CANONICAL DEGREE OF CURVES ON A MINIMAL SURFACE

YOICHI MIYAOKA

In memory of Professor Kunihiko Kodaira and Professor Nobuo Sasakura

Abstract. Given a smooth, projective, minimal complex surface X and an irreducible curve C on it, we prove that the canonical degree CK_{X} is bounded from above by an explicit function in the Chern numbers of X and the geometric genus of C (Theorem 1.1). As an immediate consequence, it follows that the number of rational or elliptic curves on a surface of general type is universally bounded if the oriented topological type of the ambient surface is fixed.

1. Introduction

Let X be a smooth, projective, complex algebraic variety. An entire curve on X is, by definition, an image of the complex line ℂ via a non-constant holomorphic mapping. A theorem of R. Brody [3] says that X is hyperbolic if and only if X does not contain any entire curve. Inspired by this result and by the measure hyperbolicity of varieties of general type (proved by K. Kodaira and S. Kobayashi-T. Ochiai [7]), Mark L. Green suggested that a surface of general type might be hyperbolic modulo rational and elliptic curves therein (see [6], (7.4.13)). Serge Lang proposed a generalization to arbitrary dimension of Green’s suggestion ([8], Chapter IV, Section 5), and conjectured that the union of all the entire curves on X could not be Zariski dense as long as X was of general type.

If an entire curve is algebraic, i.e. if its Zariski closure is a closed curve C in X, C is either a rational curve or an elliptic curve. Conversely a rational or elliptic curve is obviously the closure of an entire curve. In this sense, we may call rational and elliptic curves on X algebraic entire curves.

When X is a surface, Lang’s conjecture clearly asserts the algebraicity and finiteness of the entire curves on a given X of general type. Yet the conjecture yields a far stronger result for the algebraic entire curves on surfaces, should it be true in arbitrary dimension. Namely we can derive from it the uniform boundedness of the numbers of algebraic entire curves on a minimal surface of general type X, meaning that the number should be bounded by a constant depending only on the topological invariants c_{1}^{2} and c_{2}. (For an arithmetic analogue, see [4]).

Date: October 22, 1998 / correction of mistyping: November 6/12/23/25, 1998.

1991 Mathematics Subject Classification. Primary: 14J29; Secondary: 14C25, 32H25.

Key words and phrases. canonical degree, Lang’s conjecture, effective boundedness.

Research partially supported by International Research Exchange Fellowship of Ministry of Education and Culture, Japan.

Notwithstanding its importance, very few supporting evidences for twodimensional Lang’s conjecture or Green’s conjecture have been known so far. The known results are essentially classified into two types.

The first type of resuts gets finiteness of algebraic entire curves C under certain condition on the cotangent bundle of the ambient surface. As an extreme case, assume that Ω_{X}^{1} is ample. Fix an arbitrary ample divisor H on X. Then we can find a small positive rational number t such that Ω_{X}^{1}(−tH) is still ample. By looking at the natural homomorphism Ω_{X}^{1} → ωC̃, we see that ωC̃(−tH) is also ample on C so that 2g −2 > tCH > 0. This proves (1) the non-existence of non-hyperbolic curves and gives (2) the upper bound of CH in terms of the genus^{1} g plus a geometric quantity t = t(X, H). In 1977, F. Bogomolov [1] generalized this simple observation to the case where Ω_{X}^{1} is big. (Ω_{X}^{1} is said to big if the dimension of H^{0}(X, Sym^{n} Ω_{X}^{1}) grows cubically in n as n tends to infinity.) In particular, if c_{1}(X)^{2} > c_{2}(X), there are only finitely many algebraic entire curves on X, a surface of general type.

Unfortunately, Bogomolov’s result does not apply to some important classes of surfaces including hypersurfaces in ℙ^{3}. If it does, his method, though ingeneous, is not effective and does not provide us a uniform bound of the number of non-hyperbolic curves.

The second type of results, on the other hand, proves a uniform bound of the number of algebraic entire curves that are subject to some constraint. For instance, let C be a smooth irreducible curve of genus g on a minimal surface of general type X. Then the Bogomolov-Miyaoka-Yau inequality for open surfaces (explicitly stated first in [12]) yields the estimate

CK_{X} ≤ 3c_{2}(X) − K_{X}^{2} + 4g − 4,

implying the uniform boundedness of the number of smooth rational/elliptic curves on X. A modification of the inequality gives a bound of CK_{X}, the canonical degree (perhaps “canonical height ” sounds better), for such C that has only a bounded number of ordinary double points and ordinary triple points [9]. This kind of condition on the number of certain singularities is of course artificial and even absurd, but the method is anyhow capable to give an effective result.

In the present paper, we give an effective, uniform bound of the canonical degree of an irreducible curve of given genus on a minimal surface of positive Kodaira dimension, without any codition on the ambient space or on the singularities of curves. Our main result is the following Theorem 1.1. Let X be a smooth projective complex surface and C an irreducible curve of geometric genus g on X (with arbitrary singularities). Assume that the canonical divisor K_{X} is nef, i.e. X is minimal and that CK_{X} > max{2g − 2, 3g − 3}. Then one of the following four inequalities holds:

1Throughout the paper, the genus of a complete irreducible curve will always mean the genus of its normalization.

(1)  CK_{X} ≤ 6c_{2}(X) − 2c_{1}(X)^{2} + 6g − 6,
(2)  \frac{(1 − 4(g − 1)/CK_{X})^{3}}{(1 − 3(g − 1)/CK_{X})^{2}} CK_{X} ≤ 8c_{2}(X) − 2c_{1}(X)^{2},
(3)  \frac{(1 − 2(g − 1)/CK_{X})^{3}}{(1 − 3(g − 1)/CK_{X})^{2}} CK_{X} ≤ 8c_{2}(X) − 2c_{1}(X)^{2},
(4)  CK_{X} ≤ 14^{3}(c_{2}(X) − \frac{1}{4}c_{1}(X)^{2}).

Immediate consequences of the theorem are listed below.

Corollary 1.2. If C is an algebraic entire curve, i.e. if g(C) ≤ 1, then

CK_{X} ≤ 14^{3}(c_{2}(X) − (1/4)c_{1}(X)^{2}) = 2744c_{2}(X) − 686c_{1}(X)^{2}.

Corollary 1.3. If g(C) ≥ 2, then CK_{X} is bounded from above by the maximum of 5g(C) − 5, 32c_{2} − 8c_{1}^{2} + 4g − 4 and 14^{3}(c_{2} − (1/4)c_{1}^{2}).

When the ambient surface X is of general type, theory of Chow varieties tells us that the uniform boundedness of the canonical degrees implies the uniform boundedness of the number of algebraic entire curves thereon, which is thought of as a supporting evidence for Lang’s conjecture.

Corollary 1.4. If X is a minimal surface of general type, then the number of rational/elliptic curves on X is bounded by a constant which depends only on the oriented topological type of X.

However, the real importance of our result is exhibited by the following

Corollary 1.5. Given the equations of a pluricanonically embedded surface X ⊂ ℙ^{N}, we can find all the rational curves on X by solving a system of algebraic equations in a bounded number of variables.

Indeed, a rational curve on ℙ^{N} of degree d is given by a system of equations

X_{i}(t) = ∑_{j=0}^{d} a_{i,j}t^{j}, i = 0, 1, …, N

in a parametric representation in terms of the linear coordinate t of ℙ^{1}. This curve lies on X if and only if g(X(t)) ≡ 0 for every defining equation g, which is an algebraic relation between the a_{i,j}’s.

In [14], T. Shioda observed that the Mordell-Weil rank of a certain elliptic surface does not increase much by cyclic base changes ℙ^{1} → ℙ^{1}. The following corollary provides us a natural explanation to his observation.

Corollary 1.6. Let X be a minimal elliptic surface of Kodaira dimension one, with canonical fibre structure π : X → B. Given a multi-section C ⊂ X of genus g, the mapping degree of C over B is effectively bounded in terms of the genus of C. (This statement is trivial unless B = ℙ^{1}.) In particular, the Mordell-Weil rank of X remains universally bounded after any base change B′ → B as long as the genus of B′ is bounded.

Our universal bound also gives a characterization of surfaces of general type in terms of algebraic entire curves.

Corollary 1.7. Let X be a projective complex surface. Then

1) The number of algebraic entire curves on X is finite if and only if X is either of general type or a simple abelian surface.

2) Let C(X) be the set of irreducible curves on a polarized surface (X, H). Define a function λ_{X} : C(X) → ℝ by

λ_{X}(C) = \frac{2g(C) − 2}{(C, H)}.

