Hodge-Riemann bilinear relations for Schur classes of ample vector bundles
Julius Ross, Matei Toma
备注：
v2. Two principal changes are (1) a generalisation of higher-rank Khovanskii-Tessier inequalities to Schur classes (Theorem 1.4) and (2) the inclusion of an application to cones of Nef cycles on self-products of a very general principally polarized abelian surface (Section 6) v3. Improved statement on derived Schur classes. Answers to some previously asked questions and examples provided
Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le λ_N\le λ_{N-1} \le \cdots \le λ_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class $s_λ(E)\in H^{d-2,d-2}(X)$ has the Hard Lefschetz property and satisfies the Hodge-Riemann bilinear relations. As a consequence we obtain various new inequalities between characteristic classes of ample vector bundles, including a higher-rank version of the Khovanskii-Teissier inequalities.
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type.
Let $X$ be a smooth projective variety of dimension $n$, and let $E$ be an ample vector bundle over $X$. We show that any non-zero Schur class of $E$, lying in the cohomology group of bidegree $(n-1, n-1)$, has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton-Lazarsfeld for certain degrees.
We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.
In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.
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Let $Y$ be a smooth complex projective variety of dimension $N+1$, $L$ an invertible sufficiently ample sheaf, $X\in |L|$ a smooth hypersurface and $λ\in F^kH^N(X,C)$ a vanishing cohomology class, where $F^{*}$ is the Hodge filtration and $k\in\{1,...,[N/2]\}$. Assume that $L$ is sufficiently ample and that the codimension in $|L|$ of the Hodge variety associated to $λ$ (locally defined as the locus where the image of $λ$ by flat transport over $|L|$ remains in $F^k$) is sufficiently small. I show that this forces $N$ to be even and $k=[N/2]$, and that the class $λ$ is a linear combination wit…
Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle $E$ on a complex manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$ has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample on a projective manifold, the metric can be made smooth and Griffiths positive.
Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".
Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomol…
Let (S,H) be a rational algebraic surface with an ample divisor. We compute generating functions for the Hodge numbers of the moduli spaces of H-stable rank 2 sheaves on S in terms of certain theta functions for indefinite lattices that were introduced in the paper alg-geom/9612020 written jointly with Don Zagier. If H lies in the closure of the ample cone and has self-intersection 0, it follows that the generating functions are Jacobi forms. In particular the generating functions for the Euler numbers have a similar transformation behaviour under SL(2,Z) as that predicted in Vafa and Witten: …
The main problem addressed in the paper is the Torelli problem for n-dimensional varieties of general type, more specifically for varieties with ample canonical bundle. It asks under which geometrical condition for a variety the period map for the Hodge structure of weight n is a local embedding. We define a line bundle to be almost very ample iff the associated linear system is base point free and yields an injective morphism. We define instead a line bundle to be quasi very ample if it yields a birational morphism which is a local embedding on the complement of a finite set. Our main result …
According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…
We show that the cobordism class of a polarization of Hodge module defines a natural transformation from the Grothendieck group of Hodge modules to the cobordism group of self-dual bounded complexes with real coefficients and constructible cohomology sheaves in a compatible way with pushforward by proper morphisms. This implies a new proof of the well-definedness of the natural transformation from the Grothendieck group of varieties over a given variety to the above cobordism group (with real coefficients). As a corollary, we get a slight extension of a conjecture of Brasselet, Schürmann and Y…
Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $π:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of any proper smooth multisection are determined by the degree $d$ of that multi-section as are the Hodge numbers of any smooth complete intersection of multi-sections of degrees $\left(d_{1},\ldots,d_{r}\right)$. In this paper recursive formulae are given for those Hodge numbers in terms of the integers $\left\{ d_{1},\ldots,d_{r}\right\} $ and the Hodge numbers …
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.
We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favour for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type resu…
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This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive coho…
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We ap…
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V…
We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves. We show that for families of manifolds with ample canonical bundle this invariant is uniformly bounded. As a consequence we establish that such families over a base of arbitrary dimension verify the aforementioned Arakelov inequality, answering a question of Viehweg.
Fujita's second theorem for Kähler fibre spaces over a curve asserts that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our negative answer (\cite{cd}) to a question by Fujita: is $V$ semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group $(\mathbb Z/…
For ample vector bundles $E$ over compact complex varieties $X$ and a Schur functor $S_I$ corresponding to an arbitrary partition $I$ of the integer $|I|$, one would like to know the optimal vanishing theorem for the cohomology groups $H^{p,q}(X, S_I(E))$, depending on the rank of $E$ and the dimension $n$ of $X$. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition $I$ is a hook. Here we give a simpler proof of this theorem. We also treat the same problem under weaker positivity assumptions, in particular under …
Consider a family f:A --> U of g-dimensional abelian varieties over a quasiprojective manifold U. Suppose that the induced map from U to the moduli scheme of polarized abelian varieties is generically finite and that there is a projective manifold Y, containing U as the complement of a normal crossing divisor S, such that the sheaf of logarithmic one forms is nef and that its determinant is ample with respect to U. We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map from U to the moduli scheme or…
The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains a…
Let $X$ be a non-singular quasi-projective variety over a field, and let $\mathcal E$ be a vector bundle over $X$. Let $\mathbb G_X({d}, \mathcal E)$ be the Grassmann bundle of $\mathcal E$ over $X$ parametrizing corank $d$ subbundles of $\mathcal E$, and denote by $θ$ the Plücker class of $\mathbb G_X({d}, \mathcal E)$, that is, the first Chern class of the universal quotient bundle over $\mathbb G_X({d}, \mathcal E)$. In this short note, a closed formula for the push-forward of powers of $θ$ is given in terms of the Schur polynomials in Segre classes of $\mathcal E$, which yields a degree fo…
In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for complex Finsler vector bundles with positive Kobayashi curvature.
Constant coefficient and intersection complex $L$-classes of projective varieties
IKERBASQUE, Basque Foundation for Science（伊克尔巴斯奎，巴斯克科学基金会）；BCAM, Basque Center for Applied Mathematics（巴斯克应用数学中心）；UPV/EHU（巴斯克国立大学）；University of Cantabria（坎塔布里亚大学）；RIMS Kyoto University（京都大学数理解析研究所）
