Enumeration of rational contact curves via torus actions

TL;DR

This paper proves certain Gromov-Witten numbers for rational contact curves in complex projective spaces are truly enumerative and computes their exact values using Bott's formula, confirming consistency with known results.

Contribution

It establishes the enumerativity of specific Gromov-Witten numbers for rational contact curves and provides explicit calculations using Bott's formula.

Findings

Gromov-Witten numbers are enumerative for rational contact curves.

Exact values of these numbers are computed for low degrees in projective spaces.

Results align with previously known data.

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.