Then X is of general type if and only if 0 ∈ ℝ is away from the set of accumulation values of the function λ_{X}. (This property of surfaces of general type can be viewed as an algebraic analogue of measure hyperbolicity. ^{2} )

It is evident that there are uncountably many algebraic entire curves on a rational, ruled or elliptic surface. By [11], an algebraic K3 surface carries a one-parameter family of (singular) elliptic curves, whence follows (1). The proof of (2) thus reduces to the statement that

inf_{C∈C(X)} λ_{X}(C) = 0

for a simple abelian surface X. Any simple abelian surface X is an étale quotient of a principally polarized abelian surface A with theta divisor Θ of genus two. Consider a general translation Θ_{a} by an element a ∈ A and look at the image of Θ_{a} via the composite of the multiplication m_{N} : A → A by a positive integer N and the étale morphism A → X. This image has fixed genus two and arbitrarily large degree if N is chosen sufficiently large accordingly.

This paper is organized as follows.

In Section 2, we introduce certain subsheaves Ω^{(n)}⟨iC⟩ of Sym^{n} Ω_{X}^{1}(iC) and their natural filtrations. These filtrations essentially come from the parabolic structure of Ω_{X}^{1} attached to the natural homomorphism Ω_{X}^{1}|_{C} → ν_{∗}ωC̃, where ν : C̃ → C is the normalization.

2The function λ_{X} depends on the choice of the ample divisor H. However, if one replaces H by another ample divisor H′ to get a new function λ′_{X}, one can find positive constants a = a(H, H′), b = b(H, H′), such that aλ_{X}(C) ≤ λ′_{X}(C) ≤ bλ_{X}(C) regardless which C ∈ C(X) one picks up. In this sense, qualitative behaviour of λ_{X} does not depend on the choice of H.

In Section 3, we give estimates for the dimension of the global sections of Ω^{(n)}⟨iC⟩(−tnK_{X}). For instance, if C is an algebraic entire curve with only ordinary singularities, the dimension turns out to be positive whenever i ≥ (1 − t)n > 0 and u = 1 − t satisfies

u^{3} > \frac{c_{2}(X) − \frac{1}{4}K_{X}^{2}}{(C, K_{X})} + O(\frac{1}{n}).

Furthermore, a non-zero global section of this sheaf must have large slope function, defined in Section 4.

In Section 4, we study the interplay between the slope function and the cohomology class of an effective divisor on ℙ(E), the projective bundle attached with a 2-vector bundle E with parabolic structure.

The proof of Theorem 1.1 will be completed in Section 5, by combining the results in the previous part of the paper.^{3}

In this article, everything is over the complex number field, and we always work in the category of analytic spaces. Open neighbourhoods, local coordinates, local equations, etc. will be understood in analytic topology.

Acknowledgement. This work was completed during the author’s stay at Max-Planck-Institut für Mathematik at Bonn as International Research Exchange Fellow of Minstry of Education and Culture, Japan. The author is grateful to MPI for hospitality and to Ministry of Education and Culture for financial support.

2. Canonical filtration of the symmetric tensors associated with a curve

Let C be a reduced (but possibly reducible) curve on a smooth surface X. For simplicity of the notation, we denote by Ω the sheaf of holomorphic 1-forms Ω_{X}^{1} and by Ω^{(n)} the sheaf of the n-th symmetric tensors Sym^{n} Ω. Given a local equation f of C, df/f is a meromorphic 1-form with logarithmic pole along (the smooth locus of) C. Replacing f by a second equation gf, with g invertible, the 1-form d(gf)/(gf) differs from the original one by d log g, a holomorphic differential. Thus the sheaf

Ω⟨C⟩ = \frac{df}{f} O_{X} + Ω ⊂ Ω(C)

is globally well-defined. Ω⟨C⟩/Ω is a trivial O_{C}-module with a standard basis df/f mod Ω, where f is any local equation of C.

3It might be worth pointing out that our proof is quite elementary. Apart from fundamentals of algebraic geometry and undergraduate mathematics, what we need are: (1)

Semistable reduction theorem for Proposition 2.3; (2) K_{X}-semistability and Bogomolov semistability of Ω_{X}^{1} for Lemma 3.4; and (3) Bogomolov-Miyaoka-Yau-Sakai inequality for Proposition 5.1.

If C is smooth, this sheaf is the usual sheaf of logarithmic 1-forms Ω(log C). If C has singularities (including the case where C has only nodes), however, it is not locally free any more.

Products of n elements of Ω⟨C⟩ generate a subsheaf

Ω^{(n)}⟨nC⟩ = \frac{(df)^{n}}{f^{n}} O_{X} + \frac{(df)^{n−1}}{f^{n−1}} Ω + ⋯ + Ω^{(n)} ⊂ Ω^{(n)}(nC).

By construction, Ω^{(n)}⟨nC⟩ has a natural increasing filtration given by the subsheaves

Ω^{(n)}⟨jC⟩ = Ω^{(j)}⟨jC⟩ · Ω^{(n−j)} = Ω^{(n)}⟨nC⟩ ∩ Ω^{(n)}(jC)

= \frac{(df)^{j}}{f^{j}} Ω^{(n−j)} + \frac{(df)^{j−1}}{f^{j−1}} Ω^{(n−j+1)} + ⋯ + Ω^{(n)}

⊂ Ω^{(n)}(jC), j = 0, 1, …, n.

The associated graded module is naturally isomorphic to

⨁_{i=0,...,n} O_{C} ⊗ Ω^{(i)}.

Since Ω^{(n)}⟨jC⟩ is a subsheaf of Ω^{(n)}(jC), we obtain a series of subsheaves Ω^{(n)}⟨jC⟩(−jC) of Ω^{(n)}, which are locally of the form

(df)^{j}Ω^{(n−j)} + f(df)^{j−1}Ω^{(n−j+1)} + ⋯ + f^{j}Ω^{(n)}.

Thus we have the inclusion relation

Ω^{(n)}⟨nC⟩(−nC) ⊂ Ω^{(n)}⟨(n − 1)C⟩(−(n − 1)C) ⊂ ⋯ ⊂ Ω^{(n)}⟨0C⟩ = Ω^{(n)}.

Because they are not necessarily locally free, these subsheaves do not define a nice filtration of Ω^{(n)}. However, by taking the double duals

Ω^{(n)}⟨−jC⟩ = (Ω^{(n)}⟨jC⟩)^{∗∗}(−jC),

we get a nicer decreasing filtration

Ω^{(n)} = Ω^{(n)}⟨−0C⟩ ⊃ ⋯ ⊃ Ω^{(n)}⟨−nC⟩.

Along the smooth locus of C, we have of course Ω^{(n)}⟨−jC⟩ = Ω^{(n)}⟨jC⟩(−jC) and, if C is defined by one of the local coordinates x, it is simply the locally free O_{X}-module generated by:

dx^{n}, dx^{n−1}dy, …, dx^{j}dy^{n−j}, xdx^{j−1}dy^{n−j+1}, …, x^{j}dy^{n},

or, more graphically, by

x^{j}dy^{n},
…
xdx^{j−1}dy^{n−j+1},
dx^{n}, …, dx^{j}dy^{n−j}.

Let ν : C̃ → X denote the normalization of C ⊂ X. There is a natural O_{X}-homomorphism from Ω to ν_{∗}ωC̃, which induces a generically surjective OC̃-homomorphism ν^{∗}Ω → ωC̃. Let 𝔨 and 𝔮 denote the kernel and the image of this OC̃-homomorphism, respectively. On the smooth locus of C, we can identify C̃ with C, and then 𝔨 is the conormal bundle O_{C}(−C) whereas 𝔮 is the canonical bundle ωC̃. At each point point of C, we have

𝔨 ⊗_{OC̃} 𝔮 ≃ ν^{∗}O_{X}(K_{X}),
𝔮 ⊂ ωC̃.

The equality 𝔮 = ωC̃ globally holds if and only if C̃ → X is an immersion or, equivalently, for each singularity of C, the local analytic branches of C are all smooth. When C is a compact irreducible curve of geometric genus g, we have thus obvious relations between the degrees of 𝔨 and 𝔮:

deg 𝔨 + deg 𝔮 = CK_{X},
deg 𝔮 ≤ 2g − 2,
deg 𝔨 ≥ CK_{X} − 2g + 2.

In particular, if C is compact, irreducible and CK_{X} > 4g − 4, the vector bundle ν^{∗}Ω is unstable and 0 ⊂ 𝔨 ⊂ ν^{∗}Ω is the Harder-Narsimhan filtration introduced by [5].

The subsheaf 𝔨 ⊂ ν^{∗}Ω defines a natural filtration also on ν^{∗}Ω^{(n)}:

(HN) 0 ⊂ 𝔨^{n} ⊂ ν^{∗}𝔨^{n−1}Ω ⊂ ⋯ ⊂ 𝔨ν^{∗}Ω^{(n−1)} ⊂ ν^{∗}Ω^{(n)},

which is also the Harder-Narasimhan filtration of ν^{∗}Ω^{(n)} provided CK_{X} > 4g − 4 (plus compactness and irreducibility of C).

Notice that the natural homomorphism Ω|_{C} → ν_{∗}ωC̃ defines a (simplest) parabolic structure on Ω in the sense of C.S. Seshadri (cf. [13] for vector bundles over punctured curves; for the generalization to higher dimension, see e.g. [10]). Our filtration Ω^{(n)}⟨−jC⟩ is of course understood in terms of this parabolic structure.

