+1}}...z_{n}^{i_{n}} .

Note that there are uniquely defined x_{1}^{0}, ..., x_{s}^{0} > 0 such that (y_{1}, ..., y_{s}) = μδ̃(x_{1}^{0}, ..., x_{s}^{0}). The condition on \overline{z} = (\overline{x}, \overline{v}) ∈ (ℝ_{+}^{∗})^{n} × (S^{1})^{n} = (ℂ^{∗})^{n} such that ℂμ_{Δ}(\overline{x}, \overline{v}) ∈ U\overline{w} can be expressed as follows. For \overline{v} = (v_{1}, ..., v_{n}) it, clearly, is

|v_{1} − v_{1}^{0}| < ε, ..., |v_{n} − v_{n}^{0}| < ε,  (6)

where ε > 0 is sufficiently small. For \overline{x} = (x_{1}, ..., x_{n}) we have that μΔ̃(\overline{x}), equal to

\frac{∑_{(i_{1},...,i_{s})∈δ̃} x_{1}^{i_{1}}...x_{s}^{i_{s}} · (i_{1}, ..., i_{s}, 0, ..., 0) + ∑_{(i_{1},...,i_{n}∈Λ} x_{1}^{i_{1}}...x_{s}^{i_{s}} · |M_{i_{s+1}...i_{n}}| · (i_{1}, ..., i_{n})}{∑_{(i_{1},...,i_{s})∈δ̃} x_{1}^{i_{1}}...x_{s}^{i_{s}} + ∑_{(i_{1},...,i_{n}∈Λ} x_{1}^{i_{1}}...x_{s}^{i_{s}} · |M_{i_{s+1}...i_{n}}|} ,

is close to (y_{1}, ..., y_{s}, 0, ..., 0). This means that in the latest expression the summands over Λ are small with respect to the sums over δ̃; hence pr_{s}(μΔ̃(\overline{x})) is close to μδ̃(x_{1}, ..., x_{s}), which means that (x_{1}, ..., x_{s}) is close to (x_{1}^{0}, ..., x_{s}^{0}). Coming back to the sums over Λ, we obtain that M_{i_{s+1}...i_{n}} must be close to zero, (i_{s+1}, ..., i_{n}) ∈ pr_{n−s}(Λ). So, without loss of generality the conditions on \overline{x} can be expressed as

|x_{l} − x_{l}^{0}| < ε, l = 1, ..., s, x_{s+1}^{i_{s+1}}...x_{n}^{i_{n}} < ε, (i_{s+1}, ..., i_{n}) ∈ pr_{n−s}(Λ) .  (7)

Let U′ ⊂ (ℝ_{+}^{∗})^{n} and U′′ ⊂ (S^{1})^{n} be given by (7) and (6), respectively, and note that U′ = pr_{s}(U′) × pr_{n−s}(U′). Put σ = pr_{n−s}(Δ̃). Then μ_{σ} takes pr_{n−s}(U′) onto a set U_{0}\∂σ, where U_{0} is a neighborhood of the origin (a vertex of σ !) in σ.

Since F̃ is nondegenerate, we can assume that

\frac{∂F̃}{∂x_{1}} (x_{1}^{0}, ..., x_{s}^{0}, 0, ..., 0) = \frac{∂F̃ δ̃}{∂x_{1}}(x_{1}^{0}, ..., x_{s}^{0}) ≠ 0 .

Hence, by (5) and the implicit function theorem, the hypersurface {F = 0} can be described in U′ × U′′ (with, possibly, smaller ε) by equations

x_{1} = x_{1}^{0} + G, v_{1} = v_{1}^{0} + H ,  (8)

where G and H are vanishing at zero smooth functions of x_{2} − x_{2}^{0}, ..., x_{s} − x_{s}^{0}, v_{2} − v_{2}^{0}, ..., v_{n} − v_{n}^{0}, and M_{i_{s+1}...i_{n}}, (i_{s+1}, ..., i_{n}) ∈ pr_{n−s}(Λ), where x_{2}, ..., x_{n}, v_{2}, ..., v_{n} are arbitrary satisfying (6) and (7).

Now we show that U\overline{w} is homeomorphic to pr_{s}(U′) × U_{0} × U′′ via an extension of ℂμ_{Δ}. Indeed, U\overline{w}\∂ℂΔ is homeomorphic via ℂμ_{Δ} to pr_{s}(U′) × pr_{n−s}(U′) × U′′ ≃ pr_{s}(U′′)×(U_{0}\∂σ)×U′′. As we saw, the points in U\overline{w} ∩∂Δ are lim_{t→0} ℂμ_{Δ}(\overline{x}(t), \overline{v}(t)) for all curves \overline{x}(t) ∈ U′, \overline{v}(t) ∈ U′′, t > 0, such that lim_{t→0} \overline{v}(t) ∈ U′′, lim_{t→0}(x_{1}(t), ..., x_{s}(t)) ∈ pr_{s}(U′) and

x_{s+1}(t) = λ_{1}t^{k_{1}} + O(t^{k_{1}+1}), ..., x_{n}(t) = λ_{n−s}t^{k_{n−s}} + O(t^{k_{n−s}+1}) .

Restriction (7) means, first, that the vector (k_{1}, ..., k_{n−s}) belongs to the cone generated by the interior normal vectors to the (s − 1)-dimensional faces of σ, which contain the origin, and, second, that λ_{1}^{i_{s+1}}...λ_{n−s}^{i_{n}} < ε for all (i_{s+1}, ..., i_{n}) ∈ pr_{n−s}(Λ) orthogonal to (k_{1}, ..., k_{n−s}). On the other hand, the same data determine a point lim_{t→0} μ_{σ}(x_{s+1}(t), ..., x_{n}(t)) ∈ U_{0} ∩ ∂σ. This gives a one-to-one correspondence between U\overline{w} ∩∂Δ and pr_{s}(U′)×(U_{0} ∩∂σ)×U′′, which respects the combinatorial structure: the interior of a face ψ ⊂ U\overline{w} ∩ ∂Δ corresponds to pr_{s}(U′) × pr_{n−s}(Iψ̃) × U′′, where ψ̃ = ψ − (0, ..., 0, p_{s+1}, ..., p_{n}). The above correspondence provides a homeomorphism of U\overline{w} and pr_{s}(U′)×U_{0}×U′′, since close points in U\overline{w} ∩∂Δ come from close curves γ(t) which in turn bear close points in pr_{s}(U′)×(U_{0}∩∂σ)×U′′. Consequently, we can introduce in U_{0} coordinates θ_{1}, ..., θ_{n−s} such that

U_{0} = {0 ≤ θ_{1} < ε, −ε < θ_{j} < ε, j = 2, ..., n − s}, U_{0} ∩ ∂σ = {θ_{1} = 0} ,

and M_{i_{s+1}...i_{n}}, (i_{s+1}, ..., i_{n}) ∈ pr_{n−s}(Λ), are continuous functions of θ_{1}, ..., θ_{n−s}, vanishing at zero. Thus, formulae (8) allow us to parameterize ℂCh(F) ∩ U\overline{w} by the following subset in pr_{s}(U′) × U_{0} × U′′:

x_{1} = x_{1}^{0} + G′, v_{1} = v_{1}^{0} + H′ ,

where G′ and H′ are vanishing at zero smooth functions of x_{2}, ..., x_{s}, v_{2}, ..., v_{n}, θ_{1}, ..., θ_{n−s}. This subset is a subvariety of codimension 2 with a boundary on pr_{s}(U′) × (U_{0} ∩ ∂σ) × U′′. Hence ℂCh(F) ∩ U\overline{w} is a subvariety of codimension 2 in U\overline{w} with boundary on U\overline{w} ∩ ∂Δ.

(v) The statement of Lemma for ℝCh(F) can be proven as in the complex case. □

Corollary 1.6 If dim Δ = 2, Δ ⊂ ℝ^{2}, then ℂCh(F) is a smooth surface with boundary.

1.4 Digression: real resolution of toric varieties

Let Δ ⊂ (ℝ_{+}^{∗})^{n} be an n-dimensional polyhedron, F be a nondegenerate polynomial with Newton polyhedron Δ. Denote by Tor_{ℂ}(Δ) the toric variety over ℂ defined by Δ, and by Z(F) the (closed) hypersurface in Tor_{ℂ}(Δ) defined by F.

Proposition 1.7 There exists an equivariant surjective map ν : ℂΔ → Tor_{ℂ}(Δ) such that

• ν(ℂCh(F)) = Z(F),

• ν|_{ℂI(Δ)} takes ℂI(Δ) diffeomorphically to (ℂ^{∗})^{n} ⊂ Tor_{ℂ}(Δ),

• for any face δ, dim δ = s < n, the map ν|_{ℂI(δ)} is a submersion of ℂI(δ) onto (ℂ^{∗})^{s} ⊂ Tor_{ℂ}(δ) ⊂ Tor_{ℂ}(Δ).

Proof. Let us represent Tor_{ℂ}(Δ) as the closure of the variety

{(z_{1}^{i_{1}}...z_{n}^{i_{n}})_{(i_{1},...,i_{n})∈Δ∩ℤ^{n}} : (z_{1}, ..., z_{n}) ∈ (ℂ^{∗})^{n}} ⊂ ℂP^{N−1} ,

where N = #(Δ ∩ ℤ^{n}) (see, for example [12]). We define ν|_{ℂI(Δ)} = (ℂμ_{Δ})^{−1}, which is a required diffeomorphism of ℂI(Δ) and (ℂ^{∗})^{n} ⊂ Tor_{ℂ}(Δ), and extend it to ∂ℂΔ. Namely, given a face δ ⊂ Δ and a point \overline{w} = ℂμ_{δ}(\overline{z}) ∈ ℂI(δ), \overline{z} = (z_{1}, ..., z_{n}) ∈ (ℂ^{∗})^{n}, we put ν(\overline{w}) = (a_{i_{1}...i_{n}})_{(i_{1},...,i_{n})∈Δ∩ℤ^{n}}, where

a_{i_{1}...i_{n}} = z_{1}^{i_{1}}...z_{n}^{i_{n}}, (i_{1}, ..., i_{n}) ∈ δ, a_{i_{1}...i_{n}} = 0, (i_{1}, ..., i_{n}) ∈ Δ\δ .

Clearly, ν|_{ℂI(δ)} satisfies the required property. It remains only to explain that ν is continuous. Indeed, in the previous notation \overline{w} = lim_{t→∞} ℂμ_{Δ}(γ(t)), where

γ(t) = (z_{1}t^{k_{1}} + O(t^{k_{1}−1}), ..., z_{n}t^{k_{n}} + O(t^{k_{n}−1})), t > 0 ,

is a curve in (ℂ^{∗})^{n} with (k_{1}, ..., k_{n}) ∈ ℤ^{n}\{0} belonging to the dual cone of δ with respect to Δ. Then one can easily check that lim_{t→∞} γ(t) ∈ ℂP^{N−1} is just ν(\overline{w}) defined as above. □

1.5 Gluing of charts

Let S be a subdivision of a polyhedron Δ ⊂ ℝ^{n} into equidimensional polyhedra: Δ = Δ_{1} ∪ ... ∪ Δ_{N} (i.e., Δ_{i} ∩ Δ_{j} is empty or a common proper face), and let A = \{A\overline{i}, \overline{i} ∈ Δ ∩ ℤ^{n}\} be a collection of complex numbers such that A\overline{i} ≠ 0 if \overline{i} is a vertex of some Δ_{i}, 1 ≤ i ≤ N. Further on, speaking on subdivisions of polyhedra and corresponding collections of numbers, we always assume the above properties.

Assume that the polynomials

F_{k}(z_{1}, ..., z_{n}) = ∑_{(i_{1},...,i_{n})∈Δ_{k}} A_{i_{1}...i_{n}}z_{1}^{i_{1}} · ... · z_{n}^{i_{n}} , k = 1, ..., N,

are nondegenerate. We define the gluing of the complex charts of F_{1}, …, F_{N} (below referred to as a C-hypersurface in ℂΔ) by

ℂCh(S, A) = ⋃_{k=1}^{N} ℂCh(F_{k}) .

If all the polynomials F_{1}, …, F_{N} are real, we define the gluing of the real charts of F_{1}, …, F_{N} (which is called a C-hypersurface in ℝΔ) by

ℝCh(S, A) = ⋃_{k=1}^{N} ℝCh(F_{k}) .

Proposition 1.8 Let Δ ⊂ (ℝ_{+}^{∗})^{n}. Then the set ℂCh(S, A) is a PL-manifold of codimension 2 in ℂΔ with boundary ∂ℂCh(S, A) = ∂ℂΔ ∩ ℂCh(S, A). If all the numbers in A are real, then ℂCh(S, A) is invariant with respect to Conj, the set ℝCh(S, A) is a PL-manifold of codimension 1 in ℝΔ with boundary ∂ℝCh(S, A) = ∂ℝΔ ∩ ℝCh(S, A).

Remark 1.9 (a) Neither ℂCh(S, A), nor ℝCh(S, A) are smooth in general. Indeed, for the polynomials

F_{1}(x, y) = x(y − x − 1), F_{2}(x, y) = xy − x + 1

with the Newton triangles

Δ_{1} = [(1, 0), (1, 1), (2, 0)], Δ_{2} = [(0, 0), (1, 0), (1, 1)]

the closures of curves

μ_{Δ_{1}}(\{F_{1} = 0\}) = \{(\frac{2 + 3x}{2 + 2x} , \frac{x + 1}{2 + 2x}) : x > 0\} ,

μ_{Δ_{2}}(\{F_{2} = 0\}) = \{(\frac{2x − 1}{2x} , \frac{x − 1}{2x}) : x > 0\}

have the common limit point (1, 1/2) on the edge [(1, 0), (1, 1)], and different tangent lines

y = \frac{1}{2}, y = x − \frac{1}{2}

at this point.

(b) Similarly to Proposition 1.4, the statement of Proposition 1.8 holds for Δ intersecting coordinate hyperplanes, when substituting ℂΔ\Sing(ℂΔ) and ℝΔ\Sing(ℂΔ) for ℂΔ and ℝΔ, respectively.

Proof of Proposition 1.8. In view of Proposition 1.4, we have to study only the behavior of ℂCh(S, A) of faces δ ⊂ Δ_{k} ∩ Δ_{l}, k ≠ l, dim δ = s > 0.

