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The intersection cohomology Hodge module of toric varieties

Hyunsuk Kim, Sridhar Venkatesh

TL;DR

This work develops a combinatorial framework to compute the Hodge filtration on the intersection cohomology Hodge module $\mathrm{IC}_X^{H}$ for toric varieties. By leveraging the Ishida complex and Saito's Decomposition Theorem, it expresses the graded de Rham pieces $\mathrm{gr}_k \mathrm{DR}_X \mathrm{IC}_X^{H}$ in terms of toric data and strata, culminating in an explicit formula for the graded de Rham components $\mathrm{dR}_{\mu,\tau}(K,L)$ that depends only on the combinatorics of the fan. The approach yields algorithmic procedures to compute higher direct images of reflexive differentials and to understand the Hodge structure on singular toric varieties, with explicit calculations in low dimensions and connections to previous K-theoretic results. This provides a systematic bridge between toric geometry, Hodge theory, and combinatorics, enabling practical and theoretical advances in the study of Hodge modules on toric spaces.

Abstract

We study the Hodge filtration of the intersection cohomology Hodge module for toric varieties. More precisely, we study the cohomology sheaves of the graded de Rham complex of the intersection cohomology Hodge module and give a precise formula relating it with the stalks of the intersection cohomology as a constructible complex. The main idea is to use the Ishida complex in order to compute the higher direct images of the sheaf of reflexive differentials.

The intersection cohomology Hodge module of toric varieties

TL;DR

This work develops a combinatorial framework to compute the Hodge filtration on the intersection cohomology Hodge module for toric varieties. By leveraging the Ishida complex and Saito's Decomposition Theorem, it expresses the graded de Rham pieces in terms of toric data and strata, culminating in an explicit formula for the graded de Rham components that depends only on the combinatorics of the fan. The approach yields algorithmic procedures to compute higher direct images of reflexive differentials and to understand the Hodge structure on singular toric varieties, with explicit calculations in low dimensions and connections to previous K-theoretic results. This provides a systematic bridge between toric geometry, Hodge theory, and combinatorics, enabling practical and theoretical advances in the study of Hodge modules on toric spaces.

Abstract

We study the Hodge filtration of the intersection cohomology Hodge module for toric varieties. More precisely, we study the cohomology sheaves of the graded de Rham complex of the intersection cohomology Hodge module and give a precise formula relating it with the stalks of the intersection cohomology as a constructible complex. The main idea is to use the Ishida complex in order to compute the higher direct images of the sheaf of reflexive differentials.
Paper Structure (13 sections, 16 theorems, 130 equations)

This paper contains 13 sections, 16 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hodge modules
  4. Toric varieties
  5. Polytopes and Shellability
  6. The Decomposition Theorem
  7. Higher direct images of Kähler differentials for toric morphisms
  8. Graded de Rham complex of the Intersection Cohomology Hodge module
  9. Explicit Formulas
  10. Dimension 0, 1, and 2
  11. Dimension 3
  12. Dimension 4.
  13. A reinterpretation of Theorem \ref{['theo:main-theo1.1']}

Key Result

Theorem 1.1

With the above notation, we have Moreover, $\widetilde{H}_{\mu, \tau}$ and hence $\mathop{\mathrm{dR}}\nolimits_{\mu, \tau}$, can be computed explicitly in an algorithmic way.

Theorems & Definitions (40)

  • Theorem 1.1: Main Theorem = Theorem \ref{['theorem:main-theorem-grDR']}
  • Remark 1.2
  • Proposition 2.1: saito1990mixedHodgemodules*4.5.4
  • Theorem 2.2: Saito's decomposition theorem
  • Lemma 2.3
  • Remark 2.4
  • Proposition 2.5: dCMM-toricmaps*Lemma 2.6, Proposition 2.7
  • Proposition 2.6: dCMM-toricmaps*Theorem B, Corollary C
  • Remark 2.7
  • Remark 2.8
  • ...and 30 more