The filtration (HN) on ν^{∗}Ω^{(n)} and the filtration {Ω^{(n)}⟨−jC⟩} are apparently related to each other. More precisely, we have

Lemma 2.1. There is a natural inclusion

Ω^{(n)}⟨−jC⟩/(Ω^{(n)}⟨−jC⟩ ∩ Ω^{(n)}(−C)) ↪ 𝔨^{j}ν^{∗}Ω^{(n−j)},

which is an isomorphism along the smooth locus of C.

The proof is easy and left to the reader.

Let X^{∘} denote the open subset X \ Sing(C). The natural filtration

Ω^{(n)} ⊃ ⋯ ⊃ Ω^{(n)}(−jC)

on X^{∘} induces a finer filtration on Ω^{(n)}⟨−jC⟩. (In other words, we are considering a double filtration on Ω^{(n)}.) Namely, we define:

Ω^{(n)}⟨−jC⟩_{i} = {η ∈ Ω^{(n)}⟨−jC⟩| η|_{X^{∘}} ∈ Ω^{(n)}(−iC) + Ω^{(n)}⟨−(j + 1)C⟩|_{X^{∘}}}.

Thus the quotient Ω^{(n)}⟨−jC⟩/Ω^{(n)}⟨−(j +1)C⟩, viewed as a torsion free O_{C}- module, naturally has successive quotients 𝔔_{j,i}, i = 0, …, j. On the smooth locus of C,

𝔔_{j,i} ≃ f^{i}(df)^{j−i}ω_{C}^{n−j+i} ≃ ω_{C}^{n−j+i}(−jC).

Let 𝔔̃_{j,i} denote ν^{∗}𝔔_{j,i}/(torsion).

Lemma 2.2. If C has only ordinary double points and if j ≤ n/2, then there is a natural isomorphism

𝔔̃_{i,j} ≃ ω⊗(n−2j+i) ; C̃(jν^{∗}K_{X}).

Proof. If C is smooth at p, then ω_{C}, which is the quotient of Ω by df = f \frac{df}{f}, is canonically isomorphic to

det Ω ⊗ (f \frac{df}{f} )^{−1}O_{C} ≃ f^{−1} det Ω ⊗ O_{C} ≃ O_{C}(K_{X} + C).

This isomorphism (adjunction formula) shows the assertion at a smooth point of C.

Assume that p is a node of C, and let (x, y) be a local coordinate at p ∈ C. We may assume that f = xy. Under the condition that j ≤ n/2, direct calculation shows that Ω^{(n)}⟨−jC⟩ is generated by:

y^{j}dx^{n}  x^{j}dy^{n}
…  …
dx^{n−j}dy^{j} … dx^{j}dy^{n−j}

so that

(∗)  Ω^{(n)}⟨−jC⟩/Ω^{(n)}⟨−(j + 1)C⟩
=y^{j}dx^{n}(O/yO) ⊕ x^{j}(dy^{n})(O/xO)
⊕ y^{j−1}(dxdy)dx^{n−2}(O/yO) ⊕ x^{j−1}(dxdy)dy^{n−2}(O/xO)
⊕ ⋯
⊕ (dxdy)^{j}dx^{n−2j}(O/yO) ⊕ (dxdy)^{j}(dy^{n−2j})(O/xO).

The O_{C}-submodule Ω^{(n)}⟨−jC⟩_{i}/Ω^{(n)}⟨−(j + 1)C⟩ is generated by the first j − i + 1 lines of the above equality.

Note that dx(O/yO) ⊕ dy(O/xO) is naturally identified with ωC̃. Meanwhile, on the local branch x = 0 of C,

x(O/xO) = \frac{dx ∧ dy}{d(xy)/(xy) ∧ dy} (O/xO)

≃ (dy)^{−1}(dxdy)(O/xO) ≃ ω−1 ; C̃(ν^{∗}K_{X}).

Thus each line of the equality (*) is of the form

(dx ∧ dy)^{j}(dx)^{n−2(j−i)−i}O/xO ⊕ (dx ∧ dy)^{j}(dy)^{n−2(j−i)−i}O/yO.

Now the assertion is clear.

Thanks to the semistable reduction theorem, this result carries over to general cases.

Proposition 2.3. For general (X, C), there exists an inclusion:

𝔔̃_{i,j} ↪ ω⊗(n−2j+i) ; C̃(jν^{∗}K_{X}),

for j ≤ n/2.

Proof. We have never assumed that X or C is compact. Hence, by replacing X by a small Stein open subset, we only have to prove that

H^{0}(C̃, 𝔔̃_{i,j}) ↪ H^{0}(C̃, ω⊗(n−2j+i) ; C̃(jν^{∗}K_{X})).

We view C (locally) as the fibre over the origin of the fibration π : X → Δ, where Δ ⊂ C is a small disc with local coordinate t. Then Ω^{(n)}⟨−jC⟩ is the saturation in Ω^{(n)} of

(π^{∗}ω_{Δ})^{n} + π^{∗}t(π^{∗}ω_{Δ})^{n−1}Ω + ⋯ + (π^{∗}t^{j})Ω^{(n)}.

By the semistable reduction theorem, after a suitable cyclic base change, desingularization and blow-down, we get a new fibration g : Y → Δ′ such that the singular fibre D = g^{−1}(0) is a reduced divisor with normal crossings on a smooth surface Y. Thus we get a commutative diagram

\begin{matrix}\overline{X} & ←^{γ} & \overline{Y} \\ ↓ & & ↓ \\ X & & Y \\ ↓ & & ↓ \\ Δ & ← & Δ′. \end{matrix}

Here \overline{X} and \overline{Y} stand for blown-up surfaces of X and Y, respectively. The top line and the botom line are finite cyclic Galois covers, the latter being given simply by τ ↦ t = τ^{r}.

From the construction follow the facts

(1) the projection \overline{X} → Δ is smooth at a general point of the strict transform \overline{C} of C on \overline{X};

(2) and hence γ^{−1}(\overline{C}) is set-theoretically isomorphic to \overline{C};

(3) γ^{−1}(\overline{C}) does not collapse to a point on Y;

(4) the Galois group G = Gal(Y/X) = Gal(Δ′/Δ) ≃ ℤ/(r) naturally acts on the pair (\overline{Y} , \overline{D}), which descends to the action on (Y, D).

The filtration Ω^{(n)}⟨−jC⟩_{i} does not behave nicely under base change simply because the defining equation f of C changes to g^{r} after base change. Nevertheless Ω^{(n)}⟨−jC⟩_{i}(jC) behaves far better. In fact, Ω^{(n)}⟨−jC⟩(jC) is, by definition, the double dual of

(\frac{df}{f} )^{j}Ω^{(n)} + ⋯ + Ω^{(n)},

while

\frac{df}{f} = \frac{dt}{t} = r \frac{dτ}{τ} = r \frac{dg}{g} .

The submodule Ω^{(n)}⟨−jC⟩_{i}(jC) is the double dual of

f(\frac{df}{f} )^{j}Ω^{(n−j)} + ⋯ + f(\frac{df}{f} )^{j−i+1}Ω^{(n−j+i−1)} + (\frac{df}{f} )^{j−i}Ω^{(n−j+i)} + ⋯ + Ω^{(n)},

which is “almost” compatible with the base change.

In general, given torsion free sheaves E_{1} ⊂ E_{2}, the quotient E_{2}/E_{1} will inflate to a bigger sheaf if we replace E_{i}’s by their double duals. From this viewpoint, Lemma 2.2 is interpreted as an upper bound of the inflation:

Ω_{Y}^{(n)}⟨−jD⟩_{i}(jD)/Ω_{Y}^{(n)}⟨−jD⟩_{i+1}(jD)

≃ ω_{D}^{i}((j − i)(K + D)) ⊗ ωn−2j+i ; D̃.

This sheaf is a quotient of

∑_{i=0}^{j}(\frac{(dx ∧ dy)/g}{dg/g} )^{i} · (dx ∧ dy/g)^{j−i}Ω_{Y}^{(n−2j+i)} ⊂ Ω^{(n)}(jD),

dx ∧ dy being a local generator of det Ω_{Y}. Here we used two identifications

(dx ∧ dy/g)O ≃ det((dg/g)O + Ω_{Y})^{∗∗}),
((dx ∧ dy/g)/(dg/g))O ≃ (((dg/g)O + Ω)^{∗∗}/(dg/g)O)^{∗∗}.

By construction, there are natural isomorphisms

γ^{∗}μ_{X}^{∗}ω_{C} ≃ μ_{Y}^{∗} ω_{D},
γ^{∗}μ_{X}^{∗}(K_{X} + C) ≃ μ_{Y}^{∗} (K_{Y} + D),

where γ : \overline{Y} → \overline{X} is the cyclic Galois cover and μ stands for the blowing up. These isomorphisms send the local generators (ds ∧ dt/f)/(df/f) and (ds ∧ dt)/f to (dx ∧ dy/g)/(dg/g) and r(dx ∧ dy)/g, respectively.

