Polar Homology

Boris Khesin^{∗} and Alexei Rosly^{†}

August 12, 2000

Abstract

For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré residue on it.

1 Introduction

In this paper we introduce certain homology groups defined for complex projective manifolds that can be regarded as a complex version of singular homology groups in topology. The idea of such a geometric analogue of topological homology comes from thinking of the Dolbeault (or ¯∂) complex of (0, k)-forms on a complex manifold as an obvious analogue of the de Rham complex of k-forms on a smooth manifold. This poses an immediate question: “What is the analogue of the chain complex relevant to the context of complex manifolds?”, which we address in detail below.

It should be mentioned that the correspondence between de Rham and Dolbeault complexes, or d ↔ ¯∂, has the following natural extension.

d ↔ ¯∂
de Rham complex ↔ Dolbeault complex
smooth functions or sections ↔ smooth functions or sections
flat bundles ↔ holomorphic bundles
locally constant functions or sections ↔ local holomorphic functions or sections
cohomology of locally constant sheaves ↔ cohomology of sheaves of O_{X}-modules

(Here O_{X} denotes the sheaf of holomorphic functions on a complex (algebraic) manifold X.) Very informally, this table could be summarized in one line with

“Topology” versus “Complex Algebraic Geometry”.

Our interest in this line of thinking is related to the ideas of Arnold on complex analytic analogues of differential geometric concepts (cf., [A]). Some features of the above correspondence can also be found in the papers [FK, DT, KR]. In particular, the approach of Donaldson and Thomas [DT] of transferring differential geometric constructions into the context of complex analytic (or algebraic) geometry could lead one to a complexification of geometry in a sense similar to the complexification of topology pursued here.

∗Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada; e-mail:

khesin@math.toronto.edu

†Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117259 Moscow, Russia; e-mail: rosly@heron.itep.ru

There are also several motivations from mathematical physics: in particular, from considering any topological field theory of type B [ASL, LNS] and of BV type [AKSZ] or, e.g., a complex analogue of the Chern–Simons gauge theory suggested in Ref. [W]. The latter context leads us immediately to a search for a proper holomorphic analogue of the linking number (cf., also [Ger, FT]). 1.1 Holomorphic orientation. Let X be a compact complex manifold and u be a smooth (0, k)-form on it, 0 ⩽ k ⩽ n = dim X. We would like to treat such (0, k)-forms in the same manner as ordinary k-forms on a smooth manifold, but in the framework of complex geometry. In particular, we have to be able to integrate them over k-dimensional complex submanifolds in X. Recall that in the theory of differential forms, a form can be integrated over a real submanifold provided that the submanifold is endowed with an orientation. Thus, we need to find a holomorphic analogue of the orientation.

Obviously, if a k-dimensional submanifold W ⊂ X is equipped with a holomorphic k-form ω, one can consider the following integral

∫_{W} ω ∧ u

of the product of the (k, 0)- and (0, k)-forms. Therefore we are going to regard a top degree holomorphic form on a complex manifold as an analogue of orientation. More generally, if the form ω is allowed to have first order poles on a smooth hypersurface in W, the above integral is still well-defined.

1.2 The Cauchy–Stokes formula. The new feature brought by the presence of poles of ω shows up in the following relation. Consider the integral (1.1) with a meromorphic k-form ω having first order poles on a smooth hypersurface V ⊂ W. Let the smooth (0, k)-form u on X be ¯∂-exact, that is u = ¯∂v for some (0, k − 1)-form v on X. Then

∫_{W} ω ∧ ¯∂v = 2πi ∫_{V} res ω ∧ v .

We shall exploit this straightforward generalization of the Cauchy formula as a complexified analogue of the Stokes theorem. Here res ω denotes a (k − 1)-form on V which is the Poincaré residue of ω (see Sect. 2.1).

1.3 Boundary operator. The formula (1.2) prompts us to consider the pair (W, ω) consisting of a k-dimensional submanifold W equipped with a meromorphic form ω (with first order poles on V) as an analogue of a compact oriented submanifold with boundary. In the present paper we construct a homology theory in which the pairs (W, ω) will play the role of chains, while the boundary operator will take the form ∂ (W, ω) = 2πi(V, res ω). Note, that in the situation under consideration, when the polar set V of the form ω is a smooth (k − 1)-dimensional submanifold in a smooth k-dimensional W, the induced “orientaion” on V is given by a regular (k −1)-form res ω. This means that ∂ (V, res ω) = 0, or the boundary of a boundary is zero. The latter will be the source of the identity ∂^{2} = 0 in the homology theory discussed below. We shall call it the polar homology.

1.4 Pairing to smooth forms. It is clear that the (would-be) polar homology groups of a complex manifold X should have a pairing to Dolbeault cohomology groups H0,k ; ¯∂(X). Indeed, for a polar k-chain (W, ω) and any (0, k)-form u such a pairing is given by the integral

⟨(W, ω) , u⟩ = ∫_{W} ω ∧ u .

In other words, the polar chain (W, ω) defines a current on X of degree (n, n − k), where n = dim X. This pairing descends to (co)homology classes by virtue of the Cauchy–Stokes formula (1.2), see Sect. 4.

1.5 Example. Now we are already able to find out what could be the polar homology groups HP_{k} of a complex projective curve X. In this (and in any) case, all the 0chains are cycles. Let (P, a) and (Q, b) be two 0-cycles, where P, Q are points on X and a, b ∈ ℂ. They are polar homologically equivalent iff a = b. Indeed, a = b is necessary and sufficient for the existence of a meromorphic 1-form α on X, such that div_{∞}α = P + Q and res_{P} α = 2πi a, res_{Q} α = −2πi b. (The sum of all residues of a meromorphic differential on a projective curve is zero by the Cauchy theorem.) Then, we can write in terms of polar chain complex (to be defined in detail in Sect. 3) that (P, a) − (Q, a) = ∂ (X, α). Thus, HP_{0}(X) = ℂ.

As to polar 1-cycles, these correspond to all possible holomorphic 1-forms on X. On the other hand, there are no 1-boundaries, since there are no polar 2-chains in X. Hence HP_{1}(X) ≅ ℂ^{g}, where g is the genus of the curve X. (In particular, the polar Euler characteristics of X equals 1−g and coincides with its holomorphic Euler characteristics.)

Similar considerations show that for any n-dimensional X we have HP_{n}(X) = H^{0}(X, Ω_{X}^{n}) and, if X is connected, also HP_{0}(X) = ℂ.