Let I(δ) ⊂ I(Δ), δ = Δ_{1} ∩...∩Δ_{k}, δ ⊄ Δ_{k+1} ∪...∪Δ_{N}. Pick a point w ∈ ℂI(δ)∩ ℂCh(S, A). Following the proof of Proposition 1.4, we impose assumptions of step (iv) and obtain that a neighborhood U_{w} of w in ℂΔ is the union of neighborhoods U_{w,1}, ..., U_{w,k} of w in ℂΔ_{1}, ..., ℂΔ_{k}, respectively, so that U_{w,m} is parameterized by pr_{s}(U′_{m})×U_{0,m}×U′′, where pr_{s}(U′_{1}) = ... = pr_{s}(U′_{k}), U′′ is common for all m = 1, ..., k, and U_{0,m} is a neighborhood of the point pr_{n−s}(δ) = (p_{s+1}, ..., p_{n}) in pr_{n−s}(Δ_{m}), and, moreover, these parameterizations are compatible on the common faces of Δ_{1}, ..., Δ_{k}. In turn, ℂCh(S, A) ∩ U_{w,m} is parameterized by

∏_{j=2}^{s}{|x_{j} − x_{j}^{0}| < ε} × U_{0,m} × ∏_{j=2}^{n}{|v_{j} − v_{j}^{0}| < ε} ,

and by (4) these parameterizations are compatible along common faces of Δ_{1}, ..., Δ_{k}. Hence ℂCh(S, A) ∩ U_{w} = ⋃_{m} ℂCh(S, A) ∩ U_{0,m} is parameterized by

∏_{j=2}^{s}{|x_{j} − x_{j}^{0}| < ε} × ⋃_{m=1}^{k} U_{0,m} × ∏_{j=2}^{n}{|v_{j} − v_{j}^{0}| < ε} ,

which is homeomorphic to a (2n − 2)-ball.

Similarly one treats the case δ ⊂ ∂Δ and the real case. □

Definition 1.10 A homeomorphism or an isotopy in ℂΔ are called tame if they respect the filtration of ℂΔ by ℂδ, where δ runs over all faces of Δ, i.e., the restrictions of a tame homeomorphism or a tame isotopy to any ℂδ are a homeomorphism, or an isotopy in ℂδ. In addition, we call such objects equivariant if they commute with Conj.

Remark 1.11 For given Δ and S, and different A, A′ : Δ ∩ ℤ^{n} → ℂ the Chypersurfaces ℂCh(S, A) and ℂCh(S, A′) are tame isotopic in ℂΔ (not equivariantly, in general). Indeed, one can connect A and A′ by a family A_{t}, t ∈ [0, 1] such that the polynomials F_{k,t} are nondegenerate for all k = 1, ..., N, t ∈ [0, 1].

1.6 Projectivization of charts

Let d and n be positive integers. Denote by T_{d}^{n} the simplex in ℝ^{n} with vertices (0, 0, ..., 0), (d, 0, ..., 0), ..., (0, ..., 0, d). Observe that ℂT_{d}^{n} is homeomorphic to a closed ball in ℂ^{n}. The group S^{1} = {|v| = 1} acts freely on ∂ℂT_{d}^{n} by

v ∈ S^{1}, (z_{1}, ..., z_{n}) ∈ ∂ℂT_{d}^{n} ↦ (z_{1}v, ..., z_{n}v) ∈ ∂ℂT_{d}^{n} .

Proposition 1.12 The quotient ℂT_{d}^{n}/S^{1} is equivariantly homeomorphic to the projective space ℂP^{n} so that the faces of T_{d}^{n} naturally correspond to the coordinate planes in ℂP^{n}.

Proof. The momentum map ℂμ_{T_{d}^{n}} takes ℂ^{n} diffeomorphically onto Int(ℂT_{d}^{n}). Its inverse extends up to a surjective map ν_{d}^{n} : ℂT_{d}^{n} → ℂP^{n} in the following way. Consider a point (0, z_{1}, ..., z_{n}) ∈ ℂP^{n}\ℂ^{n}, the line γ(t) = (z_{1}t, ..., z_{n}t) ∈ ℂ^{n}, t ∈ (0, ∞), and lim_{t→∞} ℂμ_{T_{d}^{n}}(γ(t)). Then lim_{t→∞} ℂμ_{T_{d}^{n}}(γ(t)) = ℂμ_{δ}(z_{1}, ..., z_{n}), where

δ = {∑_{z_{k}≠0} i_{k} = d} ∩ ⋂_{z_{k}=0}{i_{k} = 0}

is a face of T_{d}^{n}. Then we put ν_{d}^{n}(ℂμ_{δ}(z_{1}, ..., z_{n}) = (0, z_{1}, ..., z_{n}) ∈ ℂP^{n}. One can easily verify that ν_{d}^{n} is well-defined, surjective, continuous, commutes with Conj, sends orbits of the S^{1}-action on ∂ℂT_{d}^{n} into the points of the hyperplane {z_{0} = 0} in ℂP^{n}, and establishes a one-to-one correspondence between the faces of T_{d}^{n} and the coordinate planes in ℂP^{n}. □

Remark 1.13 Further on we always assume ℂT_{d}^{n}/S^{1} to be identified with ℂP^{n} via the map ν_{d}^{n}.

Definition 1.14 In the notation of section 1.5, given a subdivision T_{d}^{n} = Δ_{1} ∪ ... ∪ Δ_{N} and a collection of complex numbers A = \{A\overline{i} : \overline{i} ∈ T_{d}^{n} ∩ ℤ^{n}\}, define the projective gluing of charts as

P ℂCh(S, A) = ν_{d}^{n}(ℂCh(S, A)) ⊂ ℂP^{n} .

This object is called a complex combinatorial hypersurface (C-hypersurface, in short) of degree d in ℂP^{n}. If the numbers A = \{A\overline{i} : \overline{i} ∈ T_{d}^{n} ∩ ℤ^{n}\} are real, the C-hypersurface is called real and its real part ν_{d}^{n}(ℝCh(S, A)) is denoted by P ℝCh(S, A).

Proposition 1.15 If F is a nondegenerate polynomial of degree d in n variables, then P ℂCh(F) ⊂ ℂP^{n} coincides with the projective closure P {F = 0} of the affine hypersurface {F = 0}.

Proof. Straightforward from ν_{d}^{n}(ℂCh(F)) = P {F = 0} and the fact that ℂCh(F) ∩ ∂(ℂT_{d}^{n}) is invariant with respect to the S^{1} action. □

Definition 1.16 A homeomorphism, or an isotopy in ℂP^{n} are called tame if they respect the filtration of ℂP^{n} by the coordinate planes, i.e., their restrictions to any coordinate plane are a homeomorphism, or an isotopy in that plane. In addition, we call such objects equivariant if they commute with Conj.

Proposition 1.17 In the previous notation, P ℂCh(S, A) is a PL-submanifold in ℂP^{n}, of codimension 2. It is invariant with respect to Conj if all the numbers in A are real.

Proof. First, we show that the C-hypersurface ℂCh(S, A) is a PL-manifold in ℂT_{d}^{n} of codimension 2 with boundary

∂ℂCh(S, A) = ℂCh(S, A) ∩ ∂ℂT_{d}^{n} ,

and if all the numbers in A are real ℂCh(S, A is invariant with respect to Conj, and Fix(Conj) = ℝCh(S, A) is a PL-manifold in ℝT_{d}^{n} of codimension 1 with boundary

∂ℝCh(S, A) = ℝCh(S, A) ∩ ∂ℝT_{d}^{n} .

By Proposition 1.8 it is enough to consider only the gluing of charts on the coordinate hyperplanes. For any polyhedron Δ ⊂ ℝ^{n}, put Δ_{t} = (t, ..., t) + Δ, t ≥ 0.

The family ℂμ_{Δ_{k,t}}, 0 ≤ t ≤ 1, connects ℂμ_{Δ_{k}} with ℂμ_{Δ_{k,1}}, k = 1, ..., N, and defines an isotopy of Cl(ℂμ_{Δ_{k,t}}(F_{k} = 0)), the closure of ℂμ_{Δ_{k,t}}(F_{k} = 0), 0 < t ≤ 1, which degenerates into ℂCh(F_{k}), k = 1, ..., N. The same holds for any proper face δ of Δ_{1}, ..., Δ_{N} and the corresponding truncation of F_{1}, ..., F_{N}.

Since Sing(ℂΔ_{k,1}) = ∅, k = 1, ..., N, the set K_{1} = ⋃_{k=1}^{N} ℂCh(z_{1}...z_{n}F_{k}) is a PL-manifold with boundary on ∂ℂT̃_{d}^{n}, according to Proposition 1.4 (where T̃_{d}^{n} = (1, ..., 1) + T_{d}^{n}). Denote some faces of T̃_{d}^{n} as follows:

T̃_{d}^{n}(k) = T̃_{d}^{n} ∩ {i_{k} = 1}, T̃_{d}^{n}(k_{1}, ..., k_{s}) = ⋂_{j=1}^{s} T̃_{d}^{n}(k_{j}) .

Put δ = T̃_{d}^{n}(k). By (3)

ℂδ ≅ ℂT_{d}^{n−1} × S^{1}.  (9)

Similarly, by (3) K_{1} ∩ ℂδ is a PL-manifold of dimension 2n − 3 with boundary on ∂ℂδ, satisfying

K_{1} ∩ ℂδ ≅ (⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ})) × S^{1},  (10)

where the product structure is compatible with that in (9). Using the product representations (9) and (10), we define

D^{n}(k) = {(w_{1}, ..., w_{n}) : (w_{1}, ..., w_{k−1}, w_{k+1}, ..., w_{n}) ∈ ℂT_{d}^{n−1}, |w_{k}| ≤ 1}
≅ ℂT_{d}^{n−1} × D^{2} ,

where D^{2} is the closed 2-dimensional unit ball, and inside D^{n}(k) the (2n − 2)- dimensional manifold M(k):

{(w_{1}, ..., w_{n}) : (w_{1}, ..., w_{k−1}, 1, w_{k+1}, ..., w_{n}) ∈ ⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ}), |w_{k}| ≤ 1}

≅ (⋃_{l=1}^{N} ℂCh(F̃_{l}^{δ})) × D^{2}

with boundary on ∂D^{n}(k) and such that M(k) ∩ ℂδ = ⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ}).

Similarly, given δ = T̃_{d}^{n}(k_{1}, ..., k_{s}), we have by (3),

ℂδ ≅ ℂT_{d}^{n−s} × (S^{1})^{s}, K_{1} ∩ ℂδ ≅ (⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ})) × (S^{1})^{s},

and one defines

D^{n}(k_{1}, ..., k_{s}) = {(w_{1}, ..., w_{n}) : (w_{j})_{j≠k_{1},...,k_{s}} ∈ ℂT_{d}^{n−s}, |w_{k_{1}}|, ..., |w_{k_{s}}| ≤ 1}
≅ ℂT_{d}^{n−s} × (D^{2})^{s} ,

and the (2n − 2)-dimensional manifold M(k_{1}, ..., k_{s}):

{(w_{1}, ..., w_{n}) : (w_{j})_{j≠k_{1},...,k_{s}} ∈ ⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ}), |w_{k_{1}}|, ..., |w_{k_{s}}| ≤ 1}

≅ (⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ})) × (D^{2})^{s}

with boundary on ∂D^{n}(k_{1}, ..., k_{s}) and such that

M(k_{1}, ..., k_{s}) ∩ ℂδ = ⋃_{l=1}^{N} ℂCh(z_{1}...z_{n}F_{l}^{δ}) .

It can easily be shown that T_{1} = ℂT̃_{d}^{n} ∪ ⋃_{s=1}^{n} ⋃_{1≤k_{1}<...<k_{s}} D^{n}(k_{1}, ..., k_{s}) is the convex hull of ℂT̃_{d}^{n} in ℂ^{n}, and K_{1} ∪ ⋃_{s=1}^{n} ⋃_{1≤k_{1}<...<k_{s}} M(k_{1}, ..., k_{s}) is a PL-manifold in T_{1} of codimension 2 with boundary on ∂T_{1}.

Now, for any t ∈ (0, 1), in the same way one constructs similar objects:

• complexifications of polyhedra

ℂΔ_{k,t} = Cl(ℂμ_{Δ_{k},t}((ℂ^{∗})^{n})), ℂT_{d}^{n}(t) = ⋃_{k=1}^{N} ℂΔ_{k,t},

• gluing of charts K_{t} = ⋃_{k=1}^{N} Cl(ℂμ_{Δ_{k},t}({F_{k} = 0})), which is a PL-manifold of codimension 2 in ℂT_{d}^{n}(t) with boundary on ∂ℂT_{d}^{n}(t),

• completion of ℂT_{d}^{n}(t) up to its convex hull

T_{t} = ℂT_{d}^{n}(t) ∪ ⋃_{s=1}^{n} ⋃_{1≤k_{1}<...<k_{s}} D_{t}^{n}(k_{1}, ..., k_{s}),

where

D_{t}^{n}(k_{1}, ..., k_{s}) = {(w_{1}, ..., w_{n}) : (w_{j})_{j≠k_{1},...,k_{s}} ∈ ℂT_{d}^{n−s}, |w_{k_{1}}|, ..., |w_{k_{s}}| ≤ t}

≅ ℂT_{d}^{n−s} × (D^{2}(t))^{s} ,

D^{2}(t) is a disc of radius t,

• a (2n − 2)-dimensional manifold

M_{t}(k_{1}, ..., k_{s}) = {(w_{1}, ..., w_{n}) :

(w_{j})_{j≠k_{1},...,k_{s}} ∈ ⋃_{l=1}^{N} Cl(ℂμ_{δ_{t}}({F_{l}^{δ} = 0})), |w_{k_{1}}|, ..., |w_{k_{s}}| ≤ t}

≅ (⋃_{l=1}^{N} Cl(ℂμ_{δ_{t}}({F_{l}^{δ} = 0}))) × (D^{2}(t))^{s}

• a PL-manifold

K_{t} ∪ ⋃_{s=1}^{n} ⋃_{1≤k_{1}<...<k_{s}} M_{t}(k_{1}, ..., k_{s})  (11)

of codimension 2 in T_{t} with boundary on ∂T_{t}.

If t varies from 1 to 0, T_{t} contracts from T_{1} to ℂT_{d}^{n} and the manifold (11) naturally contracts into the C-hypersurface ℂCh(S, A) with boundary on ∂ℂT^{n}.

To complete the proof of Proposition 1.17 we have to verify only that ∂ℂCh(S, A) is invariant with respect to the S^{1} action on ∂ℂT_{d}^{n}, and the quotient ℂCh(S, A)/S^{1} is a closed manifold.