Thus

H^{0}(X, Ω^{(n)}⟨−jC⟩_{i}(jC)) ≃ H^{0}(\overline{X}, μ_{X}^{∗}Ω^{(n)}⟨−jC⟩_{i}(jC))
⊂ H^{0}(\overline{Y} , μ_{Y}^{∗} Ω^{(n)}⟨−jD⟩_{i}(jD)) ≃ H^{0}(Y, Ω^{(n)}⟨−jD⟩_{i}(jD))

must be contained in

∑_{i=0}^{j}(\frac{(dx ∧ dy)/g}{dg/g} )^{i} · (dx ∧ dy/g)^{j−i}Ω_{Y}^{(n−2j+i)}

= ∑_{i=0}^{j}(\frac{(ds ∧ dt)/f}{df/f} )^{i} · (ds ∧ dt/f)^{j−i}Ω_{Y}^{(n−2j+i)}.

Since Ĉ, the image of γ^{−1}(\overline{C}) on Y, is bimeromorphic to C̃, the restriction of this sheaf to Ĉ maps naturally to ∑ ω_{C}^{i}((j −i)(K +C))⊗ωn−2j+i ; C̃. In particular H^{0}(X, Ω^{(n)}⟨−jC⟩_{i}(jC))/ H^{0}(X, Ω^{(n)}⟨−jC⟩_{i+1}(jC)) must be contained in the space of global sections of

ω_{C}^{i}((j − i)(K + C)) ⊗ ωn−2j+i ; C̃.

Noting that ω_{C} is canonically isomorphic to O_{C}(K_{X} + C) via the multiplication by df/f, we have the assertion.

Remark. There should be an elementary proof of Proposition 2.3 without resorting to the semistable reduction theorem. It is indeed fairly easy to directly show the assertion in case C is an immersed curve (i.e. ν : C̃ → X is an immersion).

3. Global dimension count

Let X be a compact complex surface and C a reduced, irreducible curve on X.

We use the same notation as in the previous section. Thus Ω^{(n)}⟨nC⟩ is an enlargement of Ω^{(n)} = Sym^{n} Ω by adjoining polynomials of df/f of degree ≤ n.

First we calculate the Euler characteristic of Ω^{(n)}⟨nC⟩:

Proposition 3.1. If n ≫ 0 and CK_{X} > c_{1}(X)^{2} − c_{2}(X), then the Euler characteristic χ(X, Ω^{(n)}⟨nC⟩) is large:

χ(X, Ω^{(n)}⟨nC⟩) = \frac{n^{3}}{6} (c_{1}(X)^{2} − c_{2}(X) + CK_{X}) + O(n^{2}).

Proof. By Riemann-Roch, χ(X, Ω^{(n)}) = \frac{n^{3}}{6} (c_{1}^{2} − c_{2}) up to quadratic terms in n. Then look at the increasing filtration Ω^{(n)}⟨jC⟩ on Ω^{(n)}⟨nC⟩ with successive quotients O_{C} ⊗ Ω^{(n−j)}, which has Euler characteristic

(n − j)(n − j + 1)CK_{X} + (n − j + 1)χ(C, O_{C}).

Adding up these terms, we get the equality as above. (Note that the contribution involving χ(C, O_{C}) is only up to quadratic order in n.)

The same proof shows the following:

Corollary 3.2. We have the following asymptotic equality:

χ(X, Ω^{(n)}⟨nC⟩(−tnK_{X}))

= \frac{n^{3}}{6} ((1 − 3t + 3t^{2})c_{1}(X) − c_{2}(X) + (1 − 3t)CK_{X} + O(n^{2}).

The corollary guarantees the non-vanishing of H^{0}(X, Ω^{(n)}⟨nC⟩(−tnK_{X})) for fairly large value t, roughly up to 1/3 when CK_{X} is large. However, it is not optimal by any means. We will below improve the estimate of the maximum value of rational numbers t such that H^{0}(X, Ω^{(n)}⟨nC⟩(−tnK_{X})) does not vanish.

Fix an interger j = vn and consider the subsheaf

Ω^{(n)}⟨nC⟩_{[j]} =Ω^{(n)} + \frac{df}{f} Ω^{(n−1)}⟨−(n − j)C⟩ + (\frac{df}{f} )^{2}Ω^{(n−2)}⟨−(n − j)C⟩

+ ⋯

+ (\frac{df}{f} )^{j}Ω^{(n−j+1)}⟨−(n − j)C⟩ + (\frac{df}{f} )^{j}Ω^{(n−j)}⟨−(n − j)C⟩

of Ω^{(n)}⟨jC⟩ ⊂ Ω^{(n)}⟨nC⟩. This is globally well-defined, thanks to the natural inclusion dfΩ^{(k)}⟨−jC⟩ ⊂ Ω^{(k+1)}⟨−jC⟩. If C is locally smooth and defined by x = 0, this sheaf is locally identical with Ω^{(n)}⟨−nC⟩((n − j)C) + Ω^{(n)} spanned by the local basis

dx^{n−j}dy^{j}, … dy^{n},
x^{−1}dx^{n−j+1}dy^{j−1},
…
x^{−j}dx^{n}

whereas Ω^{(n)}⟨jC⟩ is generated by

dy^{n},
x^{−1}dxdy^{n−1},
…
x^{−j}dx^{n}, …, x^{−j}dx^{n−j}dy^{j}

Let ν : C̃ → C ↪ X be the normalization map. Assume that CK_{X} > max{4g(C)−4, 0}, so that ν^{∗}Ω is unstable. Let 𝔨 ⊂ ν^{∗}Ω denote the maximal destabilizing subsheaf, to get the exact sequence

0 → 𝔨 → ν^{∗}Ω → 𝔮 → 0.

Let d = d(X, C) be the rational number (defect coefficient) defined by

(2d + 1)CK_{X} = deg 𝔨.

It is obvious that deg 𝔮 = −2dCK_{X} and that d ≥ (1 − g)/CK_{X}. With these new symbols, we get:

Proposition 3.3. (1) We have the following asymptotic inequality

χ(X, Ω^{(n)}⟨nC⟩_{[vn]}(−tnK_{X}))
≥ χ(X,Ω^{(n)}(−tnK_{X}))

+ n^{3}CK_{X} ∫_{0}^{v} x((1 − t + 2d) − (1 + 3d)x)dx + O(n^{2}).

(2) If we vary v, the integral above, viewed as a function in v, takes at

v_{0} = \frac{(1 − t) + 2d}{1 + 3d}

the maximum value

\frac{n^{3}(1 − t + 2d)^{3}CK_{X}}{6(1 + 3d)^{2}} .

In particular,

χ(X,Ω^{(n)}⟨nC⟩_{[v_{0}n]}(−tnK_{X}))

≥ \frac{n^{3}}{6} \{((1 − 3t + 3t^{2})c_{1}^{2} − c_{2}) + \frac{(1 − t + 2d)^{3}}{(1 + 3d)^{2}} CK_{X}\} + O(n^{2}),

Proof. There is an increasing filtration

(Ω^{(n)}⟨nC⟩_{[j]} + Ω^{(n)}⟨kC⟩)/Ω^{(n)}⟨nC⟩_{[j]}^{(n)}, k = 0, …, n

of Ω^{(n)}⟨nC⟩/Ω^{(n)}⟨nC⟩_{[j]}, with successive quotients being an O_{C}-submodule of the quotients

\begin{cases}ν^{∗}Ω^{(n−k)}/𝔨^{n−j}ν^{∗}Ω^{(j−k)} & if k ≤ j; \\ ν^{∗}Ω^{(n−k)} & if j < k ≤ n. \end{cases}

The Euler characteristic of these OC̃-modules are:

\binom{n − k + 1}{2}CK + \binom{j − k + 1}{2}CK − 2\binom{j − k + 1}{2} deg 𝔨 + (n − j)χ(C̃, O)

or

\binom{n − k + 1}{2}CK + (n + k + 1)χ(C̃, O)

according as k ≤ j or k > j. If j = vn, these numbers add up to

\frac{n^{3}}{6} ((1 − v^{3})CK − 3v^{2}(1 − v) deg 𝔨) + O(n^{2}).

Hence we deduce the inequality

χ(X,Ω^{(n)}⟨nC⟩_{[vn]})

≥ \frac{n^{3}}{6} (c_{1}^{2} − c_{2} + CK − (1 − v^{3})CK + 3v^{2}(1 − v) deg 𝔨) + O(n^{2})

= \frac{n^{3}}{6} (c_{1}^{2} − c_{2} + v^{3}CK + 3v^{2}(1 − v) deg 𝔨) + O(n^{2}).

By tensoring O(−tnK_{X}), everything must be suitably adjusted to yield

χ(X,Ω^{(n)}⟨nC⟩_{[vn]}(−tnK_{X}))

≥ \frac{n^{3}}{6} ((1 − 3t + 3t^{2})c_{1}^{2} − c_{2} + (v^{3} − 3v^{2}t)CK + 3v^{2}(1 − v) deg 𝔨)

up to an error term of quadratic order in n. Now the statement (1) is a tautological restatement of this, while the statement (2) is an immediate corollary to (1).

