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Enumeration of rational contact curves via torus actions

Giosuè Muratore

Abstract

We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in $\mathbb{P}^{3}$ and $\mathbb{P}^{5}$ are computed. The results are consistent with existing results.

Enumeration of rational contact curves via torus actions

Abstract

We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in and are computed. The results are consistent with existing results.

Paper Structure

This paper contains 7 sections, 13 theorems, 51 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Contact structures
  3. Contact stable maps
  4. Equivariant classes in $\mathbb{P}^{n}$
  5. Contact curves
  6. Incidence to a linear subspace
  7. Final remarks

Key Result

Proposition 2.6

Let $X$ be a manifold with a contact structure $\{U_{i},\alpha_{i}\}_{i}$, and let $C\subset X$ be a curve. Let $L$ be the contact line bundle and let $\mathcal{D}:=\ker(T_{X}\twoheadrightarrow L^{\vee})$. The following are equivalent. Moreover if $p\in C\cap U_{i}$, then $T_{C,p}$ is an isotropic vector subspace of the symplectic space $(\mathcal{D}_{p},\mathrm{d}\alpha_{i})$.

Theorems & Definitions (40)

  • Definition 2.1: kobayashi1959remarks
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 30 more