First, note that ∂ℂCh(S, A) is invariant with respect to the S^{1} action. Indeed, ∂ℂCh(S, A) consists of charts of homogeneous polynomials which are invariant with respect to the S^{1} action on ∂ℂT_{d}^{n}. Second, for any edge of T_{d}^{n} there exists a combination of an automorphism of ℤ^{n} and shifts which puts this edge on a coordinate axis and the adjacent faces of T_{d}^{n} on the corresponding coordinate planes. Since such a transformation is compatible with ν_{d}^{n} defined via the momentum map, the question on the behavior of ℂCh(S, A) on ∂ℂT_{d}^{n} is reduced to that on coordinate planes, which has been treated above. □

Remark 1.18 Similarly to Remark 1.11, two C-hypersurfaces in ℂP^{n} of the same degree and with the same subdivision S of T_{d}^{n} are tame isotopic in ℂP^{n} (not equivariantly, in general).

2 Topology of complex C-hypersurfaces

2.1 Complexification of the Viro theorem

Theorem 2.1 (Affine complex Viro theorem). In the notation of section 1.5, if Δ ⊂ (ℝ_{+}^{∗})^{n} and a subdivision S : Δ = Δ_{1} ∪ ... ∪ Δ_{N} is defined by a convex piecewise-linear function ν : T_{d}^{n} → ℝ, then ℂCh(S, A) is tame isotopic in ℂΔ (equivariantly, if all the numbers in A are real) to ℂCh(F), where

F (z_{1}, ..., z_{n}, t) = ∑_{(i_{1},...,i_{n})∈Δ} A_{i_{1}...i_{n}}t^{ν(i_{1},...,i_{n})}z_{1}^{i_{1}} · ... · z_{n}^{i_{n}}

and t = const > 0 is sufficiently small.

Proof. Our argument is similar to that from the proof of the Viro gluing theorem [32] (see also [13], Chapter 11, and [23]).

Without loss of generality one can suppose that dim Δ = n, and ν is positive integral-valued at integral points. Consider the polynomial F̂(z_{1}, ..., z_{n}, t) = (1 + t)F (z_{1}, ..., z_{n}, t) as a polynomial in t, x_{1}, ..., x_{n}, and denote by Δ̂ its Newton polyhedron in ℝ^{n+1}. The graph of ν is the “lower” part of ∂Δ̂, naturally projected onto Δ ⊂ ℝ^{n} = {i_{n+1} = 0} ⊂ ℝ^{n+1}. Denote by Δ̂_{k} the part of the graph of ν projected onto Δ_{k}, k = 1, ..., N. Denote by B the part of Δ̂ projected onto ∂Δ. Clearly, B can be viewed as ∂Δ × [0, 1].

Step 1. Introduce the set

H_{0} = {(w_{1}, ..., w_{n}, ν(|w_{1}|, ..., |w_{n}|)) : (|w_{1}|, ..., |w_{n}|) ∈ Δ} = ⋃_{k=1}^{N} Δ̂_{k} ×(S^{1})^{n} ⊂ ∂ℂΔ̂ ,

and the family of hypersurfaces H_{a} = Cl(ℂμΔ̂({t = a}) ⊂ ℂΔ̂, a ∈ (0, ε). We shall show that the family H_{a}, a ∈ [0, ε), is an isotopy in ℂΔ̂.

The map ℂμΔ̂ takes {t = a} ⊂ (ℂ^{∗})^{n+1} diffeomorphically into ℂI(Δ̂). Considering lim_{τ→0} μΔ̂(γ(τ)) for all curves

γ(τ) = (λ_{1}x_{1}^{k_{1}}, ..., λ_{n}x_{n}^{k_{n}}, a) ⊂ (ℝ_{+}^{∗})^{n+1}, τ > 0,

with λ_{1}, ..., λ_{n} > 0, (k_{1}, ..., k_{n}) ≠ 0, one can show (as in the proof of Proposition 1.4) that H_{a} being the closure of ℂμΔ̂({t = a}) is a PL-manifold with boundary ∂H_{a} = H_{a} ∩ B, which projects onto ∂Δ. Moreover, H_{a} ∩ H_{b} = ∅, a ≠ b. The projection on the first n coordinates provides a homeomorphism of H_{a}, a > 0, and ℂΔ. Indeed, in ℂI(Δ̂) this fact is reduced to the claim that

(x_{1}, ..., x_{n}) ∈ (ℝ_{+}^{∗})^{n} ↦ \frac{∑_{(i_{1},...,i_{n})∈Δ} x_{1}^{i_{1}}...x_{n}^{i_{n}}a^{ν(i_{1},...,i_{n})} · (i_{1}, ..., i_{n})}{∑_{(i_{1},...,i_{n})∈Δ} x_{1}^{i_{1}}...x_{n}^{i_{n}}a^{ν(i_{1},...,i_{n})}} ∈ I(Δ)

is a homeomorphism, which holds since the above map is a kind of the momentum map [1, 2]. Similarly one shows that the projection is one-to-one on ∂H_{a} ⊂ B, taking into account that the limit of μΔ̂ on a face of B is the momentum map of this face. To finish the proof one should note that ⋃_{a>0} H_{a} fill the intersection of ℂI(Δ̂) with the space Im(w_{n+1}) = 0, where w_{1}, ..., w_{n+1} are coordinates in (ℂ^{∗})^{n+1} ⊃ ℂΔ̂.

Step 2. Now we show that ℂCh(F̂) ∩ H_{a}, a ∈ [0, ε), is an isotopy in ℂΔ̂. For ε > 0 small enough, {F̂ = 0} ∩ {t = a} = {F = 0} ∩ {t = a} and the intersection is transverse in (ℂ^{∗})^{n+1}. This implies that the family ℂCh(F̂) ∩ H_{a}, a ∈ (0, ε), is an isotopy in ℂΔ̂.

Now we pick a point \overline{w} = (y_{1}v_{1}^{0}, ..., y_{n}v_{n}^{0}, y_{n+1}) ∈ H_{0}∩ℂCh(F̂), where (y_{1}, ..., y_{n}) ∈ I(δ) for some s-dimensional face of Δ_{1}, ..., Δ_{N}, y_{n+1} = ν(y_{1}, ..., y_{n}), and (v_{1}^{0}, ..., v_{n}^{0}) ∈ (S^{1})^{n}. Following the proof of Proposition 1.4 and using transformations

(x_{1}, ..., x_{n}, t) ↦ (t^{β_{1}} ∏_{k=1}^{n} x_{k}^{α_{k1}}, ..., t^{β_{n}} ∏_{k=1}^{n} x_{k}^{α_{kn}}, t)

with A = (α_{kj}) ∈ SL(ℤ, n), we can get δ to be parallel to the coordinate s-plane {i_{1} = ... = 0}, ν|_{δ} = ν_{0}, and ν|_{Δ\δ} > ν_{0}. Then we introduce a parameterization of a

neighborhood U\overline{w} of the point \overline{w} in ℂΔ̂ ∩ \{Im(w_{n+1}) = 0\} by pr_{s}(U′) × U_{0} × U′′ via an extension of the map ℂμΔ̂ such that, in the notation of the proof of Proposition 1.4, pr_{s}(U′) is given by the first s inequalities in (7), U′′ is given by (6), and U_{0} is a neighborhood of the point pr_{n−s+1}(δ) in pr_{n−s+1}(Δ̂). Here Int(U_{0}) is identified with the domain pr_{n−s+1}(U′) ⊂ (ℝ_{+}^{∗})^{n−s+1}, given in the coordinates x_{s+1}, ..., x_{n}, t by the relations

M_{i_{s+1},...,i_{n},i_{n+1}} = x_{s+1}^{i_{s+1}}...x_{n}^{i_{n}}t^{i_{n+1}} < ε, (i_{s+1}, ..., i_{n+1}) ∈ pr_{n−s+1}(Δ̂\Δ) ∩ ℤ^{n−s+1} ,

and ℂCh(F̂) ∩ Int(U\overline{w}) is parameterized in pr_{s}(U′) × Int(U_{0}) × U′′ by equations (8), where G and H are vanishing at zero smooth functions of x_{2}, ..., x_{s}, v_{1}, ..., v_{n} and M_{i_{s+1},...,i_{n},i_{n+1}}, (i_{s+1}, ..., i_{n+1}) ∈ pr_{n−s+1}(Δ̂\Δ) ∩ ℤ^{n−s+1}.

Denote Δ̃ = δ × pr_{n−s+1}(Δ̂). Clearly, Δ̂ and Δ̃ coincide in a neighborhood of \overline{w}. Moreover, The maps ℂμΔ̂ and ℂμΔ̃ are connected by an isotopy on the domain pr_{s}(U′) × Int(U_{0}) × U′′: such an isotopy can be written explicitly by supplying the non-common summands in the formulae for ℂμΔ̂ and ℂμΔ̃ by a parameter running over [0, 1]. This isotopy extends up to equivariant tame isotopy on pr_{s}(U′)×U_{0}×U′′, so that replacing ℂμΔ̂ by ℂμΔ̃, one replace U\overline{w} by another neighborhood Ũ\overline{w} of \overline{w} in ℂΔ̃ ∩ {Im(w_{n+1})}. Note that the map ℂμΔ̃ splits on pr_{s}(U′) × U_{0} × U′′ into the product of

μ_{δ} : pr_{s}(U′) → δ, μ_{pr_{n−s+1}(Δ̂)} → pr_{n−s+1}(Δ̂), Id : U′′ → U′′.

Together with the result of Step 1 this allows us to introduce in U_{0} coordinates t, θ_{1}, ..., θ_{n−s} so that

U_{0} = {0 ≤ t < ε, −ε < θ_{j} < ε}, U_{0} ∩ ∂(pr_{n−s+1}(Δ̂)) = {t = 0} ,

and ℂμΔ̃(H_{a}) = pr_{s}(U′) × {t = a} × U′′. In addition, μ_{pr_{n−s+1}(Δ̂)} expresses M_{i_{s+1},...,i_{n},i_{n+1}}, (i_{s+1}, ..., i_{n+1}) ∈ pr_{n−s+1}(Δ̂\Δ) ∩ ℤ^{n−s+1}, as continuous functions of t, θ_{1}, ..., θ_{n−s}, vanishing at zero. Hence the closure of ℂμΔ̃(\{F = 0\}) in Ũ\overline{w} is given by equations

x_{1} = x_{1}^{0} + G′, v_{1} = v_{1}^{0} + H′,  (12)

where G′ and H′ are vanishing at zero smooth functions of x_{2}, ..., x_{s}, v_{2}, ..., v_{n}, t, θ_{1}, ..., θ_{n−s}. The variety (12) intersects any hypersurface pr_{s}(U′) × {t = a} × U′′, a ∈ [0, ε), transversally in pr_{s}(U′) × U_{0} × U′′ ≃ Ũ\overline{w} ≃ U\overline{w}, thereby proving that ℂCh(F̂) ∩ H_{a}, a ∈ [0, ε), is an isotopy in ℂΔ̂.

Step 3. For a ≠ 0, the projection of ℂCh(F̂) ∩ H_{a} into ℂΔ is the closure of the image of {F = 0} ∩ {t = a} ⊂ (ℂ^{∗})^{n+1} by the map

ℂμ_{Δ,a}(\overline{x}, \overline{v}) = \frac{∑_{(i_{1},...,i_{n})∈Δ}(i_{1}v_{1}, ..., i_{n}v_{n}) · (x_{1}^{i_{1}}...x_{n}^{i_{n}} ∑_{(i_{1},...,i_{n},k)∈Δ̂} a^{k})}{∑_{(i_{1},...,i_{n})∈Δ} (x_{1}^{i_{1}}...x_{n}^{i_{n}} ∑_{(i_{1},...,i_{n},k)∈Δ̂} a^{k})} .  (13)

The map ℂμ_{Δ,a} is connected with ℂμ_{Δ} on {t = a} ≃ (ℂ^{∗})^{n} by the isotopy

\frac{∑_{(i_{1},...,i_{n})∈Δ}(i_{1}v_{1}, ..., i_{n}v_{n}) · (x_{1}^{i_{1}}...x_{n}^{i_{n}} (∑_{(i_{1},...,i_{n},k)∈Δ̂} a^{k}(1 − τ) + τ))}{∑_{(i_{1},...,i_{n})∈Δ} (x_{1}^{i_{1}}...x_{n}^{i_{n}} (∑_{(i_{1},...,i_{n},k)∈Δ̂} a^{k}(1 − τ) + τ))} ,  (14)

providing a tame isotopy of the projection of ℂCh(F̂)∩H_{a} with ℂCh(F |_{t=a}) ⊂ ℂΔ. Similarly, the projection of ℂCh(F̂) ∩ ℂΔ̂_{k} ∩ {Im(w_{n+1}) = 0} into ℂΔ_{k}, 1 ≤ k ≤ N, is the closure of the image of {F Δ̂_{k} = 0}∩{t = 1} by the map ℂμΔ̂_{k} which coincides with the closure of the image of {F_{k} = 0} by the map ℂμ_{Δ_{k}}, i.e., ℂCh(F_{k}).

So, Theorem is proven in the complex case. If all the numbers in A are real, then the isotopy constructed is equivariant. □

Theorem 2.2 (Projective complex Viro theorem). Let, in the notation of Theorem 2.1, Δ = T_{d}^{n} and the subdivision S be defined by a convex piecewise-linear function ν : T_{d}^{n} → ℝ. Then P ℂCh(S, A) is tame isotopic in ℂP^{n} (equivariantly, if all the numbers in A are real) to P ℂCh(F), where

F (z_{1}, ..., z_{n}, t) = ∑_{i_{1}+...+i_{n}≤d} A_{i_{1}...i_{n}}t^{ν(i_{1},...,i_{n})}z_{1}^{i_{1}} · ... · z_{n}^{i_{n}} = 0,

and t = const > 0 is sufficiently small.