1.6 Polar intersections. One can define a complex (polar) analogue of the intersection number in topology. For instance, let (X, μ) be a complex manifold equipped with a meromorphic volume form μ without zeros (its “polar orientation”). Consider two polar cycles (A, α) and (B, β) of complimentary dimensions that intersect transversely in X (here α and β are volume forms, or “polar orientations,” on the corresponding submanifolds). Then the polar intersection number is defined by the formula

⟨(A, α) · (B, β)⟩ = ∑_{P ∈A∩B} \frac{α(P) ∧ β(P)}{μ(P)} .

(For explanations, see (5.9).) At every intersection point P, the ratio in the right-hand-side is the “comparison” of the orientations of the polar cycles at that point with the orientation of the ambient manifold. This is a straightforward analogue of the use of mutual orientation of cycles in the definition of the topological intersection number. Note, that in the polar case the intersection number does not have to be an integer. (Rather, it is a holomorphic function of the “parameters” (A, α), (B, β) and (X, μ).)

Similarly, there is a polar analogue of the intersection product of cycles when they intersect over a manifold of positive dimension (see Sect. 5).

1.7 Polar links. By developing this analogy further we come to a polar analogue of the linking number. For instance, in complex dimension three we start with two smooth polar 1-cycles (C_{1}, α_{1}) and (C_{2}, α_{2}), i.e. C_{1} and C_{2} are smooth complex curves equipped with holomorphic 1-forms in a three dimensional X. Let us take the 1-cycles which are polar boundaries. This means, in particular, that there exists such a 2-chain (S_{2}, β_{2}) that (C_{2}, α_{2}) = ∂ (S_{2}, β_{2}). Suppose, the curves C_{1} and C_{2} have no common points and S_{2} is a smooth surface which intersects transversely with the curve C_{1}. Then, in analogy with the topological linking number of two curves in a three-fold, we define the polar linking number of the 1-cycles above as the polar intersection number of the 2-chain (S_{2}, β_{2}) with the 1-cycle (C_{1}, α_{1}):

ℓk_{polar} ((C_{1}, α_{1}), (C_{2}, α_{2})) := ∑_{P ∈C_{1}∩ S_{2}} \frac{α_{1}(P) ∧ β_{2}(P)}{μ(P)} .

One can show that the expression above does not depend on the choice of (S_{2}, β_{2}), and has certain invariance properties mimicking those of the topological linking number in “polar” language. We are going to discuss the properties of ℓk_{polar} in more detail in a future publication.

1.8 Remark. Most of the above discussion extends to polar chains (A, α) where the meromorphic p-form α is not necessarily of top degree, that is 0 ⩽ p ⩽ k, where k = dim_{ℂ} A. To define the boundary operator we have to restrict ourselves to the meromorphic forms with logarithmic singularities. The corresponding polar homology groups enumerated by two indices k and p (0 ⩽ p ⩽ k) will be discussed elsewhere (see, though, some remarks in Sect. 4.b below).

2 Preliminaries

a) Polar divisors and residues

The Poincaré residue is a higher-dimensional generalization of the classical Cauchy residue, where the residue at a point in a domain of one complex variable is generalized to the residue at a hypersurface.

2.1 Let M be an n-dimensional complex manifold and ω be a meromorphic n-form on M which is allowed to have first order poles on a smooth hypersurface V. Then, the form ω can be locally expressed as

ω = \frac{ρ ∧ dz}{z} + ε ,

where z = 0 is a local equation of V and ρ (respectively, ε) is a holomorphic (n − 1)- form (resp., n-form). Then the restriction ρ|_{V} is an unambiguously defined holomorphic (n − 1)-form on V.

Definition 2.2 The Poincaré residue of the n-form ω in (2.1) is the following (n − 1)- form on V

res ω := ρ|_{V} .

2.3 It is straightforward to extend this to the case of normal crossing divisors. Suppose that the meromorphic n-form ω has the first order poles on a normal crossing divisor V = ∪_{i}V_{i} in M. [Normal crossing divisor means that V has only smooth components V_{i} (each entering with multiplicity one) that intersect generically.] Analogously to the Definition 2.2 one can define a residue at each component V_{i}. The resulting (n−1)-forms res_{V_{j}}ω are then meromorphic and have first order poles at the pairwise intersections V_{ij} = V_{i} ∩ V_{j} One can now consider the repeated Poincaré residue at V_{ij}. Representing ω as ω = ϱ ∧ \frac{dz_{i}}{z_{i}} ∧ \frac{dz_{j}}{z_{j}}, where z_{i} = 0 and z_{j} = 0 are local equations of the components V_{i} and V_{j} respectively one finds that

res_{i,j}ω := res_{V_{ij}} (res_{V_{j}}ω) = res_{z_{i}=0} (res_{z_{j}=0} ϱ ∧ \frac{dz_{i}}{z_{i}} ∧ \frac{dz_{j}}{z_{j}} ) = ϱ|_{V_{ij}}.

Note that

res_{i,j}ω = −res_{j,i}ω .

Notation. Let us denote by res ω the collection of (k −1)-forms res_{V_{j}}ω, the residues of ω at the components of the normal crossing divisor div_{∞}ω = ∪_{i}V_{i}.

b) The push-forward map (see [Gr])

For a finite covering f: X → Y and a function φ on X one can define its push-forward, or the trace, f_{∗} φ as a function on Y whose value at a point is calculated by summing over the preimages taken with multiplicities. The operation f_{∗} can be generalized to p-forms and to the maps f which are only generically finite.

2.4 Suppose that f: X → Y is a proper, surjective holomorphic mapping where both X and Y are smooth complex manifolds of the same dimension n. The push-forward map is a mapping

f_{∗} : Γ(X, Ω_{X}^{p}) → Γ(Y, Ω_{Y}^{p}) .

The push-forward map is also defined for meromorphic forms, f_{∗} : Γ(X, M^{p}) → Γ(Y, M^{p}).

