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Cobordism-valued intersection theory on $\overline{\mathcal{M}}_{0,n}$

Benjamin Ellis-Bloor

Abstract

We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued psi-class intersections on $\overline{\mathcal{M}}_{0,n}$, and in particular the cobordism classes $[\overline{\mathcal{M}}_{0,n}]$, and for their images in $K$-theory. Explicit formulas are given up to $n = 8$.

Cobordism-valued intersection theory on $\overline{\mathcal{M}}_{0,n}$

Abstract

We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued psi-class intersections on , and in particular the cobordism classes , and for their images in -theory. Explicit formulas are given up to .
Paper Structure (20 sections, 18 theorems, 127 equations, 6 tables)

This paper contains 20 sections, 18 theorems, 127 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Calculations
  3. Outline of the proof
  4. Organization
  5. Algebraic cobordism
  6. Basic properties of algebraic cobordism
  7. Geometric description of algebraic cobordism
  8. Pushforwards of blow-ups and projective bundles
  9. The cobordism class of ${\mkern5mu\overline{\mkern-5mu\mathcal{M}\mkern-1.5mu}\mkern2mu}_{0,n}$
  10. Keel's blow-up description of ${\mkern5mu\overline{\mkern-5mu\mathcal{M}\mkern-1.5mu}\mkern2mu}_{0,n}$
  11. Normal bundles of Keel's blow-up centers
  12. Case 1.
  13. Case 2.
  14. The cobordism class of ${\mkern5mu\overline{\mkern-5mu\mathcal{M}\mkern-1.5mu}\mkern2mu}_{0,n}$
  15. The String equation in cobordism
  16. ...and 5 more sections

Key Result

Theorem 1.1

The psi-class intersections are uniquely determined by the recursion together with the initial condition $\int_{{\mkern5mu\overline{\mkern-5mu\mathcal{M}\mkern-1.5mu}\mkern2mu}_{0,3}} 1 = 1$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Remark 1.4
  • Lemma 1.5
  • Theorem 2.1: Levine-Morel AlgCob
  • Definition 2.2
  • Theorem 2.3: Levine-Pandharipande LevinePandharipande
  • Remark 2.4
  • Lemma 2.5
  • ...and 30 more