Proof. We multiply all the polynomials by z_{1}...z_{n} and apply Theorem 2.1 to the simplex T̃_{d}^{n} = T_{d}^{n}+(1, ..., 1) and correspondingly shifted S, A. Then we note that the isotopy of ℂCh(z_{1}...z_{n}F) and ⋃_{k=1}^{N} ℂCh(z_{1}...z_{n}F_{k}) in ℂT̃_{d}^{n} is tame and compatible with the action of (S^{1})^{n} on ∂ℂT̃_{d}^{n}. This allows us to take quotient by this action as was done in the proof of Proposition 1.17 and obtain the required isotopy. □

Proposition 2.3 Let two subdivisions S = {Δ_{k}, k = 1, ..., N} and S′ = {Δ_{kl}, l = 1, ..., r_{k}, k = 1, ..., N} of a polyhedron Δ ⊂ (ℝ_{+}^{∗})^{n} satisfy

Δ_{k} = ⋃_{l=1}^{r_{k}} Δ_{kl}, k = 1, ..., N,

so that the subdivision S′ is given by piecewise-linear function ν : Δ → ℝ, whose restrictions ν_{k} = ν|_{Δ_{k}}, k = 1, ..., N, are convex. Then the varieties ℂCh(S, A) and ℂCh(S′, A′) are tame isotopic, provided A, A′ : Δ → ℂ define nondegenerate polynomials F_{k}, F_{kl}, l = 1, ..., r_{k}, k = 1, ..., N. Similarly, given two subdivisions S, S′ of T_{d}^{n} satisfying the previous assumptions, the C-hypersurfaces of degree d constructed out of S, A and S′, A′ are tame isotopic in ℂP^{n}.

Proof. Consider the case Δ ⊂ (ℝ_{+}^{∗})^{n}. Fix a > 0. Let the maps ℂμ_{Δ_{k},a} : (ℂ^{∗})^{n} → ℂΔ_{k}, k = 1, ..., N, be defined by (13), where summations run over Δ_{k} and Δ̂_{k}, constructed as in the proof of Theorem 2.1 by means of ν_{k}. Put

F_{k,a} = ∑_{(i_{1},...,i_{n})∈Δ_{k}} A′_{i_{1}...i_{n}}z_{1}^{i_{1}}...z_{n}^{i_{n}}a^{ν(i_{1},...,i_{n})}, k = 1, ..., N.

As shown in the proof of Theorem 2.1, the closure ℂCh_{a}(F_{k,a}) of ℂμ_{Δ_{k},a}({F_{k,a} = 0}) in ℂΔ_{k} is tame isotopic to ∪_{l}ℂCh(F_{kl}). Moreover, for any face δ = Δ_{k} ∩ Δ_{j}, the isotopies in ℂΔ_{k} and ℂΔ_{l} coincide on ℂδ. Then, for each k = 1, ..., N, we connect ℂμ_{Δ_{k},a} with ℂμ_{Δ_{k}} by an isotopy (14), thereby obtaining the required isotopy of ℂCh(S, A) and ℂCh(S′, A′).

The same argument proves the statement for Δ = T_{d}^{n}. □

A subdivision Δ = Δ_{1} ∪ ... ∪ Δ_{N} is called maximal if it cannot be refined. In this case all the integral points in Δ are vertices of Δ_{1}, ..., Δ_{N}, and these polyhedra are simplices.

Corollary 2.4 Given polyhedron Δ ⊂ (ℝ_{+}^{∗})^{n}, any C-hypersurface in ℂΔ is tame isotopic to a C-hypersurface constructed out of a maximal subdivision of Δ. The same is true for C-hypersurfaces in ℂP^{n}.

Proof. Let Δ = Δ_{1} ∪ ... ∪ Δ_{N}, and f : Δ → ℝ be a smooth convex function. Define a piecewise-linear function ν : Δ → ℝ as follows: put ν(\overline{i}) = f(\overline{i}), \overline{i} ∈ Δ∩ℤ^{n}, then define the graph of ν|_{Δ_{k}} to be the “lower” part of the convex hull of \{(\overline{i}, ν(\overline{i}) : \overline{i} ∈ Δ_{k}\}. The function ν defines a maximal subdivision of Δ inscribed into the initial subdivision and satisfying the conditions of Proposition 2.3, which completes the proof. □

Corollary 2.5 Any C-curve in ℂP^{2} is tame isotopic (not equivariantly, in general) to an algebraic curve of the same degree.

Proof. By Corollary 2.4 we can assume that a C-curve in ℂP^{2} is constructed out of a maximal triangulation of T_{d}^{2}. We shall transform any given triangulation into a convex triangulation, so that in each transformation step the conditions of Proposition 2.3 hold true.

Let S be a triangulation of T_{d}^{2}, O(S) denote the star of the origin O with respect to this triangulation, and outside O(S) the triangulation S is maximal. We construct a triangulation S′ of T_{d}^{2} such that O(S′) ⊃ O(S) and O(S′) ≠ O(S). Then, in finitely many steps we come to O(S) = T_{d}^{2}, which corresponds to a convex triangulation, and the required statement will follow from Theorem 2.2 and Proposition 2.3.

Let O, P_{1}, ..., P_{r} be all the vertices of O(S) numbered successively clockwise along ∂O(S). Any segment [P_{i}, P_{i+1}] either lies on ∂T_{d}^{2}, or is an edge of a unique triangle T_{i} ∈ S, T_{i} ⊄ O(S).

(i) Assume that, for some i = 1, ..., r, the vertex Q ≠ P_{i}, P_{i+1} of T_{i} lies between the straight lines(OP_{i}) and (OP_{i+1}), and Q ∉ (OP_{i})∪(OP_{i+1}) (see Figure 2a). Then we change the subdivision of T_{d}^{2} as shown in Figure 2b,c. These changes satisfy the conditions of Proposition 2.3 and lead to a triangulation with a strongly greater star of O.

(ii) Assume that, for some i = 1, ..., r −1, T_{i} = T_{i+1}, i.e., the vertices of the latest triangle are P_{i}, P_{i+1}, P_{i+2} (see Figure 2d). Then we perform the transformation shown in Figure 2e, once again increasing the star of O.

(iii) Assume that there are no triangles T_{i} as in (i), (ii). Then any triangle T_{i} is either “left”, i.e., the vertex Q lies on (OP_{i}) or above (OP_{i}), or “right”, i.e., Q lies on (OP_{i+1}) or below (OP_{i+1}). If there exist “left” triangles, consider the “left” triangle T_{i} with the minimal i. If i = 1, we have the situation shown in Figure 2f. Then we change triangulation as shown in Figure 2g, increasing the star of O. If i > 1, then the triangle T_{i−1} must be “right”, which means that we have a situation shown in Figure 2h. Then we change triangulation as shown in Figure 2i, increasing the star of O. □

2.2 First properties of C-hypersurfaces

The real part of a real C-hypersurface M in ℂP^{n} is denoted by ℝM.

Proposition 2.6 A complex C-hypersurface M of degree d in ℂP^{n} is an orientable manifold, homologous to an algebraic hypersurface of degree d in ℂP^{n}. If M is real its real part ℝM is a closed manifold, mod 2 homologous to the real point set of a real algebraic hypersurface of degree d in ℝP^{n}.

Proof. The Jacobian of the real momentum map μ_{Δ} : (ℝ_{+}^{∗})^{n} → Int(Δ), μ_{Δ}(\overline{x}) = (μ_{Δ}^{(1)}(\overline{x}), ..., μ_{Δ}^{(n)}(\overline{x})), is positive. Hence the Jacobian of the complex momentum map Figure 2: Transformation of a triangulation

21 ℂμ_{Δ} : (ℂ^{∗})^{n} → ℂI(Δ) is positive in the coordinates

x_{1} = |z_{1}|, v_{1} = \frac{z_{1}}{|z_{1}|}, ... , x_{n} = |z_{n}|, v_{n} \frac{z_{n}}{|z_{n}|} ,

because this is a diffeomorphism, and at a point with v_{1} = ... = v_{n} = 1 one can easily compute

det (\frac{D(ℂμ_{Δ})}{D(x_{1}, v_{1}, ..., x_{n}, v_{n})}) = det (\frac{D(μ_{Δ})}{D(x_{1}, ..., x_{n})}) · ∏_{j=1}^{n} μ_{Δ}^{(j)}(x_{1}, ..., x_{n}) > 0 .

This means, in particular, that ℂμ_{Δ} canonically defines an orientation of images of complex subvarieties of (ℂ^{∗})^{n}. Therefore, the open subset

⋃_{k=1}^{N}(ℂCh(F_{k}) ∩ ℂI(Δ_{k}))

of ℂCh(S, A) is canonically orientable. So, to complete the proof of orientability, one should verify that these orientations are compatible when gluing the charts ℂCh(F_{k}), ℂCh(F_{j}) with Δ_{k} ∩ Δ_{j} = δ being a common facet (the gluing along faces of lower dimensions does not affect the orientation). In other words, the orientations of ℂCh(F_{k}^{δ}) = ℂCh(F_{j}^{δ}) induced by ℂCh(F_{k}) and ℂCh(F_{j}) are opposite.

Without loss of generality suppose that δ is contained in a hyperplane i_{1} = a > 0, Δ_{k} ⊂ {i_{1} ≤ a}, and Δ_{j} ⊂ {i_{1} ≥ a}. Then by construction, ℂΔ_{k} induces on ℂδ an orientation defined by the form

−dv_{1} ∧ dx_{2} ∧ dv_{2} ∧ ... ∧ dx_{n} ∧ dv_{n},

and ℂΔ_{j} induces on ℂδ the opposite orientation defined by

dv_{1} ∧ dx_{2} ∧ dv_{2} ∧ ... ∧ dx_{n} ∧ dv_{n}.

On the other hand, a coorienting 2-vector bundle on ℂCh(F_{k}) ∩ ℂI(Δ_{k}) can continuously be extended to a coorienting 2-bundle on ℂCh(F_{k}^{δ}) ⊂ ℂI(δ), and the same for Δ_{j}. Indeed, let the complex straight line Λ

z_{2} = ... = z_{n} = xv, z_{1} = x_{0} = const, x ∈ (0, ∞), |v| = 1,

with small z_{0} > 0 meet the hypersurfaces {F_{k} = 0}, {F_{k}^{δ} = 0} transversally. Then the family of surfaces Σ_{λ}, λ ∈ [0, 1],

\{\frac{∑_{(i_{1},...,i_{n})∈Δ_{k}} x^{i_{2}+...+i_{n}}x_{0}^{i_{1}}λ^{a−i_{1}} · (i_{1}, i_{2}v, ..., i_{n}v)}{∑_{(i_{1},...,i_{n})∈Δ_{k}} x^{i_{2}+...+i_{n}}x_{0}^{i_{1}}λ^{a−i_{1}}} : x ∈ (0, ∞), |v| = 1\}

is a diffeotopy, connecting Σ_{0} = ℂμ_{δ}(Λ), which coorients ℂCh(F_{k}^{δ}) in ℂI(δ), and Σ_{1} = ℂμ_{Δ_{k}}(Λ), which coorients ℂCh(F_{k}) in ℂI(Δ_{k}).

Comparing this with the previous remark on orientations of ℂδ, one completes the proof of the orientability of a complex C-hypersurface.

At last, [ℂCh(S, A)] ∈ H_{2n−2}(ℂP^{n}) and [ℝCh(S, A)] ∈ H_{n−1}(ℝP^{n}, ℤ/2) can easily be computed by induction considering the intersection of the complex and real charts with the coordinate planes. □

Proposition 2.7 Any (equivariant) C-hypersurface M is (equivariantly) tame isotopic to a close smooth manifold M_{sm} of codimension 2 in ℂP^{n}.

Proof. Let M = ℂCh(S, A)/S^{1}, where S = \{Δ_{1}, ..., Δ_{N}\}, A = \{A\overline{i} : \overline{i} ∈ T_{d}^{n} ∩ ℤ^{n}\}. Put F_{m} = ∑_{\overline{i}∈Δ_{m}} A\overline{i}\overline{z}\overline{i}, m = 1, ..., N. We construct two nonvanishing ℝ-linearly independent sections \overline{s} and \overline{s}′ of the bundle T ℂP^{n}|_{U(M)}, where U(M) is a neighborhood of M in ℂP^{n}, such that the 2-subbundle Span_{ℝ}\{\overline{s}, \overline{s}′\} ⊂ T ℂP^{n}|_{M} is transverse to T ℂCh(F_{m}) in T ℂP^{n}|_{ℂCh(F_{m})} for any m = 1, ..., N. This will imply the existence of a smooth 2(n − 1)-manifold M_{sm} isotopic (equivariantly, if Conj(M) = M) to M, close to M and transverse to the 2-bundle Span_{ℝ}\{\overline{s}, \overline{s}′\}.

(i) First, we shift T_{d}^{n} into T̃_{d}^{n} = (1, ..., 1) + T_{d}^{n} (shifting respectively S, A as well), and construct sections \overline{s} and \overline{s}′ of T ℂ^{n} defined in a neighborhood of ℂCh(S, A), with the above properties and an additional one: invariance with respect to the S^{1} action on ∂ℂT̃_{d}^{n}. The latter allows us to obtain the required sections of T ℂP^{n} along the procedure described in the proof of Proposition 1.17.

(ii) Fix m = 1, ..., N and consider the hypersurface F̃_{m}(\overline{z}) = F_{m}(e^{z_{1}}, ..., e^{z_{n}}) = 0 in ℂ^{n}. Define vector fields \overline{s}_{1} and \overline{s}′_{1} on ℂ^{n} by

\overline{s}_{1}(\overline{z}) = \frac{Conj(gradF̃_{m})}{|Conj(gradF̃_{m})|}, \overline{s}′_{1}(\overline{z}) = \frac{Conj(gradF̃_{m})\sqrt{−1}}{|Conj(gradF̃_{m})\sqrt{−1}|} .

They do not vanish along {F̃_{m} = 0} and span a 2-bundle orthogonal to T {F̃_{m} = 0} in T ℂ^{n}|_{\{F̃_{m}=0\}}. These vector fields are 2π\sqrt{−1}-periodic in each coordinate z_{1}, ..., z_{n}, and their normalized images \overline{s} and \overline{s}′ by the differential D(ℂμ_{Δ_{m}} ∘ exp) of the map ℂμ_{Δ_{m}} ∘ exp : ℂ^{n} → ℂΔ_{m} give a 2-bundle Span_{ℝ}\{\overline{s}, \overline{s}′\} on ℂI(Δ_{m}) ∩ ℂCh(F_{m}) transverse to the tangent bundle T (ℂI(Δ_{m}) ∩ ℂCh(F_{m})). We will show that these vector fields continuously extend to ⋃_{m=1}^{N}(∂ℂΔ_{m}).