Its construction is as follows. First note that f is generically finite, i.e., there is an analytic hypersurface D ⊂ Y such that f is finite unramified covering away from this hypersurface D. Hence, for sufficiently small open neighborhood U of any point in Y^{∗} := Y ∖ D, the inverse image f^{−1}(U) = U_{1} ⊔ … ⊔ U_{d} is a disjoint union of d open sets U_{j}, such that f|_{U_{j}} is an isomorphism with the inverse s_{j} : U → U_{j}. Given a form ω on X, one defines its push-forward

f_{∗} ω := s_{1}^{∗} ω + ... + s_{d}^{∗} ω

in U, and therefore, in Y^{∗}. One can check that the form f_{∗} ω extends across the smooth points of D and, hence, to the whole of the manifold Y, since the remaining part of D has codimension greater than one. The resulting form f_{∗} ω is holomorphic (resp. meromorphic) on Y provided the form ω was holomorphic (resp. meromorphic) on X.

The operations of push-forward and residue are related in the following way.

Proposition 2.5 Let π : X → Y be a proper, surjective holomorphic map of complex manifolds of the same dimension n. Let ω be a meromorphic form with only first order poles on a smooth hypersurface V in X. Suppose V_{o} := π(V) is a smooth hypersurface in Y. Then π_{∗}ω has first order poles on V_{o} and

res π_{∗}ω = π̄_{∗} res ω ,

where π̄: V → V_{o} is the restriction of π.

3 Polar Homology of Projective Varieties

Here we define a homological complex based on the notion of the polar boundary. The construction is analogous to the definition of homology of a topological space with replacement of continuous maps by complex analytic ones. The notion of the boundary (of a simplex or a cell) is replaced by the Poincaré residue of a meromorphic differential form. There are however important distinctions. First, we shall only have an analogue of the non-torsion part of homology. Second, unlike the topological homology, where in each dimension k one uses all continuous maps of one standard object (the standard k-simplex or the standard k-cell) to a given topological space, in polar homology we deal with complex analytic maps of a large class of k-dimensional varieties to a given one. 3.1 Polar chains. In this section we deal with complex projective varieties, i.e., subvarieties of a complex projective space. (In this setting the complex analytic considerations are equivalent to algebraic ones.) By a smooth projective variety we always understand a smooth and connected one. For a smooth variety M, we denote by Ω_{M}^{p} the sheaf of regular p-forms on M. The sheaf Ω_{M}^{dim M} of forms of the top degree on M will sometimes be denoted by K_{M}.

The space of polar k-chains for a complex projective variety X, dim X = n, will be defined as a ℂ -vector space with certain generators and relations.

Definition 3.2 The space of polar k-chains C_{k}(X) is a vector space over ℂ defined as the quotient C_{k}(X) = Ĉ_{k}(X)/R_{k}, where the vector space Ĉ_{k}(X) is freely generated by the triples (A, f, α) described in (i),(ii),(iii) and R_{k} is defined as relations (R1),(R2),(R3) imposed on the triples.

(i) A is a smooth complex projective variety, dim A = k;

(ii) f: A → X is a holomorphic map of projective varieties;

(iii) α is a rational k-form on A with first order poles on V ⊂ A, where V is a normal crossing divisor in A, i.e., α ∈ Γ(A, Ω_{A}^{k}(V)). The relations are:

(R1) λ(A, f, α) = (A, f, λα)

(R2) ∑_{i}(A_{i}, f_{i}, α_{i}) = 0 provided that ∑_{i} f_{i∗}α_{i} ≡ 0, where dim f_{i}(A_{i}) = k for all i and the push-forwards f_{i∗}α_{i} are considered on the smooth part of ∪_{i}f_{i}(A_{i});

(R3) (A, f, α) = 0 if dim f(A) < k.

Remarks to the definition

3.3 By definition, C_{k}(X) = 0 for k < 0 and k > dim X.

3.4 In what follows we sometimes will make no difference between a triple (A, f, α) and the equivalence class defined by it in C_{k}(X). An arbitrary polar chain can thus be written as a sum of triples of the form ∑_{i}(A_{i}, f_{i}, α_{i}). A chain equivalent to a single triple will be called prime. For a chain a = ∑_{i}(A_{i}, f_{i}, α_{i}), let us call the subvariety ∪_{i}f_{i}(A_{i}) in X the support of a. If the support of a chain is a smooth subvariety in X, such a chain will be called smooth. One can show that smooth chains are prime, since we suppose that “smooth” implies “connected” (see 3.1).

3.5 The relation (R2) allows us, in particular, to refer to prime polar chains as pairs replacing a triple (A, f, α) by a pair (Â, α̂), where Â = f(A) ⊂ X, α̂ is defined only on the smooth part of Â and α̂ = f_{∗}α there. Due to the relation (R2), such a pair (Â, α̂) carries precisely the same information as (A, f, α).^{1} (The only point to worry about is that such pairs cannot be arbitrary. In fact, by the Hironaka theorem on resolution of singularities, any subvariety Â ⊂ X can be the image of some regular A, but the form α̂ on the smooth part of Â cannot be arbitrary.)

3.6 The relation (R2) also represents additivity with respect to α, that is

(A, f, α_{1}) + (A, f, α_{2}) = (A, f, α_{1} + α_{2}).

Formally speaking, the right hand side makes sense only if α_{1} + α_{2} is an admissible form on A, that is if its polar divisor div_{∞}(α_{1} + α_{2}) has normal crossings. However, one can always replace A with a variety Ã, obtained from A by a blow-up, π: Ã → A, in such a way that π^{∗}(α_{1} + α_{2}) is admissible on Ã, i.e., div_{∞}(α_{1} + α_{2}) is already a normal crossing divisor. (This is again the Hironaka theorem.) The (R2) says that (A, f, α_{1}) + (A, f, α_{2}) = (Ã, f ∘ π, π^{∗}(α_{1} + α_{2})).

Definition 3.7 The boundary operator ∂ : C_{k}(X) → C_{k−1}(X) is defined by

∂(A, f, α) = 2πi ∑_{i}(V_{i}, f_{i}, res_{V_{i}} α)

(and by linearity), where V_{i} are the components of the polar divisor of α, div_{∞}α = ∪_{i}V_{i}, and the maps f_{i} = f|_{V_{i}} are restrictions of the map f to each component of the divisor.

Theorem 3.8 The boundary operator ∂ is well defined, i.e. it is compatible with the relations (R1),(R2),(R3).

