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Stringy Hodge numbers via crepant resolutions by Artin stacks

Matthew Satriano, Jeremy Usatine

Abstract

In a previous paper we showed that any variety with log-terminal singularities admits a crepant resolution by a smooth Artin stack. In this paper we prove the converse, thereby proving that a variety admits a crepant resolution by a smooth Artin stack if and only if it has log-terminal singularities. Furthermore if $\mathcal{X} \to Y$ is such a resolution, we obtain a formula for the stringy Hodge numbers of $Y$ in terms of (motivically) integrating an explicit weight function over twisted arcs of $\mathcal{X}$. That weight function takes only finitely many values, so we believe this result provides a plausible avenue for finding a long-sought cohomological interpretation for stringy Hodge numbers. Using that the resulting integral is defined intrinsically in terms of $\mathcal{X}$, we also obtain a notion of stringy Hodge numbers for smooth Artin stacks, that in particular, recovers Chen and Ruan's notion of orbifold Hodge numbers.

Stringy Hodge numbers via crepant resolutions by Artin stacks

Abstract

In a previous paper we showed that any variety with log-terminal singularities admits a crepant resolution by a smooth Artin stack. In this paper we prove the converse, thereby proving that a variety admits a crepant resolution by a smooth Artin stack if and only if it has log-terminal singularities. Furthermore if is such a resolution, we obtain a formula for the stringy Hodge numbers of in terms of (motivically) integrating an explicit weight function over twisted arcs of . That weight function takes only finitely many values, so we believe this result provides a plausible avenue for finding a long-sought cohomological interpretation for stringy Hodge numbers. Using that the resulting integral is defined intrinsically in terms of , we also obtain a notion of stringy Hodge numbers for smooth Artin stacks, that in particular, recovers Chen and Ruan's notion of orbifold Hodge numbers.

Paper Structure

This paper contains 20 sections, 60 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Dreaming of a cohomology theory
  4. Methods and the warping stack
  5. Preliminaries
  6. Coherent sheaves on twisted discs
  7. The weight function and motivic integration over twisted arcs
  8. Weight functions
  9. Measurable sets and integrable functions
  10. Height and order functions
  11. The locus of syntomic warps
  12. Triviality of forms of (truncated) twisted discs
  13. The fibers of the twisted-to-warped jet space map
  14. Initial computation of the fibers
  15. Automorphisms of truncated twisted discs
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic 0, and let $Y$ be an irreducible finite type scheme over $k$ with log-terminal singularities. Then there exists a smooth irreducible finite type Artin stack $\mathcal{X}$ over $k$ and a crepant resolution $\mathcal{X} \to Y$ with affine diag

Theorems & Definitions (153)

  • Conjecture 1.1: Batyrev1998
  • Theorem 1.1: SatrianoUsatine3
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.1
  • Definition 1.2
  • Lemma 2.1
  • ...and 143 more