(iii) Let δ be a face of Δ_{m}, \overline{w} be a point in ℂCh(F_{m}) ∩ ℂI(δ), and U\overline{w} ⊂ ℂΔ_{m} be a neighborhood of \overline{w} as introduced in the step (iv) of the proof of Proposition 1.4. We claim that, for any point \overline{z} ∈ ℂ^{n} such that ℂμ_{Δ}(e\overline{z}) ∈ U\overline{w} ∩ ℂCh(F_{m}) ∩ ℂI(Δ_{m}), one has

||\frac{D(ℂμ_{Δ_{m}} ∘ exp)\overline{z}}{||D(ℂμ_{Δ_{m}} ∘ exp)\overline{z}||} − \frac{D(ℂμ_{δ} ∘ exp)\overline{α}}{||D(ℂμ_{δ} ∘ exp)\overline{α}||}|| < Kε ,  (15)

and

|\frac{grad(F_{m} ∘ exp)}{|grad(F_{m} ∘ exp)|}(\overline{z}) − \frac{grad(F_{m}^{δ} ∘ exp)}{|grad(F_{m}^{δ} ∘ exp)|}(\overline{α})| < Kε ,  (16)

where \overline{α} ∈ ℂ^{n} is some point, satisfying ℂμ_{Δ}(exp(\overline{α})) = \overline{w}, and K > 0 depends only on the coefficients of F and on the point \overline{w}.

(iv) To show (15), note that ℂμ_{Δ_{m}} ∘ exp splits into μ_{Δ_{m}} ∘ exp : ℝ^{n} → I(Δ_{m}) and exp : ℝ^{n}\sqrt{−1} → (S^{1})^{n}. Inequality (15) for the differential of exp : ℝ^{n}\sqrt{−1} → (S^{1})^{n} instead of ℂμ_{Δ_{m}} ∘ exp immediately follows from (6). Then, acting by a transformation from SL(n, ℤ) and a shift, we move Δ_{m}, δ into Δ̃_{m}, δ̃ such that δ̃ lies in a coordinate s-plane. According to (7),

|e^{(\overline{i},Re\overline{z})} − e^{(\overline{i},Re\overline{α})}| < K_{1}ε, \overline{i} ∈ δ̃∩ ℤ^{n}, e^{(\overline{i},Re\overline{z})} < K_{1}ε, \overline{i} ∈ (Δ̃_{m}\δ̃) ∩ ℤ^{n},

where K_{1} > 0 depends only on Δ_{m}, thereby this implies (15), because D(μΔ̃_{m} ∘ exp) is represented by the matrix

(\frac{∑_{\overline{i}∈Δ̃_{m}} i_{p}i_{q}e^{(\overline{i},Re\overline{z})} · ∑_{\overline{i}∈Δ̃_{m}} e^{(\overline{i},Re\overline{z})} − ∑_{\overline{i}∈Δ̃_{m}} i_{p}e^{(\overline{i},Re\overline{z})} · ∑_{\overline{i}∈Δ̃_{m}} i_{q}e^{(\overline{i},Re\overline{z})}}{(∑_{\overline{i}∈Δ̃_{m}} e^{(\overline{i},Re\overline{z})})^{2}} )_{p,q=1,...,n} ,

and D(μδ̃ ∘ exp) is not trivial.

(v) Similarly, F_{m}(e\overline{z}) = e^{(\overline{i}_{0},\overline{z})} ∑_{\overline{i}∈Δ} A\overline{i}e^{(\overline{i},\overline{z})}, where by (6) and (7)

|e^{(\overline{i},\overline{z})} − e^{(\overline{i},\overline{α})}| < K_{2}ε, \overline{i} ∈ δ ∩ ℤ^{n}, |e^{(\overline{i},\overline{z})}| < K_{2}ε, \overline{i} ∈ (Δ\δ) ∩ ℤ^{n},

where K_{2} > 0 depends on Δ_{m}. So, we obtain

grad(F_{m} ∘ exp)(\overline{z}) = e^{(\overline{i}_{0},\overline{z})} (∑_{\overline{i}∈δ} A\overline{i}\overline{i}e^{(\overline{i},\overline{z})} + ∑_{\overline{i}∈Δ\δ} A\overline{i}\overline{i}e^{(\overline{i},\overline{z})}) ,

which immediately implies (16), because ∑_{\overline{i}∈δ} A\overline{i}\overline{i}e^{(\overline{i},\overline{α})} = grad(F_{m}^{δ} ∘ exp)(\overline{α}) ≠ 0.

(vi) Relations (15) and (16) provide continuous extension of the vector fields \overline{s} and \overline{s}′ on ⋃_{m=1}^{N}(∂ℂΔ_{m}) so that they remain ℝ-linearly independent and belong to T ℂI(δ) along ℂI(δ) for any proper face δ of Δ_{m}, m = 1, ..., N. Moreover, the restrictions of \overline{s} and \overline{s}′ on ℂI(δ) depend only on δ and F_{m}^{δ}, common for all Δ_{m} ⊃ δ, hence they are compatible with S^{1} action on ∂ℂT̃_{d}^{n}, and we are done. □

Proposition 2.8 Given a (equivariant) C-hypersurface M, the tangent bundle to its smoothing M_{sm} is (equivariantly) isotopic to a (equivariant) bundle of complex hyperplanes. In particular, M_{sm} possesses an (equivariant) almost complex structure. If M is real, the above isotopy has fixed intersection with T ℝP^{n}|_{M_{sm}} (equal to T ℝM_{sm}).

Proof. For any \overline{w} ∈ M_{sm} denote by \overline{s}_{2}(\overline{w}) the projection of \overline{s}(\overline{w}) to (T\overline{w}M_{sm})^{⊥} ⊂ T\overline{w}ℂP^{n}. This is a nonvanishing section of T\overline{w}ℂP^{n}|_{M_{sm}}, which in the equivariant case is real along ℝM_{sm} = ℝP^{n} ∩ M_{sm} and orthogonal to T\overline{w}ℝM_{sm}. Assuming that \overline{w} = ℂμ_{Δ_{m}}(\overline{z}) ∈ ℂΔ_{m}, one obtains \overline{s}_{2}(\overline{w}) = D(ℂμ_{Δ_{m}})\overline{z}(\overline{b}(\overline{z})), where \overline{b}(\overline{z}) ∈ T\overline{z}(ℂ^{∗})^{n}\T\overline{z}(\{F_{m} = const\}). Clearly, the 2-plane Span_{ℝ}\{\overline{b}(\overline{z}), \overline{b}(\overline{z})\sqrt{−1}\} is transversal to T\overline{z}(\{F_{m} = const\}) in T\overline{z}(ℂ^{∗})^{n}. Hence the two plane L2 ; \overline{w} = Span_{ℝ}\{\overline{s}_{2}(\overline{w}), \overline{s}′_{2}(\overline{w}\}, where \overline{s}′_{2}(\overline{w}) = D(ℂμ_{Δ_{m}})\overline{z}(\overline{b}(\overline{z})\sqrt{−1}), is transversal to T\overline{w}M_{sm} in T\overline{w}ℂP^{n}.

We construct a required isotopy in two steps.

First, we (equivariantly) move T\overline{w}M_{sm} to L2n−2 ; \overline{w} = (L2 ; \overline{w})^{⊥} ⊂ T\overline{w}ℂP^{n}, for any \overline{w} ∈ M_{sm}. This, evidently, is possible, since by construction, T\overline{w}M_{sm} is transverse

to L2 ; \overline{w}.

Now we show that it is possible to move the distribution L^{2n−2} on M_{sm} to the distribution of complex hyperplanes (Span_{ℝ}\{\overline{s}_{2}, \overline{s}_{2}\sqrt{−1}\})^{⊥} in T ℂP^{n}|_{M_{sm}}.

Lemma 2.9 In the above notation, for any \overline{w} ∈ M_{sm}, no vector (1 − λ)\overline{s}′_{2}(\overline{w}) + λ\overline{s}_{2}\sqrt{−1}, 0 ≤ λ ≤ 1, is ℝ-proportional to \overline{s}_{2}(\overline{w}).

Proof. Since exp : ℂ^{n} → (ℂ^{∗})^{n} preserves the complex structure on T ℂ^{n} and T (ℂ^{∗})^{n}, we can replace D(ℂμ_{Δ_{m}}) in the above formulae by D(ℂ̃μ_{Δ_{m}}), where ℂ̃μ_{Δ_{m}} : ℂ^{n} → ℂ^{n} is defined by

ℂ^{n} = ℝ^{n} ⊕ ℝ^{n}\sqrt{−1} ⟶^{log μ̃_{Δm}⊕Id} ℝ^{n} ⊕ ℝ^{n}\sqrt{−1} = ℂ^{n} ,  (17)
log μ̃_{Δ_{m}} = (log μ_{1,Δ_{m}}, ..., log μ_{n,Δ_{m}}), μ_{Δ_{m}} = (μ_{1,Δ_{m}}, ..., μ_{n,Δ_{m}}) .

Clearly exp ∘ ℂ̃μ_{Δ_{m}} = ℂμ_{Δ_{m}} ∘ exp.

Assume that \overline{s}_{2}(\overline{w}) = (1 − λ)\overline{s}′_{2}(\overline{w}) + λ\overline{s}_{2}(\overline{w})\sqrt{−1} for some λ ∈ (0, 1) and \overline{w} ∈ ⋃_{m=1}^{N} ℂI(Δ_{m}). Splitting \overline{b}(\overline{w}) into \overline{b}_{r} + \overline{b}_{i}\sqrt{−1} with \overline{b}_{r}, \overline{b}_{i} ∈ ℝ^{n}, we obtain in view of (17)

\overline{s}_{2}(\overline{w}) = D(log μ_{Δ_{m}})(\overline{b}_{r}) + \overline{b}_{i}\sqrt{−1},
\overline{s}_{2}(\overline{w})\sqrt{−1} = −\overline{b}_{i} + D(log μ_{Δ_{m}})(\overline{b}_{r})\sqrt{−1},
\overline{s}′_{2}(\overline{w}) = −D(log μ_{Δ_{m}})(\overline{b}_{i}) + \overline{b}_{r}\sqrt{−1} .

The latest equality splits in two equations for \overline{b}_{r}, \overline{b}_{i} which imply that

((λ − λ^{2})Id + 2(1 + λ + λ^{2})D(log μ_{Δ_{m}}) + (λ − λ^{2})D(log μ_{Δ_{m}})^{2})(\overline{b}_{r}) = 0, \overline{b}_{r} ≠ 0 ;

hence the operator D(log μ_{Δ_{m}}) has a negative eigenvalue

\frac{−(1 + λ + λ^{2}) ± \sqrt{(1 + λ + λ^{2})^{2} − (λ − λ^{2})^{2}}}{λ − λ^{2}} ,

but this is impossible. Indeed, the matrix of D(log μ_{Δ_{m}}) is a product of

A = (\frac{∑_{\overline{i}∈Δ} i_{p}i_{q}e^{(\overline{i},\overline{x})} · ∑_{\overline{i}∈Δ} e^{(\overline{i},\overline{x})} − (∑_{\overline{i}∈Δ} i_{p}e^{(\overline{i},\overline{x})})(∑_{\overline{i}∈Δ} i_{q}e^{(\overline{i},\overline{x})})}{(∑_{\overline{i}∈Δ} e^{(\overline{i},\overline{x})})^{2}})_{p,q=1,...,n} ,

B = diag (\frac{∑_{\overline{i}∈Δ} i_{p}e^{(\overline{i},\overline{x})}}{∑_{\overline{i}∈Δ} e^{(\overline{i},\overline{x})}} )_{p=1,...,n} .

Here both A and B are positive definite, for instance,

(A\overline{b}, \overline{b}) = \frac{∑_{\overline{i},\overline{j}∈Δ_{m}} e^{(\overline{i},\overline{x})}e^{(\overline{j},\overline{x})}(∑_{p=1}^{n}(i_{p} − j_{p})b_{p})^{2}}{(∑_{\overline{i}∈Δ_{m}} e^{(\overline{i},\overline{x})})^{2}} > 0 ,

hence AB rotates any vector by an angle < π, and we are done.

The same argument prove the required statement when \overline{w} ∈ ⋃_{m=1}^{N}(∂ℂΔ_{m}) in view of (15) and (16). □

2.3 Algebraic covering

Introduce the map Π_{m} : ℂP^{n} → ℂP^{n}, [z_{0} : …: z_{n}] ↦ [z_{0}^{m} : …: z_{n}^{m}].

Proposition 2.10 For any C-hypersurface M of degree d in ℂP^{n} there exists a number m_{0} > 0 such that for any integer m > m_{0} the preimage Π_{m}^{−1}(M) of M is a PL-variety tame isotopic to a smooth algebraic hypersurface of degree md in ℂP^{n}.

Proof. Let M be defined by polynomials F_{i}(z_{1}, ..., z_{n}), i = 1, ..., N, with Newton polyhedra Δ_{1}, ..., Δ_{N}, where Δ_{1} ∪ ... ∪ Δ_{N} = T_{d}^{n}. The polynomials

F̃_{i}(z_{1}, ..., z_{n}) = F_{i}(z_{1}^{m}, ..., z_{n}^{m}), i = 1, ..., N,

have the Newton polyhedra mΔ_{i}, i = 1, ..., N, and define a C-hypersurface M̃ of degree md.

Lemma 2.11 The variety M̃ is tame isotopic in ℂP^{n} to Π_{m}^{−1}(M).

Proof. Let Δ ⊂ ℝ^{n} be a polyhedron of dimension n. Let us show that the diffeomorphism μ_{Δ}Π_{m}μ_{mΔ}^{−1} : I(mΔ) → I(Δ), where Π_{m} : (ℝ_{+})^{n} → (ℝ_{+})^{n} acts as (x_{1}, ..., x_{n}) ↦ (x_{1}^{m}, ..., x_{n}^{m}), extends to a homeomorphism φ : mΔ → Δ such that, for any proper face σ ⊂ Δ, the map φ|_{mσ} : mσ → σ is a homeomorphism depending only on σ (and not on Δ). Indeed, since Π_{m} commutes with SL_{n}(ℤ) acting on (ℝ_{+})^{n}, we can replace Δ by its image under some g ∈ SL_{n}(ℤ) such that the given face σ will lie on the coordinate hyperplane {i_{1} = 0, i_{2} · ... · i_{n} ≠ 0}. Then μ_{t(Δ)} and μ_{mt(Δ)} extend by the formulas similar to (1) to homeomorphisms of (ℝ_{+})^{n} ∪ {x_{1} = 0, x_{2} · ... · x_{n} ≠ 0} and I(t(Δ)) ∪ I(t(σ)) and I(mt(Δ)) ∪ I(mt(σ)), respectively. This gives us a homeomorphism I(mΔ) ∪ I(mσ) → I(Δ) ∪ I(σ), which does not depend on the choice of t, and we are done.