Proof. We have to show that ∂ respects the relations (R1),(R2),(R3), in other words, ∂ maps equivalent sums of triples to equivalent ones. It is trivial with (R1). To check (R2), let us recall Proposition 2.5. Consider a sum of triples ∑_{i}(A_{i}, f_{i}, α_{i}) belonging to (R2), that is dim A_{i} = dim f_{i}(A_{i}) = k, ∀i, and ∑_{i} f_{i∗}α_{i} = 0 on the smooth part of ∪_{i}f_{i}(A_{i}). Since the irreducible components of ∪_{i}f_{i}(A_{i}) can be treated separately it is natural to consider only the case when all the triples have the same support, f_{i}(A_{i}) = Â ⊂ X, ∀i. Let V_{i} ⊂ A_{i} be the divisor of poles of α_{i} and let V̂ := ∪_{i}f_{i}(V_{i}) ⊂ Â. We want to prove that

1Note, that the consideration of triples (A, f, α) instead of pairs (Â, α̂) which we used in Sect. 1 is similar to the definition of chains in the singular homology theory: in the latter case, although one considers the mappings of abstract simplices into the manifold, but morally it is only “images of simplices” that matter.

∑_{i} f̄_{i∗}res α_{i} = 0

on the smooth part of V̂, where f̄_{i∗} : V_{i} → V̂ for each i is the restriction of the map f_{i}.

Suppose first that there exists a smooth point of V̂ which is smooth also in Â. Then the Proposition 2.5 applied in a neighborhood of that point gives us the desired vanishing ∑_{i} f̄_{i∗}res α_{i} = 0, as a consequence of the equality ∑_{i} f_{i∗}α_{i} = 0. This is however not enough for our proof since some components of V̂ may lie entirely in the set of singular points of Â. To overcome this problem we apply the Hironaka theorem replacing Â with a smooth variety Ã, a blow-up of Â, and correspondingly blowing up all A_{i}, so that the following diagram is commutative:

A_{i} ⟵^{π_{i}}  Ã_{i}
f_{i} ↓ f̃_{i} ↓
Â ⟵^{π}  Ã

Then we apply Proposition 2.5 on the blown up side.

We must recall now that the divisor V_{i} = div_{∞}α_{i} could have components that were mapped by f_{i} to subvarieties of dimension less than k−1; hence, we conclude that we have just proved the following statement (symbolically): if a ∈ (R2) then ∂a ∈ (R2)+(R3).

Now, it remains to prove the compatibility of ∂ with (R3). Let a = (A, f, α) be a degenerate triple described in (R3), i.e., dim f(A) < k = dim A. We shall show that ∂a ∈ (R2)+(R3) in this case. The polar divisor V = div_{∞}α, dim V = k − 1, is, by assumptions of Definition 3.2, a normal crossing divisor in A. Let us split the components of V into two parts: non-degenerate and degenerate ones. That is V = N ∪D where dim f(N) = k − 1 and dim f(D) < k − 1. According to this splitting, ∂a is represented as a sum of two terms corresponding to res_{N}α and res_{D}α. The second term belongs to (R3) and we have to show only that the first one belongs to (R2), i.e., that f̄_{∗}res_{N}α = 0, where f̄ = f|_{N}. Recall that we suppose that dim f(A) < k. If it happens that dim f(A) < k − 1, we have N = ∅ and there is nothing to prove. Therefore we may assume that dim f(A) = k − 1 and, by irreducibility of A, f(A) = f(N). Then, for a generic smooth point Q ∈ f(A), its preimage in A, C := f^{−1}(Q) ⊂ A, is a smooth projective curve. This curve intersects with N over the set f̄^{−1}(Q) and we may suppose that the latter consists of a finite number of points P_{i} which are smooth in N and that the intersections are transverse there.

[figure: A; f(A); Q; N; C; P_{1}; P_{2}; D]

Let β(P_{i}) denote the values of res_{N}α at the points P_{i} ∈ N ∩ C and pick up a nonvanishing (k − 1)-form β_{o} at Q (recall that Q = f(P_{i}), ∀i). Let us show that

∑_{i} \frac{β(P_{i})}{f^{∗}β_{o}(P_{i})} = 0

(this would mean that f̄_{∗}res_{N}α = 0 on the smooth part of f̄(N) = f(A) — the required result). To prove this, let us notice that there exists a meromorphic 1-differential ω on C such that

ω(P) ⊗ f^{∗}β_{o}(P) = α(P), P ∈ C .

(ω is obtained by dividing α by the non-vanishing form f^{∗}β_{o}.) This equality is understood in the sense of the natural isomorphism

K_{C} ⊗ (f^{∗}K_{f(A)^{∗}})|_{C} = K_{A}|_{C} ,

where f(A)^{∗} is the smooth part of f(A). It is easy to see now that for β(P_{i}) = res_{N}α(P_{i}) we have

∑_{i} \frac{β(P_{i})}{f^{∗}β_{o}(P_{i})} = ∑_{i} res_{P_{i}}ω = 0 .

The latter equality follows from the observation that P_{i} are the only points on C where ω has poles. Indeed, the poles of ω are located on div_{∞}α ∩ C = (N ∩ C) ∪ (D ∩ C). One part of this gives us the points P_{i}, {P_{i}} = N ∩C, while the rest, D ∩C, corresponding to the “degenerate” part D of div_{∞}α can be assumed to be empty, D ∩ C = ∅. Indeed, we could have assumed from the very beginning that C = f^{−1}(Q) does not meet D because dim f(D) < k − 1 and we might suppose that Q ∉ f(D). ■

Theorem 3.9 ∂^{2} = 0 .

Proof. We need to prove this for triples (A, f, α) ∈ C_{k}(X), i.e., for forms α with normal crossing divisors of poles. The repeated residue at pairwise intersections differs by a sign according to the order in which the residues are taken, see 2.3. Thus the contributions to the repeated residue from different components cancel out (or, the residue of a residue is zero).^{2} ■

Definition 3.10 For a smooth complex projective variety X, dim X = n, the chain complex

0 → C_{n}(X) ⟶^{∂} C_{n−1}(X) ⟶^{∂} ⋯ ⟶^{∂} C_{0}(X) → 0

is called the polar chain complex of X. Its homology groups, HP_{k}(X), k = 0, …, n, are called the polar homology groups of X.

2One can note that an example of the polar divisor {xy = 0} for the form dx∧dy/xy in ℂ^{2} should be viewed as a complexification of a polygon vertex in ℝ^{2}. Indeed, the cancellation of the repeated residues on different components of the divisor is mimicking the calculation of the boundary of a boundary of a polygon: every polygon vertex appears twice with different signs as a boundary point of two sides.

