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Semiregularity as a consequence of Goodwillie's theorem

J. P. Pridham

Abstract

We realise Buchweitz and Flenner's semiregularity map (and hence a fortiori Bloch's semiregularity map) for a smooth variety $X$ as the tangent of a generalised Abel--Jacobi map on the derived moduli stack of perfect complexes on $X$. The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie's theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex). Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if $X$ is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov--Witten and Pandharipande--Thomas invariants. We also give generalisations allowing $X$ to be singular, and even a derived stack.

Semiregularity as a consequence of Goodwillie's theorem

Abstract

We realise Buchweitz and Flenner's semiregularity map (and hence a fortiori Bloch's semiregularity map) for a smooth variety as the tangent of a generalised Abel--Jacobi map on the derived moduli stack of perfect complexes on . The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie's theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex). Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov--Witten and Pandharipande--Thomas invariants. We also give generalisations allowing to be singular, and even a derived stack.

Paper Structure

This paper contains 14 sections, 20 theorems, 72 equations.

Table of Contents

  1. The Abel--Jacobi map for rings
  2. The Abel--Jacobi map
  3. Homogeneity and obstructions
  4. Goodwillie's theorem
  5. The semiregularity map
  6. dg algebras and dg categories
  7. The Abel--Jacobi map for (derived) schemes and stacks
  8. Derived de Rham cohomology
  9. Global constructions
  10. Perfect complexes
  11. Derived de Rham cohomology
  12. Generalised Abel--Jacobi maps
  13. Horizontal sections
  14. Reduced obstructions

Key Result

Theorem 1

Take a local Artinian $\mathbb{C}$-algebra $A$, a smooth morphism $X \to \mathrm{Spec}\, A$ of Artin stacks and square-zero ideal $I \subset A$ with quotient $B= A/I$. Then for any perfect complex $\mathscr{F}$ over $X':=X\otimes_AB$, with obstruction $o(\mathscr{F}) \in \mathbb{E}\mathrm{xt}^{2}_{\ lies in $F^p\mathrm{H}^{2p}(X,\Omega^{\bullet}_{X/A})$ if and only if $o(\mathscr{F})$ maps to zero

Theorems & Definitions (67)

  • Theorem
  • Theorem
  • Definition 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • ...and 57 more