Consider now the following commutative diagram:

∐_{1≤k≤N}(ℂ^{∗})^{n} \underrightarrow{∐_{1≤k≤N} ℂμ_{mΔ_{k}}} ℂT_{md}^{n} ⟶^{ℂμ} ℂP^{n}
↓^{∐_{1≤k≤N} Π_{m}}  ↓^{φ_{m}}  ↓Π̃_{m}
∐_{1≤k≤N}(ℂ^{∗})^{n} \underrightarrow{∐_{1≤k≤N} ℂμ_{Δ_{k}}} ℂT_{d}^{n} ⟶^{ℂμ} ℂP^{n}

where ∐_{1≤k≤N}(ℂ^{∗})^{n} means the disjoint union of N copies of (ℂ^{∗})^{n}. The map φ_{m} is defined on any ℂΔ_{k} as follows:

(x_{1}, ..., x_{n}) ∈ Δ_{k} ↦ μ_{Δ_{k}}Π_{m}μ_{mΔ_{k}}^{−1}(x_{1}, ..., x_{n}) ∈ Δ_{k},
(v_{1}, ..., v_{n}) ∈ (S^{1})^{n} ↦ (v_{1}^{m}, ..., v_{n}^{m}) ∈ (S^{1})^{n} ,

with μ_{Δ_{k}}Π_{m}μ_{mΔ_{k}}^{−1} extended on the whole Δ_{k}. This definition is correct since the extensions coming from Δ_{k} and Δ_{j} with a common face are the same on the common face, as shown above. The map Π̃_{m} is defined by this diagram. Let us show that Π̃_{m} is tame isotopic to Π_{m}. First, Π_{m}(x_{0}v_{0}, ..., x_{n}v_{n}) = (x_{0}^{m}v_{0}^{m}, ..., x_{n}^{m}v_{n}^{m}) is tame isotopic to π′_{m}(x_{0}v_{0}, ..., x_{n}v_{n}) = (x_{0}v_{0}^{m}, ..., x_{n}v_{n}^{m}). On the other hand, the homeomorphism

∐_{1≤k≤N} μ_{Δ_{k}}Π_{m}μ_{mΔ_{k}}^{−1} : T_{md}^{n} → T_{d}^{n}

in the definition of φ_{m} is tame isotopic to the homothety, turning φ_{m} into

φ′_{m}(x_{1}v_{1}, ..., x_{n}v_{n}) = \frac{1}{m}(x_{1}v_{1}^{m}, ..., x_{n}v_{n}^{m}) ,

and completing the proof of Lemma, since π′_{m} = ℂμ ∘ φ′_{m} ∘ (ℂμ)^{−1}, and φ_{m}^{−1}(ℂCh(F_{k})) = ℂCh(F_{k}(z_{1}^{m}, ..., z_{n}^{m})). □

Lemma 2.12 There exists m_{0} such that, for any m ≥ m_{0}, the subdivision T_{md}^{n} = mΔ_{1} ∪ ... ∪ mΔ_{N} admits a convex refinement.

Proof. Let Γ be the graph of a smooth convex function of n variables, say f(i_{1}, ..., i_{n}) = i_{1}^{2} + ... + i_{n}^{2}, and let pr : Γ → ℝ^{n} be the projection. Denote by sk^{1}(Δ) the 1-skeleton of the subdivision Δ_{1} ∪ ... ∪ Δ_{N}. Clearly, pr^{−1} (⋃_{k} sk^{1}(Δ_{k})) lies on the boundary of its convex hull. The same is true for

pr^{−1} (⋃_{k=1}^{N} sk^{1}(Δ_{k}) ∩ \frac{1}{m}ℤ^{n}) .  (18)

If m is big enough, one can define a required refinement by the piecewise-linear convex function whose graph is the lower part of the boundary of the convex hull of the set (18). □

Now to finish the proof of Proposition 2.10 it remains to apply Theorem 2.2 and Proposition 2.3. □

Denote by χ(X) the Euler characteristic of X, by sign(X) the signature of X if X is a manifold whose dimension is divisible by 4, and by χ_{d}^{n} (resp., sign_{d}^{n}, if n is odd) the Euler characteristic (resp., the signature) of a smooth algebraic hypersurface of degree d in ℂP^{n}.

Corollary 2.13 Any C-hypersurface M of degree d in ℂP^{n} satisfies

χ(M) = χ_{d}^{n}, sign(M) = sign_{d}^{n} .

Proof.

(i) The equality χ(M) = χ_{d}^{n} follows immediately by induction from Proposition 2.10 and the behavior of the Euler characteristic under ramified coverings.

(ii) Suppose that n is odd and consider a smooth algebraic hypersurface M′ of degree d in ℂP^{n}. Let m_{0} be as in Proposition 2.10. Take any prime number p > m_{0}. Note that M̃′ = Π_{p}^{−1}(M′) ⊂ ℂP^{n} is an algebraic hypersurface of degree pd tame isotopic to M̃ = Π_{p}^{−1}(M). The deck transformation group of the coverings Π_{p} : M̃ → M and Π_{p} : M̃′ → M′ is G ≃ (ℤ/p)^{n}. By formula (19) [17]

sign(M) = \frac{1}{|G|} ∑_{g∈G} sign(g, M̃), sign(M′) = \frac{1}{|G|} ∑_{g∈G} sign(g, M̃′) .  (19)

For g = Id ∈ G, we have sign(g, M̃) = sign(M̃) = sign(M̃′) = sign(g, M̃′). Pick g ≠ Id. The twisted signatures sign(g, M̃) and sign(g, M̃′) depend on the embedding of Fix(g) in M̃ and M̃′, respectively, and the action of g in the tangent and normal bundles of Y = Fix(g) ⊂ M̃ and Y′ = Fix(g) ⊂ M̃′ (see Theorem 6.12 [3]). We do not compute the twisted signatures along Theorem 6.12 [3], but use Hirzebruch’s formula for the signature of ramified coverings. Namely, let ⟨g⟩ ⊂ G be the cyclic group of order p generated by g. According to [16],

p · sign(M̃/⟨g⟩) = ∑_{i=0}^{p−1} sign(g^{i}, M̃)

is a universal function of p and the signatures of self-intersections Y ∘ Y, Y ∘ Y ∘ Y, . . .. Note that Y = Fix(g) ⊂ M̃ and Y′ = Fix(g) ⊂ M̃′ are the intersections of M̃ and M̃′, respectively, with the same collection of coordinate planes. Hence, due to the tame isotopy of M̃ and M̃′, one has sign(Y ∘Y) = sign(Y′∘Y′), sign(Y ∘Y ∘Y) = sign(Y′ ∘ Y′ ∘ Y′), and so on. Therefore for any g ∈ G we have

∑_{i=0}^{p−1} sign(g^{i}, M̃) = ∑_{i=0}^{p−1} sign(g^{i}, M̃′) ,

which immediately implies the required equality of the right hand sides in (19), since G ≃ (ℤ/p)^{n} can be decomposed in the union of cyclic subgroups such that the only intersection of any two of these subgroups is Id ∈ G. □

Corollary 2.14 Let M be a C-hypersurface of degree d in ℂP^{n}. Then

• M is simply connected if n > 2,

• π_{1}(ℂP^{n}\M) = ℤ/dℤ if n ≥ 2.

Proof.

(i) To show that M is simply connected for n > 2, note that an algebraic hypersurface of dimension greater than 1 is simply connected. Let m > m_{0} be as in Proposition 2.10. Consider a loop γ in M. We can move it slightly so that it does not meet the coordinate hyperplanes in ℂP^{n}. Since Π_{m}^{−1}(M) is simply connected, m^{n} · [γ] is contractible in M. Similarly, (m + 1)^{n} · [γ] is contractible in M, and we are done, because m^{n} and (m + 1)^{n} are coprime.

(ii) Since an affine nonsingular algebraic hypersurface in ℂ^{n+1}, n ≥ 2, is simply connected, the previous argument shows that M̂\ℂP^{n} ⊂ ℂ^{n+1} is simply connected, where M̂ is a C-hypersurface in ℂP^{n+1} and ℂP^{n} is a coordinate hyperplane in ℂP^{n+1}.

Let a C-hypersurface M of degree d in ℂP^{n} be defined by a subdivision S : T_{d}^{n} = Δ_{1} ∪ ... ∪ Δ_{N} and collection of numbers A : T_{d}^{n} ∩ ℤ^{n} → ℂ. Embed T_{d}^{n} into T_{d}^{n+1} as the face T_{d}^{n+1} ∩ {i_{n+1} = 0}, take the subdivision S̃: T_{d}^{n+1} = Δ̃_{1} ∪ ... ∪ Δ̃_{N} with Δ̃_{k} being the cone over Δ_{k} with the vertex (0, ..., 0, d), k = 1, ..., N, and define Ã: T_{d}^{n+1} ∩ ℤ^{n+1} → ℂ by Ã_{i_{1}...i_{n}0} = A_{i_{1}...i_{n}}, Ã_{0...0d} = −1, Ã_{i_{1}...i_{n}i_{n+1}} = 0 if 0 < i_{n+1} < d. These data define nondegenerate polynomials F̃_{k}(z_{1}, ..., z_{n+1}) = F_{k}(z_{1}, ..., z_{n}) − z_{n+1}^{d} with Newton polyhedra Δ̃_{k}, k = 1, ..., N, whose charts can be glued into a Chypersurface M̃ of degree d in ℂP^{n+1}. The required statement π_{1}(ℂP^{n}\M) = ℤ/dℤ is a corollary of the following statement.

Lemma 2.15 There is a ℤ/dℤ-covering M̃\{z_{n+1} = 0} → ℂP^{n}\M.

Proof. We construct a ℤ/dℤ-covering Φ : ℂCh(S̃, Ã)\{w_{n+1} = 0} → ℂT_{d}^{n}\ℂCh(S, A) and then show that it commutes with the S^{1}-action on ∂ℂCh(S̃, Ã)\{z_{n+1} = 0} and ∂ℂT_{d}^{n}\ℂCh(S, A), thereby defining the required covering.

Let (w_{1}, ..., w_{n}, w_{n+1}) ∈ ℂCh(S̃, Ã), w_{n+1} ≠ 0. Then (w_{1}, ..., w_{+1}) ∈ ℂI(δ̃), where δ̃ is a cone over a face δ of Δ_{k}, 1 ≤ k ≤ N, with the vertex at (0, ..., 0, d). Hence (w_{1}, ..., w_{n+1}) = ℂμδ̃(z_{1}, ..., z_{n+1}) for some (z_{1}, ..., z_{n+1}) ∈ (ℂ^{∗})^{n+1}. So, we define

Φ(w_{1}, ..., w_{n+1}) = ℂμ_{δ}(z_{1}, ..., z_{n}) .

The map Φ is well defined. If δ = Δ_{1}, …, Δ_{N}, then ℂμδ̃ and ℂμ_{δ} are diffeomorphisms. If δ lies in an (n−1)-plane α_{1}i_{1} +...+α_{n}i_{n} = β, i_{n+1} = 0, then δ̃ lies in an n plane α_{1}i_{1} + ... + α_{n}i_{n} + βi_{n}/d = β. This means that (ℂμδ̃)^{−1}(w_{1}, ..., w_{n+1}) contains the family (z_{1}t^{α_{1}}, ..., z_{n}t^{α_{n}}, z_{n+1}^{β/d}), t ∈ ℝ_{+}^{∗}, but ℂμ_{δ} takes the family (z_{1}t^{α_{1}}, ..., z_{n}t^{α_{n}}), t ∈ ℝ_{+}^{∗}, to one point.

The map Φ is continuous. If, for some k = 1, ..., N and a curve γ̃(t) ∈ (ℂ^{∗})^{n+1}:

(λ_{1}t^{k_{1}} + O(t^{k_{1}+1}), ..., λ_{n}t^{k_{n}} + O(t^{k_{n}+1}), λ_{n+1}t^{k_{n+1}} + O(t^{k_{n+1}+1})), t > 0,

one has lim_{t→0} ℂμΔ̃_{k}(γ̃(t)) = ℂμδ̃(λ_{1}, ..., λ_{n}, λ_{n+1}), then lim_{t→0} ℂμ_{Δ_{k}}(γ(t)) = ℂμ_{δ}(λ_{1}, ..., λ_{n}), where

γ(t) = (λ_{1}t^{k_{1}} + O(t^{k_{1}+1}), ..., λ_{n}t^{k_{n}} + O(t^{k_{n}+1})) ∈ (ℂ^{∗})^{n}, t > 0 .

The map Φ is surjective. Indeed, if (w_{1}, ..., w_{n}) ∈ ℂT_{d}^{n}\ℂCh(S, A), then (w_{1}, ..., w_{n}) = ℂμ_{δ}(z_{1}, ..., z_{n}) for a face δ of Δ_{k} (1 ≤ k ≤ N), where (z_{1}, ..., z_{n}) ∈ (ℂ^{∗})^{n}, F_{k}^{δ}(z_{1}, ..., z_{n}) ≠ 0. Then there exists z_{n+1} ≠ 0 such that

F_{k}^{δ}(z_{1}, ..., z_{n}) = z_{n+1}^{d}; hence ℂμδ̃(z_{1}, ..., z_{n}, z_{n+1}) belongs to ℂCh(F δ̃ ; k)\{w_{n+1} = 0}

and Φ(ℂμδ̃(z_{1}, ..., z_{n}, z_{n+1})) = (w_{1}, ..., w_{n}).

The map Φ is a ℤ/dℤ-covering. Indeed, for any point (w_{1}, ..., w_{n}) ∈ ℂT_{d}^{n}\ℂCh(S, A) its preimage Φ^{−1}(w_{1}, ..., w_{n}) consists of d distinct points (w′_{1}, ..., w′_{n}, w′_{n+1}ω) with some fixed w′_{1}, ..., w′_{n+1} and any d-th root of unity ω.

At last, note that if Φ(w′_{1}, ..., w′_{n}, w′_{n+1}) = (w_{1}, ..., w_{n}), then Φ(w′_{1}v, ..., w′_{n}v, w′_{n+1}v) = (w_{1}v, ..., w_{n}v) for arbitrary v ∈ S^{1}, which completes the proof. □

Proposition 2.16 Let M ⊂ ℂP^{n} be a C-hypersurface and M′ ⊂ ℂP^{n} a nonsingular algebraic hypersurface, both of degree d. Then H_{∗}(M) and H_{∗}(M′) are isomorphic as graded groups.