3.11 Example. For a projective curve of genus g the polar homology groups are as follows: HP_{0} = ℂ , HP_{1} = ℂ^{g}, and HP_{k} = 0 for k ≥ 2. One can readily see that the approach with triples coincides with the consideration of Introduction.

3.12 Remark. The functorial properties of polar homology are straightforward. A regular morphism of projective varieties h: X → Y defines a homomorphism h_{∗}: P H_{k}(X) → P H_{k}(Y).

3.13 Remark. The definitions of polar chains can be generalized to the case of pforms on k-manifolds, i.e., for the forms of not necessarily top degree, p ≤ k. Instead of meromorphic k-forms with poles of the first order we have to restrict ourselves by p-forms with logarithmic singularities. The definition of the boundary operator ∂, the property ∂^{2} = 0, and the definition of the polar homology groups can be carried over to this, more general, situation. The polar homology groups are then enumerated by two indices: HP_{k,p}(M). The definition above corresponds to the p = k case. We will discuss the more general polar homology groups elsewhere [iKR].

3.14 Relative polar homology. Let Z be a projective subvariety in a projective X. Analogously to the topological relative homology we can define the polar relative homology of the pair Z ⊂ X.

Definition 3.15 The relative polar homology HP_{k}(X, Z) is the homology of the following quotient complex of chains:

C_{k}(X, Z) = C_{k}(X)/C_{k}(Z).

Here we use the natural embedding of the chain groups C_{k}(Z) ↪ C_{k}(X). This leads to the long exact sequence in polar homology:

… → HP_{k}(X) → HP_{k}(X, Z) ⟶^{∂} HP_{k−1}(Z) → HP_{k−1}(X) → …

3.16 Systems of coefficients. One can introduce the notion of a homological system of coefficients appropriate for the polar complex. The most geometrical example would be, perhaps, to supply projective varieties A, f : A → X, with coherent sheaves F_{A,f} obeying certain relations between F_{A_{1},f_{1}} and F_{A_{2},f_{2}} when f_{1}(A_{1}) = f_{2}(A_{2}) and related by some homomorphisms playing the role of the residue. We do not study this in the present paper, but let us mention that the homology groups appearing in Sects. 3.13 above and 4.7 below can be viewed as an example. On the other hand, the simplest case of a polar homological system of coefficients corresponds to F_{A,f} = f^{∗}F ⊗ K_{A}, where F is a sheaf on X and α in the triple (A, f, α) is understood as a global section of f^{∗}F ⊗ K_{A}(V). Let us denote the corresponding homology as HP_{k}(X, F). This case is mentioned in Sects. 4.4, 4.5.

4 Polar Chains and Differential Forms

a) Dolbeault cohomology as polar de Rham cohomology As we discussed in the Introduction, the Dolbeault complex of (0, k)-forms should be related to the polar homology in the same way as the de Rham complex of smooth forms is related to the topological homology (e.g., singular homology). Now, after the definitions of Sect. 3 are given, we are able to make this point more explicit.

4.1 In a smooth projective variety X, consider a polar k-chain, for instance, a prime one, i.e. (an equivalence class of) a triple a = (A, f, α). Such a triple can be regarded as a linear functional on the space of smooth (0, k)-forms on X. Let u be a smooth (0, k)-form on X, then the pairing is given by the following integral:

⟨a , u⟩ := ∫_{A} α ∧ f^{∗}u .

The integral is well defined since α has only first order poles on a normal crossing divisor. It is now straightforward to show that the pairing ⟨ , ⟩ descends to the space of equivalence classes of triples C_{k}(X), i.e., that it is compatible with the relations (R1), (R2), (R3) of Definition 3.2. Indeed, (R1) is obvious, compatibility with (R3) follows from noticing that f^{∗}u = 0 if dim f(A) < k, and the compatibility with (R2) follows from the relation ∫_{A} α ∧ f^{∗}u = ∫_{f(A)} f_{∗}α ∧ u if dim f(A) = k, where the last integral is taken over the smooth part of f(A).

4.2 Remark. Let us notice that the last considerations can be used^{3} as an alternative definition of the polar chain complex on a smooth projective variety X (or any smooth compact complex manifold). The pairing above can be thought of as a map φ̂ : Ĉ_{k}(X) → D^{n,n−k}(X), where Ĉ_{k}(X) is the vector space freely generated by the triples (A, f, α) (see Definition 3.2) and D^{n,n−k}(X) is the space of currents of degree (n, n − k) on X which is defined as a space of certain linear functionals on the smooth (0, k)-forms (see [GH]). Then the relations (R1),(R2),(R3) in the Definition 3.2 correspond to the kernel of the map φ̂. In other words, the space of polar chains C_{k}(X) can be defined as a subspace of currents — the image of φ̂. We have thus an embedding

φ : C_{k}(X) ↪ D^{n,n−k}(X) .

Moreover, the Cauchy–Stokes formula (1.2) shows that C_{k}(X) is a subcomplex of the ¯∂-complex of currents D^{n,n−k}(X), i.e. for a ∈ C_{k}(X) we have φ(∂a) = ¯∂φ(a). (This is in fact shown also in the proof of 4.3 below.)

Proposition 4.3 The pairing (4.1) defines the following homomorphism in (co)homology:

ρ: HP_{k}(X) → Hn,n−k ; ¯∂(X) ,

where n = dim X.

Proof. By the Serre duality, ρ is the map HP_{k}(X) → (H0,k ; ¯∂(X))^{∗} and it is sufficient to verify that the pairing (4.1) vanishes if ∂ a = 0 and u = ¯∂v, or if ¯∂u = 0 and a = ∂ b. This follows immediately from the Cauchy–Stokes formula (1.2):

∫_{A} α ∧ f^{∗}(¯∂u) = 2πi ∫_{div_{∞}α}(res α) ∧ f^{∗}(u) ,

3We thank A. Levin for emphasizing this point.

that is ⟨a , ¯∂u⟩ = ⟨∂a , u⟩. ■

A number of examples suggests that, for projective manifolds, the homomorphism (4.3) should be in fact an isomorphism.

Conjecture 4.4 (Polar de Rham theorem) For a smooth projective manifold X the mapping ρ : HP_{k}(X) → Hn,n−k ; ¯∂(X) is an isomorphism of the polar homology and Dolbeault cohomology groups. Equivalently, in terms of dual cohomology groups,

HP^{k}(X) ≅ H0,k ; ¯∂(X) .