Lemma 3.4. Let the notation be as above. If u = 1 − t < \frac{1}{2}, then

dim H^{2}(X,Ω^{(n)}⟨nC⟩_{[vn]}(−tnK_{X}))

≤ dim H^{2}(X, Ω^{(n)}(−tnK_{X}))

= dim H^{0}(X, Ω^{(n)}(−unK_{X})) + O(n^{2})

≤ \frac{n^{3}(1 − 2u)^{2}}{8} c_{1}(X)^{2} + O(n^{2}).

Proof. The first inequality is trivial if one notes that Ω^{(n)}⟨nC⟩_{[vn]} contains Ω^{(n)}. The second equality is a direct consequence of the Serre duality and the natural isomorphism Ω^{∗} ≃ Ω(−K_{X}). Let us prove the third inequality.

If the Kodaira dimension κ(X) of X is one or less, it is more or less trivial to show that h^{0}(X, Ω^{(n)}(−unK_{X})) is at most quadratic order in n. Therefore we may assume that X is of general type. In this case, it is known that the cotangent sheaf Ω is K_{X}-semistable. (It is a corollary of S.-T. Yau’s famous theorem [16] on the existence of Kähler Einstein metric when K_{X} is ample. Also in case K_{X} is nef and big, semistability follows from a theorem of H. Tsuji on the existence of degenerate Kähler Einstein metric [15].) In particular, if Γ is a general member of |(\frac{1}{2} − u)nK_{X}| and n sufficiently large, Ω|_{Γ} is semistable. Hence so is Ω^{(n)}(−unK_{X})|_{Γ}, with rank n + 1 and with average degree n^{2}(1 − 2u)^{2}K_{X}^{2}/4. Noting that the genus of Γ is n^{2}(1 − 2u)^{2}/8 modulo a linear correction term, we get

dim H^{0}(Γ, Ω^{(n)}(−unK_{X})) ≤ \frac{n^{3}(1 − 2u)^{2}}{8} + O(n^{2}).

Now look at the exact sequence

0 → Ω^{(n)}(−\frac{n}{2}K_{X}) → Ω^{(n)}(−unK_{X}) → Ω^{(n)}(−unK_{X})|_{Γ} → 0.

The dimesion of the global sections of the third term is bounded as above, whilst the first term has, by Bogomolov semi-stability, global sections of dimension at most of linear order in n.

Corollary 3.5. Assume that u = 1 − t < 1/2. Then

dim H^{0}(X, Ω^{(n)}⟨nC⟩_{[v_{0}n]}(−tnK_{X}))

≥ \frac{n^{3}}{6} \{((1 − u)^{3} + u^{3})c_{1}^{2} − c_{2} + \frac{(u + 2d)^{3}}{(1 + 3d)^{2}} CK − \frac{3(1 − 2u)^{2}}{4} K^{2}\} + O(n^{2})

= \frac{n^{3}}{6} \{\frac{1}{4}c_{1}^{2} − c_{2} + \frac{(u + 2d)^{3}}{(1 + 3d)^{2}} CK_{X}\} + O(n^{2}).

Here v_{0} is a rational number defined by

v_{0} = \frac{u + 2d}{1 + 3d}

as in Lemma 3.3.

From this corollary immediately follows:

Corollary 3.6. Let u be the minimum rational number ∈ (1/n)ℤ such that

H^{0}(X, Ω^{(n)}⟨nC⟩(−(1 − u)nK_{X})) ≠ 0.

Then the inequality

\frac{(u + 2d)^{3}}{(1 + 3d)^{2}} CK_{X} + \frac{1}{4}c_{1}^{2} − c_{2} ≤ O(\frac{1}{n})

holds.

Finally, we note that, if the hypothesis of the above lemma is satisfied, then the global sections are necessarily contained in a small subsheaf:

Lemma 3.7. Put

j = j(u, n) = ⌈\frac{u + 2d}{1 + 2d}n⌉,

where the symbol ⌈∗⌉ denotes the round-up of ∗, i.e. the smallest integer larger than or equal to ∗. Then:

H^{0}(X, Ω^{(n)}⟨nC⟩(−(1 − u)nK_{X})) = H^{0}(X, Ω^{(n)}⟨jC⟩(−(1 − u)nK_{X}))

= H^{0}(X, Ω^{(n)}⟨nC⟩_{[j]}(−(1 − u)nK_{X})).

If 1 − u > (1/2) + max{0, g − 1}, then, in view of Proposition 2.3, we have a stronger result:

Lemma 3.8. Assume that 1 − u > (1/2) + max{0, g − 1}. Then

H^{0}(X,Ω^{(n)}⟨nC⟩(−(1 − u)nK_{X}))

= H^{0}(X, (Ω^{(n)}⟨nC⟩_{[j]} ∩ Ω^{(n)}⟨−(j + \frac{n}{2})C⟩(jC))(−(1 − u)nK_{X}))

The proofs of Lemmas 3.7 and 3.8 are similar as that of Lemma 3.2.

4. Parabolic structure, slopes and Newton polygon

Let X be a smooth surface and E a vector bundle of rank two on X. Let H = H_{E} denote the tautological line bundle on the projective bundle π : ℙ(E) → X. Given an effective divisor D on ℙ(E) such that π(D) = X, there exist a positive integer n and a ℚ-divisor Δ on X such that D is linearly equivalent to n(H + π^{∗}Δ); n is of course the mapping degree of the projection from D to X.

Let C ⊂ X be a compact, irreducible, reduced curve (note that we did not assume that X is compact). A parabolic structure on E with respect to C is, by definition, an invertible subsheaf L_{C} ⊂ E|_{C} of rank one such that L_{C} locally extends to a saturated invertible subsheaf of E at each point p ∈ C. (By a saturated subsheaf, we mean an subsheaf L such that E/L is a torsion free O_{X}-module.)

Let L_{C} be a parabolic structure of E. Then, locally around a smooth point of C, L_{C} is a line subbundle of of E|_{C}, and we can locally extend L_{C} ⊂ E|_{C} to a subbundle L. Let L′ be the quotient E/L:

0 → L → E → L′ → 0.

Thus E is locally a direct sum s_{0}O ⊕ s_{1}O, s_{0} and s_{1} being local bases of L and of L′, respectively. We regard (s_{0}, s_{1}) as a homogeneous coordinate of the fibre ≃ ℙ^{1} over p ∈ X.

Recall that a geometric realization of ℙ(E) is the quotient of E^{∗}\(zero-section) via the natural ℂ^{∗}-action (fibrewise scalar multiplication). The subbundle L, which is naturally identified with the annihilator of L^{′∗} ⊂ E^{∗}, defines a relative hyperplane divisor ∼ H − π^{∗}L_{1}, locally given by a single equation s_{0} = 0. The extension L is not unique, so that the s_{0} involves some indeterminacy apart from non-vanishing holomorphic multiplicative factor. However, such additional indeterminacy vanishes modulo the ideal of C.

An effective divisor D ∼ n(H + π^{∗}Δ) on ℙ(E) is, locally around a given smooth point p ∈ C, defined by a homogeneous polynomial ∑ a_{i}s_{0}^{n−i}s_{1}^{i} in (s_{0}, s_{1}) of degree n with coefficients in O_{X,p}. Though the choice of s_{0} is fairly rigid, that of s_{1} is almost arbitrary. One can easily check that, after a suitable linear transformation by an upper triangular matrix with entries in the holomorphic functions on X, the coefficient a_{0} of s_{0}^{n} can be chosen to be one, unless D contains the fibre π^{−1}(p).

In what follows, we always fix an irreducible curve C on X together with a parabolic structure L_{C} ⊂ E|_{C}. Given an effective divisor D ⊂ ℙ(E), we assume that D does not contain any vertical component and normalize the first coefficient a_{0} to 1 (at a general point of C).

Pick up a general point p ∈ C and let (x, y) be a local coordinate system of X at p such that x = 0 defines C. The minimum slope, of an effective divisor D at p (with respective to the pair (C, L_{C})) is defined as the non-negative rational number

σ_{min}(D) = σ_{min}(D; C, L_{C})

= min\{1, min_{i}\{\frac{ord_{C} a_{i}}{i} \}\},

where D is given by the homogeneous equation ∑ a_{i}(x, y)s_{0}^{n−i}s_{1}^{i} and the symbol ord_{C}(∗) is defined to be the integer α such that x^{α} exactly divides ∗ (i.e. the vanishing order of ∗ along C). The positive integer n is, in the meantime, called the covering degree of D. (The choice of local extensions L and its bases s_{0} leaves the indeterminacy of ord a_{i} up to order i, so that σ_{min}(D) makes sense only up to one. In case the subsheaf L_{C} is globally given in the r-th infinitesimal neighbourhood of C, the minimal slope will be well-defined to be a rational number less than or equal to r. )

The zero-orders of the coefficients a_{i} also determine the Newton polygon Newt_{(C,L_{C};p)}(D) of the effective divisor D ∼ n(H + π^{∗}Δ) at p (with respect to the pair (C, L_{C})), which is, by definition, the convex closure of the union of the n + 1 half lines

{(T, S) ∈ ℝ^{2}; T = i, S ≥ min{ord a_{i}, i}}, i = 0, 1, …, n

in the (T, S)-plane ≃ ℝ^{2}.

