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Simple obstructions and cone reduction

F. Qu

TL;DR

This work shows that for a Deligne-Mumford stack $X$ locally of finite type over a field of characteristic zero, the intrinsic normal cone $C_X$ is contained in the abelian cone $\\mathbb{V}(\\Omega_X^1[-1])$, enabling cone reduction by cosections without requiring a global obstruction theory. It provides a concise, global perspective on obstructions by relating simple obstructions to $\\Ext^1_k(\\xi^*\\Omega_X^1,k)$ and extends the reduction to the relative setting via $C_f$ for maps $f: X\\to Y$ with $Y$ smooth. In addition, the paper constructs a global semiregularity map for sheaves using the Atiyah class, and proves vanishing results for simple extensions under this map, placing local obstruction data into a global derived-framework context. Together, these results unify obstruction theory and cone-reduction methods, with implications for virtual class constructions and relative obstruction theories in moduli problems.

Abstract

Let $X$ be a Deligne-Mumford stack locally of finite type over an algebraically closed field $k$ of characteristic zero. We show that the intrinsic normal cone $C_X$ of $X$ is supported in the subcone $\mathbb{V}(Ω_X[-1])$ ($h^1/h^0((Ω^1_X)^\vee)$) of its intrinsic normal sheaf $N_X$. This leads to an alternative proof of cone reduction by cosections for $C_X$. We also discuss vanishing of simple obstructions under the Buchweitz-Flenner semiregularity map for sheaves.

Simple obstructions and cone reduction

TL;DR

This work shows that for a Deligne-Mumford stack locally of finite type over a field of characteristic zero, the intrinsic normal cone is contained in the abelian cone , enabling cone reduction by cosections without requiring a global obstruction theory. It provides a concise, global perspective on obstructions by relating simple obstructions to and extends the reduction to the relative setting via for maps with smooth. In addition, the paper constructs a global semiregularity map for sheaves using the Atiyah class, and proves vanishing results for simple extensions under this map, placing local obstruction data into a global derived-framework context. Together, these results unify obstruction theory and cone-reduction methods, with implications for virtual class constructions and relative obstruction theories in moduli problems.

Abstract

Let be a Deligne-Mumford stack locally of finite type over an algebraically closed field of characteristic zero. We show that the intrinsic normal cone of is supported in the subcone () of its intrinsic normal sheaf . This leads to an alternative proof of cone reduction by cosections for . We also discuss vanishing of simple obstructions under the Buchweitz-Flenner semiregularity map for sheaves.
Paper Structure (9 sections, 8 theorems, 44 equations)

This paper contains 9 sections, 8 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarse sheaf and support
  4. $\pi_0$
  5. Support of closed substacks
  6. The functor $h^1/h^0((-)^\vee)$
  7. Simple extensions and obstructions
  8. Cone reduction for $C_X$
  9. semiregularity map for sheaves

Key Result

Lemma 2.1

Let $M \to N$ be a fully faithful map of stacks, then \begin{tikzcd} M \ar[r]\ar[d] & N \ar[d]\\ \pi_0(M) \ar[r] & \pi_0(N) \end{tikzcd}is a cartesian diagram in the $(2,1)$-category of stacks over $\mathbf{Sch}_k$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 13 more