An analogous conjecture that HP_{k}(X, F) ≅ H^{n−k}(X, K_{X} ⊗ F) sounds reasonable also for polar (co)homology with coefficients in locally free sheaves on X (see Remark 3.16). 4.5 Example. If X is a complex curve of genus g one has HP_{0}(X) ≅ ℂ ≅ H1,1 ; ¯∂(X) and HP_{1}(X) ≅ ℂ^{g} ≅ H1,0 ; ¯∂(X) (see Example 1.5). Considering also HP_{k}(X, F) in case of F = T_{X} (the tangent sheaf of X) one can check that HP_{0}(X, T_{X}) ≅ H0,1 ; ¯∂(X) and

HP_{1}(X, T_{X}) ≅ H0,0 ; ¯∂(X).

4.6 Remark. Consider the polar Euler characteristic,

χ_{pol}(X) := ∑_{k=0}^{n}(−1)^{k} dim HP_{k}(X) ,

of an n-dimensional variety X. Then, if the conjecture (4.4) is true, for a smooth projective X one obtains the equality χ_{pol}(X) = χ_{hol}(X) of the polar and holomorphic Euler characteristics, where

χ_{hol}(X) = ∑_{k=0}^{n}(−1)^{k} dim H0,k ; ¯∂(X) .

b) Forms of any degree

4.7 So far we considered polar chains with complex volume forms. More generally, one could consider polar (k, p)-chains (A, f, α), where α is a meromorphic p-form of not necessarily maximal degree, p ⩽ k, on A that can have only logarithmic singularities on a normal crossing divisor.^{4} The requirement of log-singularities is needed to have a convenient definition of the residue and, hence, the boundary operator.

The Cauchy–Stokes formula (1.2) extends to this case as well. As a consequence, the natural pairing between polar (k, p)-chains and smooth (k − p, k)-forms on X gives us as before the homomorphism (cf. (4.3))

ρ: HP_{k,p}(X) → Hn−k+p,n−k ; ¯∂(X) .

However, unlike the case p = k, the map ρ is not, in general, an isomorphism for other values of p, 0 ⩽ p < k. For instance, at least in the case of p = 0, this is easy to see for the following reason.

4An important property of such forms on projective varieties is that they are closed, see [De].

4.8 Polar chains with p = 0. In this case the triples (A, f, α) involve 0-forms (i.e., just functions) α on projective varieties A. Then the requirement of log-singularities amounts here to saying that these functions are holomorphic on A. Since A is compact, these functions must be constant. In particular, we conclude that all polar (k, 0)-chains are polar cycles.

Thus, the space of polar (k, 0)-cycles in a projective manifold X is the same as the vector space generated over ℂ by all k-dimensional algebraic cycles in X. (Note that the replacement of the triples (A, f, α) by the pairs (A, α) with A ⊂ X is especially convenient when α’s are 0-forms.) In this case one can show that the homomorphism ρ maps HP_{k,0}(X) to the algebraic part of H^{r,r}(X), where r = n − k, or more precisely, to

H_{alg}^{r,r}(X, ℂ) := (Hr,r ; ¯∂(X) ∩ H^{2r}(X, ℚ)) ⊗ ℂ ⊂ Hr,r ; ¯∂(X) .

This allows us to conclude that ρ is not surjective, in general. Indeed, there are examples where H_{alg}^{r,r}(X, ℂ) is strictly smaller than Hr,r ; ¯∂(X). For instance, for a generic algebraic K3 surface one has that dim H^{1,1}(X) = 20, while dim H_{alg}^{1,1}(X, ℂ) = 1, see [Tju]. We also note that by the Hodge conjecture, the image of ρ coincides with H_{alg}^{r,r}(X, ℂ).

4.9 Remark. It would be, certainly, very interesting to describe the polar homology groups HP_{k,p}(X) for all values of p. In particular, it is not clear whether the groups HP_{k,p}(X) are finite-dimensional.^{5}

5 Intersection in Polar Homology

We define here a polar analogue of the topological intersection product. In particular, for polar cycles of complimentary dimensions one obtains a complex number, called the polar intersection number.

Recall that in topology, one considers a smooth oriented closed manifold M and two oriented closed submanifolds A, B ⊂ M of complementary dimensions, i.e., dim A + dim B = dim M. Suppose, A and B intersect transversely at a finite set of points. Then to each intersection point P one assigns ±1 (local intersection index) by comparing the mutual orientations of the tangent vector spaces T_{P} A, T_{P} B, and T_{P} M.

5.1 Polar oriented manifolds. Let now M be a compact complex manifold of dimension n, on which we would like to define a polar intersection theory. It has to be polar oriented, i.e., equipped with a complex volume form. As the discussion below shows, the n-form μ defining its polar orientation has to have no zeros on M, since we are going to consider expressions in which μ, the orientation of the ambient manifold, enters a denominator. Therefore we adopt the following terminology.

Definition 5.2 i) A compact complex manifold M, endowed with a nowhere vanishing holomorphic volume form μ, is said to be a polar oriented closed manifold. ii) If the volume form μ on a compact complex manifold M is nonvanishing and meromorphic with only first order poles on a normal crossing divisor N ⊂ M, then M is called a polar oriented manifold with boundary. The hypersurface N is then endowed with a polar orientation ν := res μ ≠ 0 and (N, ν) is called the polar boundary of (M, μ).

5It should be mentioned, that there is a map of the complex of polar chains to the complex considered in [BO]. The corresponding (co)homology groups are, however, quite different, as already the simplest example of a complex curve shows. We are grateful to S. Bloch for illuminating discussions on this relation.

5.3 Remark. By definition, polar oriented closed manifolds are complex manifolds whose canonical bundle is trivial (Calabi–Yau, Abelian manifolds or, for example, any complex tori, if we do not restrict ourselves to algebraic manifolds). We have just defined the notion of the polar orientation in a more restrictive sense than before, when we considered the definition of chains. In fact, polar chains with their orientations are to be compared to oriented piece-wise smooth submanifolds in differential topology, while the ambient space on which we want to have Poincaré duality has to be everywhere smooth and oriented. Zeros of a volume form could be regarded as a complex analogue of singularities of a real manifold.^{6}

a) Polar intersection number

5.4 Let (M, μ) be a polar oriented closed manifold of dimension n. In such a case we define the following natural pairing between its polar homology groups HP_{p}(M) and HP_{n−p}(M) of complimentary dimension.