One easily checks that the minimum slope, or the Newton polygon, of D is well-defined by the parabolic structure of E, i.e. independent of the choice of L, s_{i}, etc. Furthermore, they are independent also of the choice of a general point p ∈ C (recall that C is irreducible).

The ring of holomorphic functions on X at a general point of C ⊂ X is identified with the ring O_{C}{{x}} of the convergent power series in x with coefficients in O_{C}. The field of meromorphic functions in a small neighbourhood of C is ℂ(C)((x)), the field of convergent Laurent series in x with coefficients in the function field of C. Its algebraic closure is the field of the Puiseux series \overline{ℂ(C)}((x))[x^{1/∞}] obtained by adjoining all the roots of x to the field of the Laurent series with coefficients in the algebraic closure of ℂ(C). The ring of integers in this field is \overline{O_{C}}\{\{x\}\}[x^{1/∞}], \overline{O_{C}} being the integral closure of O_{C} in \overline{ℂ(C)}.

Any analytically locally irreducible divisor D ⊂ ℙ(E) over a small neighbourhood of p ∈ C defines a finite extension of O_{C}{{x}}. Therefore D can be written in the form

s_{0}^{n} + x^{b}s_{2}^{n} = 0

or

x^{b}s_{0}^{n} + (1 − x^{b})(s_{0} − s_{1})^{n} = 0

after a suitable local analytic coordinate change at a general point p ∈ C. Here b is a positive integer with (b, n) = 1. In this case, the slope of this locally irreducible divisor D at p is defined to be min{1, b/n} in the former case and 0 in the latter case and is denoted by σ(D) (of course it coincides with σ_{min}(D) in this situation). The Newton polygon of D is the region given by 0 ≤ T ≤ n, S ≥ σ(D)T or 0 ≤ T ≤ n, S ≥ 0, accordingly.

Given a general point p of C, the locally irreducible divisor D intersects π^{−1}(p) at a single point. σ(D) is positive if and only if the intersection point coincides π^{−1}(p)∩{s_{0} = 0} at p or, equivalently, D passes through the curve π^{−1}(C) ∩ {s_{0} = 0}. The intersection multiplicity D ∩ {s_{1} = 0} along this curve is at least min{nσ(D), 1} (exactly nσ(D) if nσ(D) < 1).

Let D be an arbitrary effective divisor on ℙ(E). Choose a general point p ∈ C and decompose D into the sum of analytically locally irreducible components D_{α} near p. Let n_{α} and σ_{α} denote the covering degree and the slope of D_{α}. Choose the indices α such that σ_{α} is increasing.

Then we define the slope function σ = σ_{D} : [0, n] → ℝ to be the monotone increasing step function σ(T) = σ(D_{α}), if n_{1}+⋯+n_{α−1} < T ≤ n_{1}+⋯+n_{α}.

Lemma 4.1. Assume that X is projective. Let D be an irreducible divisor ∼ n(H + π^{∗}Δ) with covering degree n ≥ 2. Let A be a general hyperplane section on X. Then

nA(2Δ + det E) ≥ \frac{1}{n − 1} ∑_{i=1}^{n} σ_{D}(i)(I(i) − 1)AC

> \frac{1}{n} ∑_{i=1}^{n} σ(i)(I(i) − 1)AC

≥ AC(\frac{1}{n} ∫_{0}^{n} σ(T)(n − T)dT − ε(D)),

where I(i), called the counting function, is the number of the integers j such that σ(j) ≥ σ(i), and ε(D) is the correction term:

ε(D) = \begin{cases}1 & if σ_{D}(n − 1) < 1 and σ_{D}(n) = 1 \\ 0 & otherwise. \end{cases}

Proof. For simplicity, we assume that det E is numerically trivial. Take a suitable ramified Galois cover Y → X such that the inverse image of D decomposes into a union of sections S_{i}. Fix one irreducible component S_{i} once for all. Choose A generally so that A meets C and the branch locus of f transversally at smooth points. Let q ∈ Y be an arbitrary point of the inverse image of A·C. A small neighbourhood of S_{i} of q is mapped to a unique local irreducible component D_{α(i)}. Hence S_{i} meets the other components S_{j} at {q} × {s_{0} = 0} with intersection multiplicity ≥ min{Nσ(D_{α(i)}), Nσ(D_{α(j)}}, where N is the branch index of f along C. Hence if we write S_{i} ∼ H + Δ_{i}, we have

S_{i}(∑_{j≠i} S_{j})π^{∗}f^{∗}A = (∑_{j≠i} Δ_{j} + (n − 1)Δ_{i}))f^{∗}A ≥ ∑_{i=1}^{m} σ(i)(I(i) − 1)f^{∗}Af^{∗}C,

On the other hand, one can easily check that

(1) NS(Y) ⊗ ℚ is a direct sum of NS(X) ⊗ ℚ and its orthogonal complement, and that

(2) Δ_{i} − Δ_{j} sits in the orthogonal compliment (here we use the fact that D is irreducible).

Now the assertion is streightforward, if one notices that the slope function σ_{D} takes value a/b at b or more integer points (here the integers a, b are of course assumed to have no common factor).

Call D a special irreducible divisor if the correction term ε(D) is non-zero.

Corollary 4.2. Let D ∼ n(H + π^{∗}Δ) be an irreducible divisor of covering degree ≥ 2. Let \overline{σ} = \overline{σ}(D) denote the mean slope:

\overline{σ}(D) = \frac{1}{n} ∫_{0}^{n} σ(T)dT.

Assume that the slope function σ(T) = σ_{D}(T) is bounded from below by σ_{min} < 1. Then the inequality

nA(2Δ + det E) ≥ \frac{n}{2} · \frac{(\overline{σ} − σ_{min})^{2} + σ_{min}(1 − σ_{min})}{1 − σ_{min}}AC + \frac{ε(D)}{2}AC

holds.

Proof. The slope function σ(T) is monotone increasing with values between σ_{min} and 1, while the weight function n − T is monotone decreasing. If the mean value \overline{σ} is given, the integral

∫_{0}^{n} σ(T)(n − T)dT

takes the minimum value when σ(T) is the two-step function

σ(T) = \begin{cases}σ_{min}, & 0 ≤ T ≤ T_{0}, \\ 1, & T_{0} < T ≤ n. \end{cases}

Here T_{0} is of course the solution of σ_{min}T_{0} + (n − T_{0}) = n\overline{σ}. By computing the integral of this step function, we get the estimate.

Corollary 4.3. Let D be an effective divisor on ℙ(E) of which each irreducible component has covering degree ≥ 2. If D ∼ nH + π^{∗}Δ of average slope \overline{σ} and minimum slope σ_{min} < 1, then

nA(2Δ + det E) ≥ [\frac{n}{2} \frac{(\overline{σ} − σ_{min})^{2} + σ_{min}(1 − σ_{min})}{1 − σ_{min}} + \frac{ν(D)}{2}]AC

where ν(D) denotes the number of special irreducible components contained in D.

Proof. Decompose D into irreducible components D_{α}, with covering degrees n_{α}, average slopes \overline{σ}_{α}. Then we have

nA(2Δ+ det E) = ∑_{α} n_{α}A(2Δ_{α} + det E)

≥ ∑_{α}[\frac{n_{α}}{2} \frac{(\overline{σ}_{α} − σ_{min})^{2} + σ_{min}(1 − σ_{min})}{1 − σ_{min}} + \frac{ε(D_{α})}{2}]AC.

The right hand side is bounded from below by

[\frac{n}{2} \frac{(\overline{σ} − σ_{min})^{2} + σ_{min}(1 − σ_{min})}{1 − σ_{min}} + \frac{ν(D)}{2}]AC,

by the Schwarz inequality.

Note that the inequality above remains true after replacing A by an arbitrary ample or, more generally, nef, divisor. In particular, if C^{2} ≥ 0 (which implies that C is nef) and if deg L_{C} is bounded from above, we have:

Corollary 4.4. Assume that C^{2} ≥ 0 and that

−2 deg L_{C} + C det E ≥ \frac{C^{2}}{2} .

Let D ∼ n(H + π^{∗}Δ) be an arbitrary effective divisor, with slope function σ(x) bounded from below by σ_{min} > 0. Then the inequality

nC(2Δ + det E) ≥ \frac{nC^{2}}{2} · \frac{(\overline{σ} − σ_{min})^{2} + σ_{min}(1 − σ_{min})}{1 − σ_{min}} + \frac{ν(D)C^{2}}{2}

holds. Here \overline{σ} stands for the average slope of D.

