Counting curves with local tangency constraints

TL;DR

This paper introduces new invariants for counting rational curves with local tangency constraints in symplectic manifolds, providing recursive formulas and computations related to Gromov--Witten invariants and punctured curves.

Contribution

It develops a framework for invariants counting curves with local tangency constraints and relates them to existing Gromov--Witten invariants through recursive formulas.

Findings

Derived formulas for invariants as point constraints coalesce in dimension four.

Computed all invariants in terms of Gromov--Witten invariants of blowups.

Studied invariants counting punctured curves with negative ends on ellipsoids.

Abstract

We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency constraints. We give a formula describing how these invariants arise as point constraints are pushed together in dimension four, and we use this to recursively compute all of these invariants in terms of Gromov--Witten invariants of blowups. As a key tool, we study analogous invariants which count punctured curves with negative ends on a small skinny ellipsoid.