According to Proposition 4.3, the above groups can be mapped to the Dolbeault cohomology groups Hn,n−p ; ¯∂(M) and Hn,p ; ¯∂(M), respectively. On a polar oriented closed manifold we are given a nowhere vanishing section μ of the line bundle Ω_{M}^{n} . Hence,

we have the isomorphism Hn,n−p ; ¯∂(M) ⟶^{μ^{−1}} H0,n−p ; ¯∂(M). Using this and the product in Dolbeault cohomology we obtain the following pairing:

Hn,p ; ¯∂(M) ⊗ Hn,n−p ; ¯∂(M) ⟶^{id⊗μ^{−1}} Hn,p ; ¯∂(M) ⊗ H0,n−p ; ¯∂(M) → Hn,n ; ¯∂(M) →^{∼} ℂ .

Together with the homomorphism ρ: HP_{k}(X) → Hn,n−k ; ¯∂(X), this yields the pairing

⟨ · ⟩: HP_{p}(M) ⊗ HP_{n−p}(M) → ℂ .

Here, in fact, we interchanged the order of factors (see the explicit formula (5.5) below).

5.5 Consider two polar cycles, [a] ∈ HP_{p}(M) and [b] ∈ HP_{n−p}(M). Let t_{a} and t_{b} be the Dolbeault forms representing ρ([a]) and ρ([b]) respectively. Then, the above product, which we denoted by ⟨a · b⟩, can be written as

⟨a · b⟩ = ∫_{M} t_{b} ∧ \frac{t_{a}}{μ} .

Note that t_{a} is an (n, n−p)-form and thus, t_{a}/μ is a (0, n−p)-form that can be integrated against an (n, p)-form t_{b}.

6For instance, on a complex curve X of genus g one has HP_{1}(X) = ℂ^{g}, and a holomorphic 1differential representing a generic element in HP_{1}(X) has 2g − 2 zeros. From this point of view the complex genus g curve is like a graph which has g loops joined by g −1 edges and having 2g −2 trivalent (i.e., “non-smooth”) points. The “smooth cases” are ℂℙ^{1}, which corresponds to a real segment, and an elliptic curve, which is a complex counterpart of the circle in this precise sense.

Definition 5.6 The pairing ⟨a · b⟩ of polar cycles is called the polar intersection index.

5.7 Remark. If the conjecture (4.4) is true, this pairing is non-degenerate.

5.8 Let us consider now the case when the cycles a and b are smooth and transverse. That is a = (A, α) and b = (B, β), where A is a smooth p-dimensional variety and α a holomorphic p-form on it (and similarly for (B, β) in dimension n − p) and it is assumed that A and B intersect transversely. Then, we have the following formula for the polar intersection index.

Theorem 5.9 The polar intersection index of two smooth transverse cycles (A, α) and (B, β) is given by the following sum over the set of points in A ∩ B:

⟨(A, α) · (B, β)⟩ = ∑_{P ∈A∩B} \frac{α(P) ∧ β(P)}{μ(P)}

Here α(P) and β(P) are understood as exterior forms on T_{P} M = T_{P} A × T_{P} B obtained by the pull-back from the corresponding factors.

The ratio in the right-hand-side can be understood as the comparison of the polar orientations brought to the intersection point P by the two cycles with the polar orientation μ(P) of the ambient manifold at that point.

Proof. As we have already mentioned, the homomorphism ρ of Sect. 4.3 can be conveniently described in terms of the following natural map of polar chains:

φ : C_{k}(M) → D^{n,n−k}(M) ,

where D^{p,q}(M) is the space of currents of degree (p, q) (i.e., a space of linear functionals on smooth (n − p, n − q)-forms, see [GH]). As a matter of fact, this map is described by the integral (4.1). For a p-dimensional submanifold A ⊂ M, let the current δ_{A} ∈ D^{n−p,n−p}(M) denote the linear functional on (p, p)-forms corresponding to the integration over A. The current δ_{A} is supported on A. Therefore, for a p-form α defined on A, the product δ_{A} ∧ α makes sense and defines a current in D^{n,n−p}(M). Recalling the isomorphism of the cohomology of currents with the cohomology of smooth forms,

H^{j}(D^{i,•}(M), ¯∂) ⟶^{∼} Hi,j ; ¯∂(M) ,

we can use δ_{A} ∧ α and δ_{B} ∧ β in place of t_{a} and t_{b} in the integral (5.5). Thus, for a transverse intersection of smooth polar cycles we derive that^{7}

⟨(A, α) · (B, β)⟩ = ∫_{M} δ_{B} ∧ β ∧ \frac{δ_{A} ∧ α}{μ} = ∫_{M} (\frac{α ∧ β}{μ} ) · δ_{A} ∧ δ_{B} .

7Although the (exterior) product of currents is not in general defined, the product of the cohomology classes of ¯∂-closed currents is always well defined. In some cases it is easy to find a representative of the product of cohomology classes. For instance, for two submanifolds V and W in a generic position the product of cohomology classes of the corresponding currents δ_{V} and δ_{W} is represented by δ_{V ∩W}. In this sense we can write δ_{V} ∧ δ_{W} = δ_{V ∩W}.

The second equality can be checked in local coordinates. This proves the theorem, since δ_{A} ∧ δ_{B} is supported on A ∩ B. ■

b) Polar intersection product

5.10 Now consider the case when on a polar oriented closed manifold (M, μ) we have two polar cycles of arbitrary dimensions p and q (not necessarily complimentary ones). Similarly to the pairing (5.4), we may consider the following chain of homomorphisms:

HP_{p}(M) ⊗ HP_{q}(M) ⟶^{s_{12}∘(ρ⊗ρ)} Hn,n−q ; ¯∂(M) ⊗ Hn,n−p ; ¯∂(M)

⟶^{id⊗μ^{−1}} Hn,n−q ; ¯∂(M) ⊗ H0,n−p ; ¯∂(M) → Hn,2n−p−q ; ¯∂(M) ,

where s_{12} is the transposition of tensor factors and the last term is understood as zero unless p + q ⩾ n. Let Λ denote the resulting composition:

Λ: HP_{p}(M) ⊗ HP_{q}(M) → Hn,2n−p−q ; ¯∂(M) .

5.11 If the conjecture (4.4) holds, the above homomorphism and the inverse of ρ in (4.3) define an intersection product on polar homology,

HP_{p}(M) ⊗ HP_{q}(M) → HP_{p+q−n}(M) .