Proof. What is not covered by Corollary 4.3 is the contribution from the irreducible components of covering degree one, which is taken care of by the degree bound of L_{C}.

5. Proof of Theorem 1.1

Let X be a minimal complex surface and C an irreducible curve of genus g on it. Let n be a sufficiently large integer and let u denote the minimal rational number such that H^{0}(X, Ω^{(n)}⟨nC⟩(−(1−u)nK_{X}) stays non-vanishing.

We prove the theorem in three separate cases:

Case 0. KC ≤ max{2g − 2, 3g − 3}. Case 1. C^{2} ≤ 0. Case 2. C^{2} > 0, u ≥ (1/2) − (g − 1)/CK_{X} or u ≥ 1/2. Case 3. C^{2} > 0, u < (1/2) − (g − 1)/CK_{X}, u < 1/2.

Case 0 is a trivial case, and there is nothing to prove.

In what follows, we always assume that KC > max{2g − 2, 3g − 3}.

Case 1 is taken care of by the following

Proposition 5.1. If C^{2} ≤ 0, then the inequality

CK_{X} ≤ 6c_{2}(X) − 2c_{1}(X)^{2} + 6g − 6

holds.

Proof. Take a minimal embedded resolution μ : X̃ → X such that the total transform of C is supported by a divisor with simple normal crossings \overline{C} = C̃ + F. Here C̃ is the smooth strict transform of C and F is the union of the exceptional curves.

Apply the Bogomolov-Miyaoka-Yau-Sakai inequality to the pair (X̃, \overline{C}):

(KX̃ + C̃ + F)^{2} ≤ 3c_{2}(X̃) − 3e(\overline{C})
= 3c_{2}(X) − 3e(C).

The topological Euler number e(C) turns out to be equal to 2 − 2g − F C̃ + # Sing(C) by easy calculation. In the meantime, the left-hand side is the sum of three terms: Namely, C̃(C̃ + KX̃), F (F + KX̃) + 2F C̃ and KX̃(\overline{C} + KX̃).

The first term is nothing but 2g − 2.

Each connected component of F is a tree of ℙ^{1} of arithmetic genus 0, so that F (F +KX̃) = −2# Sing(C), meaning that the second term is −2e(C)− 4g + 4.

Since our resolution μ is minimal, the third term is bounded from below by CK_{X} + K_{X}^{2}.

Thus the left-hand side is bounded from below by CK_{X} + K_{X}^{2} − 2g + 2 − 2e(C), and hence we get

CK_{X} ≤ 3c_{2}(X) − c_{1}(X)^{2} − e(C) + 2g − 2.

It is easy to show the inequality

−e(X) ≤ \frac{1}{2}(C^{2} + CK_{X}) + g − 1.

(The equality is attained if and only if C has only ordinary double points.) Noticing the hypothesis C^{2} ≤ 0, we eventually get

CK_{X} ≤ 3c_{2}(X) − c_{1}(X)^{2} + \frac{CK_{X}}{2} + 3g − 3,

whence the assertion follows.

In Case 2, we have the following

Proposition 5.2. (1) If C^{2} > 0 and u ≥ 1/2, then

\frac{(1 − \frac{4(g−1)}{CK_{X}})^{3}}{8(1 − \frac{3(g−1)}{CK_{X}})^{2}} CK_{X} ≤ c_{2} − \frac{1}{4}c_{1}^{2}.

(2) If C^{2} > 0 and u ≥ (1/2) + g − 1, then

\frac{(1 − \frac{2(g−1)}{CK_{X}})^{3}}{8(1 − \frac{3(g−1)}{CK_{X}})^{2}} CK_{X} ≤ c_{2} − \frac{1}{4}c_{1}^{2}.

Proof. d ≥ (1 − g)/CK_{X} being the deficiency number, Corollary 3.6 tells us that

\frac{(u + 2d)^{3}}{(1 + 3d)^{2}} CK_{X} ≤ c_{2} − \frac{1}{4}c_{1}^{2}.

Therefore, if u ≥ 1/2, then

\frac{(1 + 4d)^{3}}{8(1 + 3d)^{2}} CK_{X} ≤ c_{2} − \frac{1}{4}c_{1}^{2}.

Putting x = (1 + 3d), we rewrite the coefficient of CK_{X} into

\frac{8}{27} · \frac{(x − \frac{1}{4})^{3}}{x^{2}} ,

which is, viewed as a function defined on the interval (0, ∞), monotone increasing. However, since d ≥ (1 − g)/CK_{X}, the minimum possible value of x is 1 − (3(g − 1)/CK_{X}), and we obtain

\frac{(1 − \frac{4(g−1)}{CK_{X}})^{3}}{8(1 − \frac{3(g−1)}{CK_{X}})^{2}} CK_{X} ≤ c_{2} − \frac{1}{4}c_{1}^{2}.

Similarly, if u ≥ (1/2) + g − 1, we get

\frac{(1 − \frac{2(g−1)}{CK_{X}})^{3}}{8(1 − \frac{3(g−1)}{CK_{X}})^{2}} CK_{X} ≤ c_{2} − \frac{1}{4}c_{1}^{2}.

What is left is Case 3. We will prove

Proposition 5.3. In Case 3, the inequality

CK_{X} ≤ 14^{3}(c_{2} − \frac{1}{4}c_{1}^{2})

holds.

Assume hereafter that C^{2} > 0, CK_{X} > max{2g − 2, 3g − 3}, u < min{1/2, 1/2 − (g − 1)/CK}. The last condition is of course equivalent to t = (1 − u) > max{1/2, 1/2 + (g − 1)}.

These conditions imply that a non-zero global section of Ω^{(n)}⟨nC⟩(−unK_{X}) is contained in a much smaller subsheaf. For instance, it is contained in

Ω^{(n)}⟨−\frac{n}{2}C⟩ + (df/f)Ω^{(n−1)}⟨−\frac{n}{2}C⟩

+ ⋯ + (df/f)^{n/2}Ω^{(n/2)}⟨−\frac{n}{2}C⟩

in view of Proposition 2.3.

The subsheaf

L_{C} = Ker(Ω → ν_{∗}ωC̃) = (dfO_{X} + fΩ)/fΩ

defines a natural parabolic structure on Ω. When we speak about a parabolic structure below, we always mean this structure on Ω.

Lemma 5.4. Let D ⊂ ℙ(Ω) be an irreducible effective divisor ∼ H + π^{∗}Δ, i.e. D is a rational section of π. If the slope of D with respect to the parabolic structure of Ω is one, then CΔ ≥ C^{2}.

Proof. D ∼ H + π^{∗}Δ is effective and irreducible if and only if O_{X}(−Δ) is a saturated subsheaf of Ω. σ(D) = 1 if and only if the restriction of O_{X}(−Δ) to C sits in the subsheaf L_{C} ≃ O_{C}(−C). Now the assertion is trivial.

Lemma 5.5. Let D ⊂ ℙ(Ω) be an irreducible effective divisor ∼ n(H + π^{∗}Δ), n ≥ 2. If D is a special irreducible divisor, then

O_{X}(−nΔ) ↪ dfΩ^{(n−1)} + fΩ^{(n)}.

Proof. By definition, at each smooth point p of C, there is a unique analytically local irreducible component D_{0} which has slope one. This component is independent of the choice of p, so that we infer that D_{0} defines an irreducible component of the restriction of D to a small analytic neighbourhood of C. This effective divisor D_{0} clearly defines an invertible subsheaf of dfΩ + fΩ near C, so that D gives a subsheaf of (dfΩ+fΩ)Ω^{(n−1)} near C. This implies the assertion.

By dividing out the both sides by the defining equation f, we rephrase the lemma above in terms of the filtration introduced in Section 1.

Corollary 5.6. Let D ∼ n(H +π^{∗}Δ) be an effective divisor. If ν = ν(D) is the number of the special irreducible components of D, then the vector space H^{0}(X, Ω^{(n)}⟨νC⟩(−νC − nΔ)) does not vanish.

Fix a sufficiently large integer n. Let u be a positive rational number such that

1) un is an integer;

2) H^{0}(X, Ω^{(n)}⟨nC⟩(−n(1 − u)K)) ≠ 0;

3) H^{0}(X, Ω^{(n)}⟨nC⟩(−(n(1 − u) + a)K)) = 0. Here a = a(X) is a fixed positive integer such that |aK| ≠ ∅ (we may assume that a = 1 unless p_{g} = 0).

Let v be the smallest non-negative rational number such that

H^{0}(X, Ω^{(n)}⟨nC⟩(−n(1 − u)K)) = H^{0}(X, Ω^{(n)}⟨vnC⟩(−n(1 − u)K)).

Any non-zero global section of Ω^{(n)}⟨nvC⟩(−n(1 − u)K) defines a non-zero section of Ω^{(n)}(−(1 − u)nK + vnC) or, equivalently, an effective divisor D ∼ n(H + π^{∗}(vC − (1 − u)K)).

