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Twisted jets, motivic measure and orbifold cohomology

Takehiko Yasuda

Abstract

We introduce the notion of twisted jet on a Deligne-Mumford stack. and study the motivic measure on the space of the twisted $\infty$-jets on a smooth Deligne-Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology with Hodge structure.

Twisted jets, motivic measure and orbifold cohomology

Abstract

We introduce the notion of twisted jet on a Deligne-Mumford stack. and study the motivic measure on the space of the twisted -jets on a smooth Deligne-Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology with Hodge structure.

Paper Structure

This paper contains 19 sections, 25 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Motivic measure; review
  3. Completing Grothendieck rings
  4. Jets on schemes
  5. Motivic measure
  6. The transformation rule
  7. The motivic Gorenstein measure
  8. Twisted jets
  9. Non-twisted jets on stacks
  10. Twisted jets
  11. The formal neighborhood of $I(\mathscr X)$ and its canonical automorphism
  12. Shift number and orbifold cohomology
  13. The motivic measure on twisted $\infty$-jets
  14. Main theorem
  15. General results on Deligne-Mumford stacks
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ and $X'$ be smooth complete varieties. Suppose that there are proper birational morphisms $Z\to X$ and $Z\to X'$ such that $K_{Z/X}=K_{Z/X'}$. Then the rational cohomologies of $X$ and $X'$ have the same Hodge structures.

Theorems & Definitions (89)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5: =Corollary \ref{['cor-of-main']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 79 more