However, even without this hypothesis we will show that if a and b are two smooth transverse cycles, [a] ∈ HP_{p}(M), [b] ∈ HP_{q}(M), their polar intersection product can be represented in HP_{p+q−n}(M) by a smooth cycle c.

5.12 ℂ -orientations of vector spaces. Let W be an n-dimensional complex vector space, and μ be a non-zero complex volume form on W (μ ∈ ⋀^{n} W^{∗}). Let V_{A}, V_{B} ⊂ W be vector subspaces of dimensions dim V_{A} = p, dim V_{B} = q, p + q ⩾ n, which intersect transversely, i.e. V_{A} + V_{B} = W (or, r := dim V_{A} ∩ V_{B} = p + q − n). Suppose we are also given complex volume forms on each of V_{A} and V_{B}, that is α ∈ ⋀^{p} V_{A}^{∗} and β ∈ ⋀^{q} V_{B}^{∗}. (We may say that all three spaces W, V_{A}, and V_{B} are ℂ -oriented.) Then, one can define a complex volume form γ on the intersection V_{A} ∩ V_{B} (i.e., one can ℂ -orient it) as follows.

Let λ_{A} ∈ ⋀^{n−p} W^{∗} be a non-zero exterior form conormal to V_{A} (i.e., λ_{A} vanishes on any vector from V_{A} and is non-zero as an element of ⋀^{n−p}(W/V_{A})^{∗}). Similarly, let λ_{B} ∈ ⋀^{n−q} W^{∗} be a non-zero exterior form conormal to V_{B}. Note that in this case λ_{A} ∧ λ_{B} is a form conormal to V_{A} ∩ V_{B}.

Definition-Lemma 5.13 Given complex orientations (i.e., complex volume forms) α, β, and μ of the vector subspaces V_{A}, V_{B} and the space W respectively, the following complex orientation γ of the intersection V_{A} ∩ V_{B} is defined by the following relation:

λ_{A} ∧ λ_{B} ∧ γ = (\frac{λ_{A} ∧ α}{μ}) · (\frac{λ_{B} ∧ β}{μ} ) · μ .

Here α, β, and γ are understood as arbitrary extensions of these forms to the whole space W. The r-form γ on V_{A} ∩ V_{B} depends neither on these extensions, nor on the choice of the auxiliary forms λ_{A} and λ_{B}.

Proof. A straightforward verification. ■

Corollary 5.14 For a transverse intersection of subspaces of complimentary dimensions (p + q = n and W = V_{A} ⊕ V_{B}), the 0-form γ is just the following complex number:

γ = \frac{α ∧ β}{μ} ,

where α and β are now understood as exterior forms on W = V_{A} × V_{B} obtained by pull-backs from the corresponding factors.

5.15 Let a = (A, α) and b = (B, β) be two smooth polar cycles of dimension p and q respectively in a polar oriented closed manifold (M, μ), p + q ⩾ n = dim M. Suppose they intersect transversely. Then we can define a (p + q − n)-cycle c = (C, γ), where C = A ∩ B is a smooth subvariety in M and γ is a holomorphic (p + q − n)-form on it defined by (5.13). Let us denote this as [c] = [a] · [b] and call the intersection product. (Of course, the product [a] · [b] equals zero if p + q < n.)

Theorem 5.16 The polar intersection product [a] · [b] of two smooth transverse cycles a = (A, α) and b = (B, β) defined above agrees with the homomorphism (5.10), i.e.,

Λ([a] ⊗ [b]) = ρ([a] · [b]) ,

where ρ : HP_{k}(X) → Hn,n−k ; ¯∂(X) was defined in (4.1)-(4.3).

Proof. This will be similar to the proof of Theorem 5.9 and will use the same notations. We first represent the polar cycles a and b by the currents δ_{A} ∧α and δ_{B} ∧β respectively, then

Λ([a] ⊗ [b]) = [δ_{B} ∧ β ∧ \frac{δ_{A} ∧ α}{μ}] ,

where [ ] on the right is understood as taking the ¯∂-cohomology class. On the other hand, for c = (C, γ) introduced in (5.15), the current representing c is δ_{C} ∧ γ and it is easy to show that

δ_{C} ∧ γ = δ_{B} ∧ β ∧ \frac{δ_{A} ∧ α}{μ} ,

what implies the statement of the Theorem. The last equality is easily checked by noticing that δ_{A} is an (n − p, n − p)-form (in fact, a current) conormal to A and similarly for δ_{B}, while δ_{C} = δ_{A} ∧ δ_{B}. This is to be compared to λ_{A} and λ_{B} in (5.13). One has to note only that, e.g., δ_{A} is conormal to A over ℝ (that is in the sense of (n−p, n−p)-forms) while λ_{A} is conormal to it over ℂ (that is in the sense of (n − p, 0)-forms). ■

5.17 Remark. We have defined the polar intersection on any complex manifold M that can be equipped with a holomorphic non-vanishing volume form μ. This is analogous to the topological intersection theory on a compact smooth oriented manifold without boundary. (Note that the Poincaré duality in this context should correspond to the Serre duality.) Furthermore, the consideration above easily extends to the case of a complex manifold possessing a meromorphic non-vanishing form μ (in particular, to a complex projective space), i.e., to the case of a polar oriented manifold (M, μ) with boundary (N, res μ) (cf. 5.2). The latter setting is similar to the topological intersection theory on manifolds with boundary. In this case the above formulas can be used to define the pairing between polar homology HP_{k}(M) and polar homology relative to the boundary HP_{n−k}(M, N).

Acknowledgments. B.K. and A.R. are grateful for hospitality to the Max-Planck-Institut für Mathematik in Bonn where this work was conceived and later completed. B.K. is also grateful to the Institute for Advanced Study in Princeton, and A.R. is also grateful to the Erwin Schrödinger International Institute for Mathematical Physics in Vienna where a part of this work was done. The present work was partially sponsored by PREA and McLean awards.

A.R. is extremely grateful to A. Bondal and A. Levin for explaining some valuable tools of algebraic geometry and very instructive discussions. A number of valuable remarks by A. Levin saved us from many missteps in the course of this work.

The work of B.K. was partially supported by an Alfred P. Sloan Research Fellowship, by the NSF and NSERC research grants. The work of A.R. was supported in part by the Grants RFBR-98-01-00344, INTAS-97-0103 and the Grant 00-15-96557 for the support of scientific schools.

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