Hodge-Riemann Bilinear Relations Research Articles

Equivariant log-concavity and equivariant Kähler packages

We show that the exterior algebra ΛR[α1,⋯,αn], which is the cohomology of the torus T=(S1)n, and the polynomial ring R[t1,…,tn], which is the cohomology of the classifying space B(S1)n=(CP∞)n, are Sn-equivariantly log-concave. We do so by explicitly giving the Sn-representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting.

Hodge-Riemann bilinear relations for Schur classes of ample vector bundles

Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class $s_{\lambda}(E)\in H^{d-2,d-2}(X)$ 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.

Hodge–Riemann Relations for Schur Classes in the Linear and Kähler Cases

Abstract We prove a version of the Hodge–Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.

Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies

Very recently J. Kotrbatý[14] has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge-Riemann bilinear relations in Kähler geometry. The goal of this note is to prove several inequalities for mixed volumes of convex bodies using Kotrbatý’s theorem as the main tool.

On higher-rank Khovanskii-Teissier inequalities

We shall discuss a higher-rank Khovanskii-Teissier inequality, generalizing a theorem of Li in [14]. In the course of the proof, we develop new Hodge-Riemann bilinear relations in certain mixed and degenerate settings, which in themselves slightly extend the existing results and imply new Khovanskii-Teissier type inequalities and log-concavity results.

Mixed Hodge-Riemann bilinear relations and m-positivity

Motivated by our previous work on Hodge-index type theorems, we give a form of mixed Hodge-Riemann bilinear relation by using the notion of m-positivity, whose proof is an adaptation of the works of Timorin (1998) and Dinh and Nguyên (2006). This mixed Hodge-Riemann bilinear relation holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate along some directions. In particular, it applies to fibrations of compact Kähler manifolds.

Relative Hard Lefschetz theorem for fans

We prove the Relative Hard Lefschetz theorem and the Relative Hodge-Riemann bilinear relations for combinatorial intersection cohomology sheaves on fans.

Homogeneity of cohomology classes associated with Koszul matrix factorizations

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$. To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

Local Hodge theory of Soergel bimodules

We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kubel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.

Cauchy–Schwarz-type inequalities on Kähler manifolds-II

We establish in this note some Cauchy-Schwarz-type inequalities on compact K\"{a}hler manifolds, which generalize the classical Khovanskii-Teissier inequalities to higher-dimensional cases. Our proof is to make full use of the mixed Hodge-Riemann bilinear relations due to Dinh and Nguy$\hat{\text{e}}$n. A proportionality problem related to our main result is also proposed.

Hochster's theta invariant and the Hodge–Riemann bilinear relations

Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor i R ( M , N ) ≅ Tor i + 2 R ( M , N ) for i ≫ 0 ). Since R has only an isolated singularity, these torsion modules are of finite length for i ≫ 0 . The theta invariant of the pair ( M , N ) is defined by Hochster to be length ( Tor 2 i R ( M , N ) ) − length ( Tor 2 i + 1 R ( M , N ) ) for i ≫ 0 . H. Dao has conjectured that the theta invariant is zero for all pairs ( M , N ) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj ( R ) .

Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

The Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in various contexts: they impose restrictions on the cohomology algebra of a smooth compact Kahler manifold; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the f-vectors of convex polytopes. While the statements of these theorems depend on the choice of a Kahler class, or its analog, there is usually a cone of possible choices. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note, we present a unified approach to proving the mixed HLT and HRR, generalizing the known results, and proving it in new cases, such as the intersection cohomology of nonrational polytopes.

The mixed Hodge–Riemann bilinear relations for compact Kähler manifolds

We prove the Hodge–Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes

The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [9]. This theorem implies the conjecture of R. Stanley on the unimodularity of the generalized h-vector. In this paper we strengthen Karu's theorem by introducing a canonical bilinear form (.,.) Φ on the intersection cohomology IH(Φ) of a complete fan Φ and proving the Hodge-Riemann bilinear relations for (.,.) Φ .

The Hard Lefschetz Theorem and the topology of semismall maps

A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky–MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge–Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps, Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.

The mixed Hodge-Riemann bilinear relations in the linear situation

The mixed Hodge-Riemann bilinear relations in the linear situation

Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds

A manifold (M;I;J;K) is called hypercomplex if I;J;K are complex structures satisfying quaternionic relations. A quaternionic Hermitian hypercomplex manifold is called HKT (hyperkahler with torsion) if the (2,0)-form associated with the corresponding Sp(n)-structure satises @ = 0. A Hermitian metric ! on a complex manifold is called bal- anced if d ! = 0. We show that balanced HKT metrics are precisely the quaternionic Calabi-Yau metrics dened in terms of the quaternionic Monge-Amp ere equation. In particular, a balanced HKT-metric is unique in its cohomology class, and it always exists if the quaternionic Calabi-Yau theorem is true. We investigate the cohomological properties of strong HKT metrics (the quaternionic Hermitian metrics, satisfying, in addition to the HKT condition, the relation dd c ! = 0), and show that the space of strong HKT metrics is nite-dimensional. Using Howe's duality for rep- resentations of Sp(n), we prove a hyperkahler version of Hodge-Riemann bilinear relations. We use it to show that a manifold admitting a balanced HKT-metric never admits a strong HKT-metric, if dimRM > 12.