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The Atiyah class of DG manifolds of amplitude $+1$

Seokbong Seol

TL;DR

This work identifies the Atiyah class of DG manifolds of amplitude $+1$ with a geometric criterion: the class vanishes exactly when the derived intersection encoded by $(E[-1],\iota_s)$ is clean. The authors establish locality of the Atiyah class in positive amplitude, compute the cocycle globally and in local coordinates, and prove that nowhere-vanishing sections trivialize the class. A central application shows that the Atiyah class of the derived intersection of two submanifolds $X,Y$ vanishes precisely when $X$ and $Y$ intersect cleanly, linking derived geometry to classical intersection theory. The results pave the way for extending such analyses to higher positive amplitudes and exploring invariance under DG-equivalences. The methodology relies on explicit local normal forms, horizontal lifts from affine connections, and a partition-of-unity argument to glue local data into a global invariant.

Abstract

A DG manifold of amplitude $+1$ encodes the derived intersection of a section $s$ and the zero section of a vector bundle $E$. In this paper, we compute the Atiyah class of DG manifolds of amplitude $+1$. In particular, we show that the Atiyah class vanishes if and only if the intersection of $s$ with the zero section is a clean intersection. As an application, we study the Atiyah class of DG manifolds that encodes the derived intersection of two smooth manifolds.

The Atiyah class of DG manifolds of amplitude $+1$

TL;DR

This work identifies the Atiyah class of DG manifolds of amplitude with a geometric criterion: the class vanishes exactly when the derived intersection encoded by is clean. The authors establish locality of the Atiyah class in positive amplitude, compute the cocycle globally and in local coordinates, and prove that nowhere-vanishing sections trivialize the class. A central application shows that the Atiyah class of the derived intersection of two submanifolds vanishes precisely when and intersect cleanly, linking derived geometry to classical intersection theory. The results pave the way for extending such analyses to higher positive amplitudes and exploring invariance under DG-equivalences. The methodology relies on explicit local normal forms, horizontal lifts from affine connections, and a partition-of-unity argument to glue local data into a global invariant.

Abstract

A DG manifold of amplitude encodes the derived intersection of a section and the zero section of a vector bundle . In this paper, we compute the Atiyah class of DG manifolds of amplitude . In particular, we show that the Atiyah class vanishes if and only if the intersection of with the zero section is a clean intersection. As an application, we study the Atiyah class of DG manifolds that encodes the derived intersection of two smooth manifolds.
Paper Structure (13 sections, 27 theorems, 145 equations)

This paper contains 13 sections, 27 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Background on DG manifolds
  3. The Atiyah class and clean intersection
  4. Clean intersection
  5. Locality of the Atiyah class in positive amplitude
  6. The Atiyah class of DG manifolds of amplitude $+1$
  7. Vector fields and affine connections
  8. Computation of the Atiyah class: global
  9. Computation of the Atiyah class: local
  10. Proof of Theorem \ref{['thm:Main']}
  11. Illustration of the proof
  12. Proof of \ref{['item:A']} $\Rightarrow$ \ref{['item:B']}
  13. The case of derived intersections

Key Result

Theorem 2

Let $E$ be a vector bundle and $s$ a section of $E$. Then the Atiyah class of the DG manifold $(E[-1], \iota_{s})$ vanishes if and only if the intersection of $s$ with the zero section is clean.

Theorems & Definitions (60)

  • Theorem 2: Theorem \ref{['thm:Main']}
  • Theorem 3: Theorem \ref{['thm:DerivedIntersection']}
  • Remark 2.1
  • Example 2.2
  • Proposition 2.3: MR3319134
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 50 more