Proof. Take m_{0} as in Proposition 2.10. Choose m > m_{0} in such a way that m is coprime with all the orders of elements in TorsH_{∗}(M), and consider M_{m} = Π_{m}^{−1}(M). Pick a homology class α ∈ H_{i}(M). Note that m^{n}α belongs to the image of (Π_{m})_{∗} : H_{i}(M_{m}) → H_{i}(M). For an odd i different from n − 1 we have H_{i}(M_{m}) = 0 and, hence, H_{i}(M) = 0. In particular, H_{∗}(M) has no torsion. Any group H_{i}(M) with even i (0 ≤ i ≤ 2n − 2) contains a nontrivial element: the fundamental class of the intersection of M with coordinate hyperplanes taken in appropriate number. Thus, H_{i}(M) is isomorphic to H_{i}(M_{m}) ≃ ℤ if i is even and different from n − 1. The fact that the groups H_{n−1}(M) and H_{n−1}(M′) are isomorphic follows now from the equality χ(M) = χ(M′) proven in Corollary 2.13. □

Proposition 2.17 Let n be a positive odd number, M a C-hypersurface of degree d in ℂP^{n} and M′ a nonsingular algebraic hypersurface of degree d in ℂP^{n}. Then the lattices (H_{n−1}(M), B_{M}) and (H_{n−1}(M′), B_{M′}) (where B_{M} : H_{n−1}(M) × H_{n−1}(M) → ℤ and B_{M′} : H_{n−1}(M′) × H_{n−1}(M′) → ℤ are the intersection forms on M and M′, respectively) are isomorphic.

Proof. The lattices H_{n−1}(M) and H_{n−1}(M′) are unimodular and have the same rank and signature (Corollary 2.13). It remains to show that these two lattices have the same parity, since for d = 1 we have H_{n−1}(M) ≃ H_{n−1}(M′) ≃ ℤ, and for d ≥ 2 the lattice H_{n−1}(M′) is indefinite. Let us show, first, that the lattices H_{n−1}(M) and H_{n−1}(M̃) have the same parity, where M̃ = Π_{m}^{−1}(M), m_{0} is as in Proposition 2.10 and m > m_{0} is odd.

(i) If α^{2} is odd, α ∈ H_{n−1}(M), then ((Π_{m})^{!}α)^{2} = m^{2n}α^{2} is odd as well.

(ii) If β^{2} is odd, β ∈ H_{n−1}(M̃), then γ^{2} is odd, where γ = ∑_{g∈G} g_{∗}β, and G = (ℤ/m)^{n} is the deck transformation group of Π_{m}. Furthermore, γ = (Π_{m})^{!}α for certain α ∈ H_{n−1}(M), and α^{2} is odd.

Similarly, the lattices H_{n−1}(M′) and H_{n−1}(M̃′), where M̃′ = Π_{m}^{−1}(M′), have the same parity. Since M̃ and M̃′ are isotopic in ℂP^{n}, we obtain that the lattices H_{n−1}(M) and H_{n−1}(M′) have the same parity. □

Corollary 2.18 A C-surface of degree d in ℂP^{3} is homeomorphic to a nonsingular algebraic surface of degree d in ℂP^{3}.

Proof. Note that a C-surface of degree d in ℂP^{3} is simply connected (see 2.13), smoothable (see Proposition 2.7) and have the same intersection form as an algebraic surface of degree d in ℂP^{3} (see 2.17). It remains to apply Freedman’s theorem (see, for example, [11]). □

2.4 Ramified double coverings

Proposition 2.19 Let M be a C-hypersurface of even degree d in ℂP^{n}. Then there exists a closed simply connected 2n-dimensional PL-manifold Y and a map Π : Y → ℂP^{n} which is a double covering ramified along M.

Proof. The existence of a ramified double covering Π : Y → ℂP^{n} with Y being a PL-manifold follows generically from the fact that [M] ∈ H_{2n−2}(ℂP^{n}) is an even class. However, we prefer to give an explicit construction of Y.

Namely, we repeat the construction of step (ii) in the proof of Corollary 2.14, taking the simplex T_{d,2}^{n+1} ⊂ ℝ^{n+1} with vertices (0, ..., 0), (d, 0, ..., 0), ..., (0, ..., 0, d, 0), (0, ..., 0, 2) instead of T_{d}^{n+1}. So, T_{d}^{n} embeds into T_{d,2}^{n+1} as the face T_{d,2}^{n+1} ∩ {i_{n+1} = 0}, the subdivision T_{d,2}^{n+1} = Δ̃_{1} ∪ ... ∪ Δ̃_{N} is defined as the cone over the subdivision T_{d}^{n} = Δ_{1} ∪ ... ∪ Δ_{N} with the vertex at (0, ..., 0, 2), and Ã : T_{d,2}^{n+1} ∩ ℤ^{n+1} → ℂ is defined by A_{i_{1}...i_{n}0} = A_{i_{1}...i_{n}}, Ã_{i_{1}...i_{n}1} = 0, Ã_{0...02} = −1. These data define a PLmanifold ℂCh(S̃, Ã) with boundary ∂ℂCh(S̃, Ã) = ℂCh(S̃, Ã) ∩ ∂ℂT_{d,2}^{n+1}, which is the gluing of the charts of the polynomials F̃_{k}(z_{1}, ..., z_{n}, z_{n+1}) = F_{k}(z_{1}, ..., z_{n}) − z_{n+1}^{2}, k = 1, ..., N, and, as in the proof of Lemma 2.15, there exists a double covering Φ : ℂCh(S̃, Ã) → ℂT_{d}^{n} ramified along ℂCh(S, A).

Now note that S^{1} acts on ∂ℂT_{d,2}^{n+1} by

(v, (w_{1}, ..., w_{n}, w_{n+1}) ↦ (w_{1}v, ..., w_{n}v, w_{n+1}v^{d/2}), v ∈ S^{1} .

The manifold ∂ℂCh(S̃, Ã) is invariant with respect to this action, because it is defined by quasihomogeneous polynomials f(z_{1}, ..., z_{n+1}) satisfying

f(z_{1}τ, ..., z_{n}τ, z_{n+1}τ^{d/2}) = τ^{d}f(z_{1}, ..., z_{n+1}),

which is compatible with the S^{1} action. By the same reason

Φ(w_{1}v, ..., w_{n}v, w_{n+1}v^{d/2}) = (w′_{1}v, ..., w′_{n}v), v ∈ S^{1} ,

as far as Φ(w_{1}, ..., w_{n}, w_{n+1}) = (w′_{1}, ..., w′_{m}), (w_{1}, ..., w_{n+1}) ∈ ∂ℂCh(S̃, Ã). Hence Φ reduces to a double covering Π : Y → ℂP^{n}, where Y = ℂCh(S̃, Ã)/S^{1}, ramified along M. It remains to show that Y is a closed PL-manifold. Indeed, for any edge of T_{d,2}^{n+1} there exists a combination of an automorphism of ℤ^{n+1}, leaving the (n + 1)-st axis fixed, and shifts which puts this edge on a coordinate axis and the adjacent faces of T_{d,s}^{n} on the corresponding coordinate planes. One can easily verify that the orbits of the S^{1} action on that edge and adjacent faces contract into points, whose neighborhoods in Y are homeomorphic to ℝ^{2n} as shown in the proof of Proposition 1.17.

At last, we show that π_{1}(Y) = 0. Any loop in ℂP^{n} through a point \overline{w} ∈ M lifts to a loop on Y through \overline{w}, because \overline{w} is covered in Y by itself only. Since any loop in ℂP^{n} is contractible, so does its lifting in Y. □

Proposition 2.20 Let n and d be positive even numbers, M be a C-hypersurface of degree d in ℂP^{n}, and Y be a double covering of ℂP^{n} ramified along M. Then

χ(Y) = χ_{d,2}^{n}, sign(Y) = sign_{d,2}^{n}, H_{∗}(Y) ≃ H_{∗}(Y′),

and the lattices H_{n}(Y) and H_{n}(Y′), equipped with the intersection forms, are isomorphic, where Y′ is the double covering of ℂP^{n} ramified along a nonsingular algebraic hypersurface of degree d, and χ_{d,2}^{n} and sign_{d,2}^{n} are, respectively, the Euler characteristic and the signature of Y′.

Proof. The equalities χ(Y) = χ(Y′) and sign(Y) = sign(Y′) follow immediately from the additivity of the Euler characteristic, the Hirzebruch formula for the signature of ramified coverings and Corollary 2.13. The isomorphism H_{∗}(Y) ≃ H_{∗}(Y′), by [10], reduces to H_{2i−1}(Y) = 0, i = 1, ..., n, H_{2i}(Y) = ℤ, i = 0, ..., n, i ≠ n/2, H_{n}(Y) ≃ ℤ^{r}, r = rkH_{n}(Y′). It can be proven as Proposition 2.16, using a ramified covering of Y similar to that constructed in Proposition 2.10. The same ramified covering applied as in the proof of Proposition 2.17, gives a lattice isomorphism H_{n}(Y) ≃ H_{n}(Y′). □

Proposition 2.21 Let n and d be positive even numbers and M be a real Chypersurface of degree d in ℂP^{n}. Then ℝP^{n} = ℝM_{+} ∪ ℝM_{−}, where ℝM_{+} and ℝM_{−} are compact manifolds with boundary such that

∂ℝM_{+} = ∂ℝM_{−} = ℝM_{+} ∩ ℝM_{−} = ℝM .

The ramified double covering Π : Y → ℂP^{n} admits an action of the group ℤ/2ℤ ⊕ ℤ/2ℤ = {Id, τ, Conj_{+}, Conj_{−}}, where τ is the deck transformation of Π and Π ∘ Conj_{+} = Π ∘ Conj_{−} = Conj ∘ Π. In addition,

Fix(τ) = M, ℝY_{±} =^{def} Fix(Conj_{±}) = Π^{−1}(ℝM_{±}) .

Proof. In the framework of the construction in the proof of Proposition 2.19, we define ℝCh(S, A)_{+} (resp., ℝCh(S, A)_{−}) as the union of the closures of ℂμ_{Δ_{k}}({F_{k} ≥ 0} ∩ (ℝ^{∗})^{n}) (resp., ℂμ_{Δ_{k}}({F_{k} ≤ 0} ∩ (ℝ^{∗})^{n})), k = 1, ..., N. The action of S^{1} on ∂ℂT_{d}^{n} reduces to the action of S^{0} = {±1} on ∂ℝT_{d}^{n} which preserves ℝCh(S, A)_{+} and ℝCh(S, A)_{−}, thus defining ℝM_{±} = ℝCh(S, A)_{±}/S^{0}. The involutions τ, Conj_{+}, Conj_{−} on Y one gets from the involutions on ℂCh(S̃, Ã):

(w_{1}, ..., w_{n}, w_{n+1}) ↦ (w_{1}, ..., w_{n}, −w_{n+1}) ,
(w_{1}, ..., w_{n}, w_{n+1}) ↦ Conj(w_{1}, ..., w_{n}, w_{n+1}) ,
(w_{1}, ..., w_{n}, w_{n+1}) ↦ Conj(w_{1}, ..., w_{n}, −w_{n+1}) □

Now we have to make a digression on the topology of smooth (not necessarily algebraic) hypersurfaces in ℝP^{n}. Following Viro [37], we define a rank of a connected smooth hypersurface in ℝP^{n} to be the maximal integer r such that the homomorphism induced in r-dimensional homology with ℤ/2ℤ coefficients by the inclusion of the hypersurface into the projective space is nontrivial. It is easy to see that the intersection of a hypersurface of rank r in ℝP^{n} with a transversal projective subspace P of dimension k ≥ n − r is a hypersurface of rank r − n + k in P. Similarly, we define a rank of a connected component of the complement of a hypersurface in ℝP^{n}. Clearly, the rank of a connected component C of the complement of a two-sided hypersurface (i.e., a hypersurface dividing its tubular neighborhood) is greater or equal to the rank of each component of the boundary ∂C.

A component C of the complement of a two-sided hypersurface is called principal if the rank of C is greater then the rank of each component of ∂C.

Proposition 2.22 Let S be a two-sided hypersurface in ℝP^{n}. Then there exists at most one principal component of ℝP^{n} \ S.

Proof. Suppose that ℝP^{n} \ S has two principal components C_{1} and C_{2}. Let r_{1} (resp., r_{2}) be the maximal rank of a connected component of ∂C_{1} (resp., ∂C_{2}). Assume that r_{1} ≤ r_{2}. Consider a projective subspace P of ℝP^{n} transversal to S and of dimension n − r_{1}. All the connected components of P ∩ ∂C_{1} are of rank 0 in P. Note that the ranks of P ∩ C_{1} and P ∩ C_{2} in P are positive. Each connected component of P ∩ ∂C_{1} divides P into two parts. One of them has rank 0 in P, and is called the interior of the component. The other one contains P ∩ C_{1} (and has a positive rank). Therefore, P ∩ C_{2} is contained in the interior of some component of P ∩ ∂C_{1}, and thus should have rank 0. □

Clearly, the real parts of a real algebraic hypersurface of even degree and of a real C-hypersurface of even degree are two-sided in ℝP^{n}. Switching if necessary ℝM_{+} and ℝM_{−}, we suppose from now on that ℝM_{−} contains the principal component of ℝP^{n} \ ℝM, if this principal component does exist. For any manifold X denote by b_{∗}(X) the total Betti number dim_{ℤ/2ℤ} H_{∗}(X; ℤ/2ℤ) of X.

Theorem 2.23 Let n be a positive even number, and M a real C-hypersurface of degree d in ℂP^{n}. Then b_{∗}(ℝY_{+}) = b_{∗}(ℝM) and b_{∗}(ℝY_{−}) = b_{∗}(ℝM) + 2.

Proof. From the Smith exact sequence (see, for example, [5, 38]) applied to the deck transformation of ℝY_{±}, it easily follows that b_{∗}(ℝY_{±}) = b_{∗}(ℝM) + 2a, where a is the number of principal components in ℝM_{±}. It remains to use Proposition 2.22 and note that in our case ℝP^{n} \ ℝM should have a principal component. Indeed, if ℝP^{n} \ ℝM does not have a principal component, one has b_{∗}(ℝY_{+}) = b_{∗}(ℝM) and b_{∗}(ℝY_{−}) = b_{∗}(ℝM), and thus, χ(ℝY_{+}) ≡ χ(ℝY_{−}) mod 4. The last congruence is impossible, because χ(ℝY_{±}) = 2χ(ℝM_{±}) and χ(ℝM_{+}) + χ(ℝM_{−}) = χ(ℝP^{n}) = 1. □

Remark 2.24 In particular, the statement of Theorem 2.23 holds for real algebraic hypersurfaces of even degree in even dimensional projective space.

