Reduced genus-two Gromov-Witten Invariants for complex manifolds

TL;DR

This paper constructs reduced genus-two Gromov-Witten invariants for specific almost Kähler manifolds, including projective spaces up to dimension 7, counting genus-two holomorphic curves under certain conditions.

Contribution

It introduces a new method for defining genus-two Gromov-Witten invariants for almost Kähler manifolds with integrable complex structures and regularity conditions.

Findings

Invariants are well-defined for projective spaces up to dimension 7.

The invariants count simple genus-two J-holomorphic curves with constraints.

The construction extends Gromov-Witten theory to new classes of manifolds.

Abstract

In this article, we construct the reduced genus-two Gromov-Witten invariants for certain almost K\"{a}hler manifold $(X, \omega, J)$ such that $J$ is integrable and satisfies some regularity conditions. In particular, the standard projective space $(\P^n, \omega_0, J_0)$ of dimension $n \le 7$ satisfies these conditions. This invariant counts the number of simple genus-two $J$-holomorphic curves that satisfy appropriate number of constraints.