[Paper Review] Low degree Hodge theory for klt varieties

Martin Schwald|arXiv (Cornell University)|Dec 6, 2016

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper establishes that for complex projective varieties with klt singularities, the first and second singular cohomology groups carry pure Hodge structures, with Hodge components isomorphic to spaces of reflexive differential forms. It proves a Lefschetz (1,1) theorem and a weak Hodge-Riemann bilinear relation for klt varieties, extending classical Hodge theory to singular settings via reflexive forms and resolution techniques.

ABSTRACT

If X is a complex projective variety with klt singularities, then the mixed Hodge structures on the first two singular cohomology groups are pure. We describe the pieces of the Hodge decomposition in terms of reflexive differential forms. Applications include a Lefschetz (1,1) Theorem and a weak analogue of the Hodge-Riemann bilinear relations for klt varieties.

Motivation & Objective

• To extend classical Hodge theory to singular complex projective varieties with klt singularities.
• To describe the Hodge decomposition of $\mathrm{H}^1(X,\mathbb{C})$ and $\mathrm{H}^2(X,\mathbb{C})$ in terms of reflexive differential forms $\Omega^{[p]}_X$.
• To establish a Lefschetz (1,1) theorem for klt varieties, identifying integral classes in $\mathrm{H}^{1,1}(X)$ with first Chern classes of line bundles.
• To prove a weak analogue of the Hodge-Riemann bilinear relations for klt varieties, restricted to forms of degree one and two.

Proposed method

• Using the extension theorem for reflexive differential forms on klt varieties to lift holomorphic forms from the smooth locus to the whole variety.
• Applying resolution of singularities and pullback functors to relate cohomology classes on $X$ to those on a resolution $\widetilde{X}$.
• Employing the local Hodge-Riemann bilinear relations on smooth manifolds to derive global inequalities on singular varieties via pullbacks.
• Defining a sesquilinear form $\psi_{X,a}$ on cohomology using cup products with an ample class $a$ and volume forms.
• Proving that the positivity of the Hodge-Riemann form holds on the primitive part of $\mathrm{H}^{p,q}(X)$ when pulled back to a resolution.
• Using the canonical isomorphisms $\kappa_{p0}: \mathrm{H}^{p,0}(X) \to \mathrm{H}^0(X, \Omega^{[p]}_X)$ and $\kappa_{0p}: \mathrm{H}^{0,p}(X) \to \mathrm{H}^p(X, \mathcal{O}_X)$ to relate Hodge components to reflexive forms.

Experimental results

Research questions

• RQ1 How do the Hodge structures on $\mathrm{H}^1(X,\mathbb{C})$ and $\mathrm{H}^2(X,\mathbb{C})$ decompose for a klt variety $X$?
• RQ2 Can the Lefschetz (1,1) theorem be extended to klt singularities?
• RQ3 Is there a meaningful analogue of the Hodge-Riemann bilinear relations in the singular klt setting?
• RQ4 How do reflexive differential forms $\Omega^{[p]}_X$ relate to the Hodge decomposition of cohomology groups on klt varieties?

Key findings

• The Hodge structure on $\mathrm{H}^p(X,\mathbb{C})$ for $p=1,2$ is pure of weight $p$ when $X$ has klt singularities.
• There are canonical isomorphisms $\kappa_{p0}: \mathrm{H}^{p,0}(X) \to \mathrm{H}^0(X, \Omega^{[p]}_X)$ and $\kappa_{0p}: \mathrm{H}^{0,p}(X) \to \mathrm{H}^p(X, \mathcal{O}_X)$ for $p=1,2$.
• The Lefschetz (1,1) theorem holds for klt varieties: $\mathrm{H}^{1,1}(X) \cap H^2(X,\mathbb{Z})$ consists precisely of first Chern classes of line bundles on $X$.
• A weak Hodge-Riemann bilinear relation holds for klt varieties: $i^{p-q} \cdot \psi_{X,a}(\alpha,\alpha) > 0$ for non-zero $\alpha$ in the primitive part of $\mathrm{H}^k(X,\mathbb{C})$ that is a cup product of classes from $\mathrm{H}^1(X,\mathbb{C})$ and $\mathrm{H}^2(X,\mathbb{C})$.
• The result implies that every non-trivial fibration of an irreducible symplectic variety $X$ is a Lagrangian fibration with $\dim B = \frac{1}{2}\dim X$ and $\rho(B) = 1$.