Lemma 5.7. Assume that C^{2} > 0. In the notation above, we have an estimate of the number of the special irreducible components of D:

ν(D) ≤ nv + O(1).

Proof. Put ν = ν(D). Then by Lemma 5.6, we get

H^{0}(X, Ω^{(n)}⟨νC⟩(−νC − n(1 − u)K + nvC)) ≠ 0.

Since C^{2} > 0, we can find a positive integer c = c(X, C), independent of n, such that |cC − K| ≠ ∅. Therefore, by replacing cC by K, we get

H^{0}(X, Ω^{(n)}⟨νC⟩((nv − ν + ac)C − (n(1 − u) + a)K)) ≠ 0.

This is possible only when nv − ν + ac ≥ 0, i.e. ν ≤ nv + ac.

Proof of Proposition 5.3.

Let us recall results obtained in Sections 2 and 3 once again. We defined a rational number d by

deg 𝔨 = (1 + 2d)KC,

or, equivalently, −2d is the degree of the image of the natural map ν^{∗}Ω → ωC̃. Then (A) By Corollary 3.6, u, being defined as above, satisfies:

\frac{(u + 2d)^{3}}{(1 + 3d)^{2}} KC + \frac{1}{4}K_{X}^{2} − c_{2}(X) ≤ 0.

(B) By Lemma 3.8, v satisfies

v ≤ \frac{u + 2d}{1 + 2d}.

Pick up a non-zero element from H^{0}(X, Ω^{(n)}⟨vnC⟩(−(1−u)nK)). We can choose an element which is not contained in H^{0}(X, Ω^{(n)}⟨(vn − 1)C⟩(−(1 − u)nK)). This non-zero element determines an effective divisor D ≃ n(H + vC − (1 − u)K) on ℙ(E).

The Newton polygon of the divisor D ⊂ ℙ(Ω) is contained in the region given by

S ≥ \begin{cases}T, & 0 ≤ T ≤ nv \\ vn, & nv ≤ T ≤ n/2 \\ T − \frac{n}{2} + vn, & n/2 ≤ T ≤ n. \end{cases}

This region is not convex. Its convex hull is given by

S ≥ \begin{cases}2vT, & 0 ≤ T ≤ n/2 \\ T − \frac{n}{2} + vn, & n/2 ≤ T ≤ n. \end{cases}

Hence the minimum slope σ_{min} and the average slope \overline{σ} of D satisfy

σ_{min} ≥ 2v
\overline{σ} ≥ v + \frac{1}{2}.

By applying Corollary 4.4 and Lemma 5.7 to D, we get

2vnC^{2} ≥ 2(vnC − n(\frac{1}{2} − u)K)C

≥ \frac{C^{2}}{n} ∫_{0}^{n/2} σ_{min}(n − x)dx + \frac{C^{2}}{n} ∫_{n/2}^{n}(n − x)dx − \frac{ν(D)}{2}C^{2}

≥ \frac{3vnC^{2}}{4} + \frac{n}{8}C^{2} − \frac{vn}{2}C^{2} + O(1)

= (\frac{n}{4}v + \frac{n}{8})C^{2} + O(1).

Thus we get

v ≥ \frac{1}{14} + O(\frac{1}{n}),

provided u ≤ \frac{1}{2}. Hence by (B),

\frac{u + 2d}{1 + 2d} ≥ \frac{1}{14} + O(\frac{1}{n}),

or, equivalently,

u + 2d ≥ \frac{1 + 2d}{14} + O(\frac{1}{n}).

Then by the property (A) above,

\frac{(1 + 2d)^{3}}{14^{3}(1 + 3d)^{2}} CK + \frac{1}{4}K^{2} − c_{2} ≤ O(\frac{1}{n}).

The rational number d takes value in the interval [\frac{1−g}{CK} , ∞). It is easy to see that the function (1 + 2d)^{3}/(1 + 3d)^{2} defined on (−1/3, ∞) is monotone decreasing on (−1/3, 0] and monotone increasing on [0, ∞), and its minimum value is 1. Thus we eventually arrive at the inequlity as stated.

6. Concluding remarks

Our result above is obtained by crude estimates and by no means optimal. The coefficient 14^{3} = 2744 in our bound for the canonical degree of rational/elliptic curves is indeed fairly large, but it is not absurdly large.

Example. T. Shioda showed that an elliptic surface birational to the affine surface

y^{2} = x^{3} + at^{n}x + bt^{m}, 2m ≠ 3m, ab ≠ 0

has Mordell-Weil rank at most 56 and the maximum 56 is attained if and only if

m ≡ 0 mod 2
2m − 3n ≡ 0 mod 2^{3} · 3^{2} · 5 · 7

(see [14], Corollary 9). If we take m = 17, n = 7 and general coefficients a, b ∈ ℂ, the minimal model X of this surface is an elliptic surface of Kodaira dimension one and has numerical invariants c_{1}^{2} = 0, c_{2} = 36, with canonical divisor linearly equivalent to a single closed fibre of the elliptic fibration.

By a cyclic base change ℙ^{1} → ℙ^{1}, u ↦ t = u^{r}, the Mordell-Weil rank increases, and attains the maximum if and only if r is divisible by 2^{3} ·3^{2} ·5·7. Hence if one takes a “general” section of the fibre product ℙ^{1}×_{ℙ^{1}}X, its image C on the original surface X is a rational curve which is a 2^{3} · 3^{2} · 5 · 7-sheeted cover of the base curve ℙ^{1} or, equivalently,

CK_{X} = 2^{3} · 3^{2} · 5 · 7 = 70c_{2}(X).

Thus the coefficient of c_{2} must be large in a general estimate of canonical degree of curves. However, once one imposes suitable constraints on the singularity of C, then one gets a much better estimate.

For instance, Proposition 6.1. Let X be a minimal surface and C an irreducible curve with only ordinary double points as its singularities. Suppose that CK_{X} > 0 ( i.e. C is not a (−2)-curve.) Then

CK_{X} − 6(g − 1) − 8c_{2}(X) + 2c_{1}(X)^{2}

is bounded from above by a term of of order O(\frac{|g−1|}{CK_{X}} ). More precisely, we have:

(1) If g(C) = 0, then x = CK_{X} satisfies

x + O(1/x) = \frac{(1 + (2/x))^{3}}{(1 + (3/x))^{2}} x ≤ 8c_{2}(X) − 2c_{1}(X)^{2}.

(2) If g(C) = 1, then

CK_{X} ≤ 8c_{2}(X) − 2c_{1}(X)^{2}.

(3) If g(C) = g ≥ 2, x = CK_{X} satisfies one of the following two equations

x ≤ 6c_{2}(X) − 2c_{1}(X)^{2} + 6g − 6,
x − 6(g − 1) + O(\frac{g − 1}{x}) = \frac{(1 − 4(g − 1)/x)^{3}}{(1 − 3(g − 1)/x)^{2}} x ≤ 8c_{2} − 2c_{1}^{2}.

To prove this statement, it suffices to show that Case 3 in the beginning of Section 5 does not occur. By our assumption that C has only nodes, one checks that

Ω^{(n)}⟨−nC⟩ ≃ Sym^{n} Ω(log C)(−nC).

If Case 3 occurs, then it follows that

h^{0}(X, Ω^{(n)}⟨−nC⟩(\frac{n}{2}(C − K)))
= h^{0}(X, Sym^{n}(Ω(log C))(−\frac{n}{2}(C + K))) > O(n^{2}),

which contradicts the Bogomolov-semistability of the sheaf of the logarithmic 1-forms [2].

References

[1] F.A. Bogomolov, Families of curves on a surface of general type, Soviet Math. Dokl. 18 (1977), 1294–1297.

[2] —, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Izv. 13 (1978), 499–555.

[3] R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219.

[4] L. Caporaso, J. Harris and B. Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), 1–35.

[5] G. Harder and M.S. Narasimhan, On the cohomology groups of moduli space of vector bundles on curves, Math. Ann. 212 (1975), 215–248.

[6] S. Kobayashi, Hyperbolic complex spaces, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1997.

[7] S. Kobayashi and T. Ochiai, Meromorphic mappings into compact complex spaces of general type, Invent. Math. 31 (1975), 7–16.

[8] S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1987.

[9] S. Lu and Y. Miyaoka, Bounding curves in algebraic surfaces by genus and Chern numbers, Math. Research Letters 2 (1995), 663–676.

[10] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293 (1992), 77–99.

[11] S. Mori and S. Mukai, Mumford’s theorem on curves on K3 surfaces, Appendix to: The uniruledness of the moduli space of curves of genus 11, Springer Lect. Notes in Math. 1016 (1981), 334–353.

[12] F. Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), 89–120.

[13] C.S. Seshadri, Moduli of vector bundles with parabolic structures, Bull. Amer. Math. Soc. 83 (1977), 124–126.

[14] T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), 415–432.

[15] H. Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology 27 (1988), 429–442.

[16] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure & Appl. Math. 31 (1978), 339–411.

RIMS, Kyoto University, Kyoto 606-8502, Japan Current address: Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26,

53225 Bonn, Germany