Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classical and irregular Hodge numbers

Yichen Qin, Dingxin Zhang

Abstract

Let $U$ be a smooth quasi-projective complex variety with a regular function $f$. The twisted de Rham cohomology groups $\mathrm{H}^k_{\mathrm{dR}}(U, f)$ carry the decreasing irregular Hodge filtration, whose graded pieces have dimensions known as the irregular Hodge numbers. In this paper, we prove that the irregular Hodge numbers admit an explicit characterization in terms of classical Hodge numbers, closely related to Hodge-theoretic numbers constructed by Katzarkov, Kontsevich, and Pantev for Landau--Ginzburg models. As direct applications, we show that irregular Hodge numbers of non-degenerate functions are independent of the choice of non-degenerate functions, and we give a concrete formula for irregular Hodge numbers for unipotent non-degenerate functions.

Classical and irregular Hodge numbers

Abstract

Let be a smooth quasi-projective complex variety with a regular function . The twisted de Rham cohomology groups carry the decreasing irregular Hodge filtration, whose graded pieces have dimensions known as the irregular Hodge numbers. In this paper, we prove that the irregular Hodge numbers admit an explicit characterization in terms of classical Hodge numbers, closely related to Hodge-theoretic numbers constructed by Katzarkov, Kontsevich, and Pantev for Landau--Ginzburg models. As direct applications, we show that irregular Hodge numbers of non-degenerate functions are independent of the choice of non-degenerate functions, and we give a concrete formula for irregular Hodge numbers for unipotent non-degenerate functions.
Paper Structure (22 sections, 15 theorems, 95 equations)

This paper contains 22 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Hodge numbers of Landau--Ginzburg models
  4. Deformation invariance of irregular Hodge numbers
  5. Organization
  6. Preliminaries
  7. Mixed Hodge modules
  8. Rees modules
  9. Exponential mixed Hodge structures
  10. Irregular Hodge filtrations on the de Rham fibers
  11. Stationary phase formula
  12. Proof of the stationary phase formula
  13. Classical, irregular, and Landau--Ginzburg Hodge numbers
  14. Classical and irregular Hodge numbers
  15. A conjecture of Katzarkov--Kontsevich--Pantev
  16. ...and 7 more sections

Key Result

Theorem 1.1.1

With notation as above, for any $\alpha$ in $[0,1)$ and $p \in \mathbb{Z}$, we have where $\lambda=\exp(-2\pi\mathrm{i}\alpha)$.

Theorems & Definitions (43)

  • Theorem 1.1.1: \ref{['thm:KKP-numbers']}
  • Remark 1.2.1
  • Corollary 1.2.2: \ref{['prop:kkp']}
  • Example 1.2.3
  • Theorem 1.3.1
  • Proposition 3.0.1
  • Lemma 3.1.1
  • proof
  • proof
  • Corollary 3.1.2
  • ...and 33 more