3 Topology of real C-hypersurfaces

3.1 Generalized Harnack inequalities

Let M′ be a nonsingular algebraic hypersurface of degree d in ℂP^{n}. Denote by b_{d}^{n} the total Betti number b_{∗}(M′) = dim_{ℤ/2ℤ} H_{∗}(M′; ℤ/2ℤ) of M′. One has (see, for example, [8])

b_{d}^{n} = \frac{(d − 1)^{n+1} − (−1)^{n+1}}{d} + d + (−1)^{n+1}.

Theorem 3.1 For a real C-hypersurface M of degree d in ℂP^{n} one has

b_{∗}(ℝM) = b_{d}^{n} − 2a(ℝM),  (20)

where a(ℝM) is a nonnegative integer. Furthermore, if n and d are both even, then

b_{∗}(ℝY_{−}) = b_{d,2}^{n} − 2a(ℝM),  (21)
b_{∗}(ℝY_{+}) = b_{d,2}^{n} − 2a(ℝM) − 2,  (22)

where Y is the double covering of ℂP^{n} ramified along M, ℝY_{±} are the fixed point sets of two liftings Conj_{±} to Y of the complex conjugation Conj in ℂP^{n}, and b_{d,2}^{n} is the total Betti number of the double covering of ℂP^{n} ramified along a nonsingular hypersurface of degree d.

Proof. The statement follows from the Smith-Floyd inequality

dim_{ℤ/2ℤ} H_{∗}(Fix(τ); ℤ/2ℤ) ≤ dim_{ℤ/2ℤ} H_{∗}(X; ℤ/2ℤ)  (23)

(where X is a compact CW-complex and τ : X → X is an involution) and the congruence

dim_{ℤ/2ℤ} H_{∗}(X; ℤ/2ℤ) ≡ dim_{ℤ/2ℤ} H_{∗}(Fix(τ); ℤ/2ℤ) mod 2;  (24)

see for details [24, 38]. We apply the Smith inequality and the above congruence to the involutions Conj on M and Conj_{±} on Y. In order to get (20) we use Proposition 2.16. To prove (21) and (22) we notice that according to Theorem 2.23 one has b_{∗}(ℝY_{+}) = b_{∗}(ℝM) and b_{∗}(ℝY_{−}) = b_{∗}(ℝM) + 2. In addition,

b_{∗}(Y) = χ(Y) = 2χ(ℂP^{n}) − χ(M) = 2(n + 1) − (2n − b_{∗}(M)) = 2 + b_{∗}(M) ,

and it remains to apply Proposition 2.20. □

3.2 Congruences

Theorem 3.2 Let M be a real C-hypersurface of degree d in ℂP^{n}. If n is odd then

χ(ℝM) ≡ sign_{d}^{n} + \{ \begin{matrix}0, if a(ℝM) = 0, \\ ±2, if a(ℝM) = 1 \end{matrix} mod 16 .  (25)

If n and d are even, then

χ(ℝM_{−}) ≡ \frac{sign_{d,2}^{n}}{2} + \{ \begin{matrix}0, if a(ℝM) = 0, \\ ±1, if a(ℝM) = 1 \end{matrix} mod 8 ,  (26)

χ(ℝM_{+}) ≡ \frac{sign_{d,2}^{n}}{2} ± 1 mod 8, if a(ℝM) = 0.  (27)

Proof. The statement can be proven as the Rokhlin and Gudkov-Krahnov-Kharlamov congruences in the algebraic case [14, 21, 24, 38].

(i) Let n = 2k + 1 and a(ℝM) = 0. As in [24] (see also [14, 38]), the latter implies the splitting of H_{n−1}(M) into the orthogonal sum H_{+} ⊕ H_{−} of unimodular eigenlattices corresponding to the eigenvalues ±1 of Conj_{∗} : H_{n−1}(M) → H_{n−1}(M).

Lemma 3.3 The signature of involution sign(Conj_{∗}) = sign(H_{+})−sign(H_{−}) is equal to (−1)^{k}χ(ℝM).

Proof. By the Atiyah-Singer formula (see [3, 24, 38]) sign(Conj_{∗}) is equal to ℝM ∘ ℝM, the selfintersection of ℝM in M. Let us show that ℝM ∘ ℝM = (−1)^{k}χ(ℝM). First, we smooth M as in Proposition 2.7. Then we take a tangent vector field V on ℝM_{sm} having only finitely many singular points which are all nondegenerate and lie outside ℝδ for any proper face δ of the polyhedra Δ_{1}, ..., Δ_{N} in the subdivision of T_{d}^{n}. Extend the vector field J(V) (where J is the almost complex structure on M_{sm} defined in Proposition 2.8) to a neighborhood of ℝM_{sm} in M_{sm} and slightly move ℝM_{sm} along geodesics in M_{sm} tangent to the field obtained. The result has transversal intersection points with ℝM_{sm} at the singular points of V. The intersection indices are equal to the multiplied by (−1)^{k} indices of the singular points of V (see [24, 38]). □

Denote by H_{e} the lattice H_{+} (resp., H_{−}) if k is odd (resp., k is even). From the existence of an almost complex structure on a smoothing of M (Proposition 2.8), it follows that H_{e} is even (see, for example, [38]). Let σ_{e} be the signature of H_{e}. Since H_{e} is unimodular and even, the signature σ_{e} is divisible by 8. To finish the proof in the case a(ℝM) = 0, it remains to note that according to Lemma 3.3 we have sign(M) − χ(ℝM) = 2σ_{e}.

The prove (27) and (26) in the case a(ℝM) = 0, we apply the same arguments to (Y, Conj_{−}) and (Y, Conj_{+}) and use the relations χ(ℝY_{±}) = 2χ(ℝM_{±}), b_{∗}(ℝY_{+}) = b_{∗}(ℝM) and b_{∗}(ℝY_{−}) = b_{∗}(ℝM) + 2.

(ii) Let now n = 2k + 1 and a(ℝM) = 1. In this case (see, for example, [21, 38]), the discriminants of H_{+} and H_{−} are ±2 (from the Smith theory we get the inequality | discr(H_{±}) | ≤ 2 and then use the congruence χ(M) ≡ (−1)^{k}sign(M) mod 4 to show that | discr(H_{±}) | ≠ 1). Using Lemma 3.3 and the fact that the signature of an even lattice with discriminant ±2 is congruent to ±1 mod 8 we immediately obtain the statement required.

To prove (26) in the case a(ℝY_{−}) = 1, we again apply the previous arguments to (Y, Conj_{−}). □

3.3 Comessatti inequality for real C-surfaces

Theorem 3.4 Let M be a real C-hypersurface of degree d in ℂP^{n} and M′ be a nonsingular algebraic hypersurface of degree d in ℂP^{n}. Then

2 − h^{1,1}(M′) ≤ χ(ℝM) ≤ h^{1,1}(M′).

Proof. The arguments are completely similar to the proof of the Comessatti inequality in the case of real algebraic surfaces.

Let again H_{+} and H_{−} be eigensublattices of H_{2}(M) corresponding to the eigenvalues ±1 of Conj_{∗} : H_{2}(M) → H_{2}(M). Denote by a_{±}^{+} (resp., a_{±}^{−}) the number of positive (resp., negative) squares in the diagonal form over ℚ of the restriction of B : H_{2}(M) × H_{2}(M) → ℤ to H_{±}. We have

a_{+}^{+} + a_{+}^{−} + a_{−}^{+} + a_{−}^{−} = dim H_{2}(M),
a_{+}^{+} − a_{+}^{−} + a_{−}^{+} − a_{−}^{−} = sign(M),
a_{+}^{+} + a_{+}^{−} − a_{−}^{+} − a_{−}^{−} = χ(ℝM) − 2,
a_{+}^{+} − a_{+}^{−} − a_{−}^{+} + a_{−}^{−} = −χ(ℝM).

The first two equations are clear. The third and fourth ones follow from the Lefschetz fixed point theorem and Atiyah-Singer theorem, respectively. We obtain that

a_{+}^{−} = (dim H_{2}(M) − sign(M) + 2χ(ℝM) − 2)/4 ≥ 0,
a_{−}^{−} = (dim H_{2}(M) − sign(M) − 2χ(ℝM) + 2)/4 ≥ 0.

These inequalities together with the Hodge index relations give the required statement. □

3.4 Topology of real C-curves

Let M be an oriented smooth connected closed surface in ℂP^{2}. Then M is called a flexible curve of degree d (see [35]) if

• it realizes d[ℂP^{1}] ∈ H_{2}(ℂP^{2}), • the genus of M is equal to (d − 1)(d − 2)/2,

• M is invariant under the complex conjugation,

• the field of tangent planes to M on M ∩ ℝP^{2} can be equivariantly deformed to the field of lines in ℂP^{2} tangent to M ∩ ℝP^{2}.

According to Propositions 2.6, 2.8 and Corollary 2.13 (a smoothing of) a real C-curve of degree d in ℂP^{2} is a flexible curve of degree d. Thus, all the restrictions on the topology of flexible curves are applicable to real C-curves. We formulate here in the framework of real C-curves the principal known restrictions on flexible curves. A big list of restrictions on the topology of flexible curves can be found in [35].

Let us start from definitions. The standard definitions applicable to real algebraic curves can be naturally extended to real C-curves. A real C-curve A of degree d in ℂP^{2} is called an M-curve or maximal if ℝA has (d − 1)(d − 2)/2 + 1 connected components. A real C-curve A of degree d in ℂP^{2} is called an (M − i)-curve if ℝA has (d − 1)(d − 2)/2 + 1 − i connected components. A connected component of the real part of a real C-curve of degree d in ℂP^{2} is called an oval if it divides ℝP^{2} into two parts. The part homeomorphic to a disk is called the interior of the oval. All the connected components of the real part of a real C-curve of an even degree in ℂP^{2} are ovals. Exactly one connected component of the real part of a real C-curve of an odd degree in ℂP^{2} is not an oval. This component is called nontrivial. An oval is even (resp., odd) if it lies inside of an even (resp., odd) number of other ovals of the curve. The numbers of even and odd ovals of a curve are denoted by p and n, respectively. The Euler characteristic of a connected component of the complement in ℝP^{2} of the real part of a real C-curve is called the characteristic of an oval bounding the component from outside.

Theorem 3.5 • Harnack inequality. The number of connected components of the real part of a real C-curve of degree d in ℂP^{2} is at most (d−1)(d−2)/2+1.

• Gudkov-Rokhlin congruence. For a maximal real C-curve of degree d = 2k

in ℂP^{2} one has

p − n ≡ k^{2} mod 8.

• Gudkov-Krahnov-Kharlamov congruence. Let A be a real C-curve of degree d = 2k in ℂP^{2}. If A is an (M − 1)-curve, then

p − n ≡ k^{2} ± 1 mod 8.

• Strengthened Petrovsky inequalities. For a real C-curve A of degree d = 2k in ℂP^{2} one has

p − n^{−} ≤ 3k(k − 1)/2 + 1, n − p^{−} ≤ 3k(k − 1),

where p^{−} (resp., n^{−}) is the number of even (resp., odd) ovals of ℝA with negative characteristic.

• Strengthened Arnold inequalities. For a real C-curve A of degree d = 2k in ℂP^{2} one has

p^{−} + p^{0} ≤ (k^{2} − 3k + 3 + (−1)^{k})/2, n^{−} + n^{0} ≤ (k^{2} − 3k + 2)/2,

where p^{0} (resp., n^{0}) is the number of even (resp., odd) ovals of ℝA with characteristic 0.

A real C-curve A in ℂP^{2} is said to be of type I if its real part ℝA divides A into two parts; otherwise, the curve is of type II. For a curve A of type I, the orientations of two halves of A \ ℝA induce on ℝA two opposite orientations which are called complex orientations. Note that a real C-curve A is of type I if and only if all the algebraic curves used in the gluing of A are of type I and complex orientations on the real parts of these curves can be chosen in such a way that they glue into an orientation of ℝA.

A pair of ovals of the real part of a real C-curve in ℂP^{2} is injective if one of them is inside of the other one. A collection of ovals is called a nest if any two of them form an injective pair. An injective pair of ovals of a real C-curve is positive (resp., negative) if the complex orientations of the ovals are induced (resp., are not induced) from some orientation of the annulus bounded by the ovals. Take an oval of a real C-curve of type I and of an odd degree in ℂP^{2}, and consider the Möbius band which is the complement in ℝP^{2} of the interior of the oval. The oval is called positive (resp., negative) if the integer homology class realized in the Möbius band by the oval equipped with a complex orientation differs in sign (resp., coincides) with the class of the doubled nontrivial component equipped with the complex orientation.

Theorem 3.6 • Klein congruence. Let A be a real C-curve of type I in ℂP^{2}. If A is an (M − i)-curve, then i ≡ 0 mod 2.

• Arnold congruence. A real C-curve of even degree d = 2k satisfies

p − n ≡ k^{2} mod 4 .

• Rokhlin-Mishachev formulae. For a real C-curve A of type I and of degree d =

2k in ℂP^{2} one has

2(Π^{+} − Π^{−}) = l − k^{2},

where l is the number of ovals of ℝA, and Π^{+} and Π^{−} are the numbers of positive and negative injective pairs, respectively. For a real C-curve A of type I and of degree d = 2k + 1 in ℝP^{2} one has

2(Π^{+} − Π^{−}) + Λ^{+} − Λ^{−} = l − k(k + 1),

where Λ^{+} and Λ^{−} are the numbers of positive and negative ovals, respectively.

• Kharlamov- Marin congruence. Let A be a real C-curve of degree d = 2k in ℂP^{2}. If A is an (M − 2)-curve and p − n ≡ k^{2} + 4 mod 8, then A is of type I.

Harnack inequality in the case of real C-curves constructed using a primitive (i.e., such that all its triangles are of area 1/2) triangulation was, first, proved in [18] and then in a different way in [15]. Rokhlin-Mishachev formulae in the case of real C-curves constructed using a primitive triangulation was proved in [22].

It is interesting that one restriction on real C-curves is not proved in the case of flexible curves. This restriction was proved in [9]:

consider a real C-curve A of degree d in ℂP^{2}; if A is glued from threenomials, then the sum of the depths of any two nests of ℝA is at most d/2.

In the case of real algebraic curves this statement is known as Hilbert’s theorem and is an immediate corollary of the Bezout theorem.

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