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Virtual classes via vanishing cycles

Tasuki Kinjo

Abstract

We develop a new method to construct the virtual fundamental classes for quasi-smooth derived schemes using the perverse sheaves of vanishing cycles on their $-1$-shifted contangent spaces. It is based on the author's previous work that can be regarded as a version of the Thom isomorphism for $-1$-shifted cotangent spaces. We use the Fourier-Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend-Fantechi and Li-Tian, under the quasi-projectivity assumption. We also discuss an approach to construct DT4 virtual classes for $-2$-shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.

Virtual classes via vanishing cycles

Abstract

We develop a new method to construct the virtual fundamental classes for quasi-smooth derived schemes using the perverse sheaves of vanishing cycles on their -shifted contangent spaces. It is based on the author's previous work that can be regarded as a version of the Thom isomorphism for -shifted cotangent spaces. We use the Fourier-Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend-Fantechi and Li-Tian, under the quasi-projectivity assumption. We also discuss an approach to construct DT4 virtual classes for -shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.

Paper Structure

This paper contains 30 sections, 26 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. History
  3. Results
  4. Strategy of the proof
  5. Microlocal interpretation
  6. Joyce's conjecture
  7. Structure of the paper
  8. Preliminaries on sheaf operations
  9. Base change maps
  10. Natural transformations between four functors
  11. Vanishing cycle and nearby cycle functors
  12. Fourier--Sato transforms
  13. Specialization functor and specialization map
  14. Specialization functors
  15. Specialization maps
  16. ...and 15 more sections

Key Result

Theorem 1.1

Assume ${\mathbb L}_{\boldsymbol{Y}} |_Y$ has a global resolution by a two-term complex of locally free sheaves. Then the following equality in $\mathop{\mathrm{\mathrm{H}}}\nolimits^{\mathrm{BM}}_{2\mathop{\mathrm{vdim}}\nolimits{\boldsymbol{Y}}}(Y)$ holds: where $\mathop{\mathrm{cl}}\nolimits_Y \colon A_{\mathop{\mathrm{vdim}}\nolimits \boldsymbol{Y}}(Y) \to \mathop{\mathrm{\mathrm{H}}}\nolimit

Theorems & Definitions (50)

  • Theorem 1.1: = Theorem \ref{['thm:main']}
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 3.1: Ver81
  • Proposition 3.2: Ver81
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • ...and 40 more