Holomorphic jets in symplectic manifolds

TL;DR

This paper develops a framework for studying holomorphic discs in symplectic manifolds, establishing smooth moduli spaces, and deriving inequalities related to singularities and non-immersed discs, advancing understanding in symplectic geometry.

Contribution

It introduces pointwise partial differential relations for holomorphic discs, constructs smooth moduli spaces, and proves new inequalities and parity results for non-immersed discs.

Findings

Constructed smooth moduli spaces for holomorphic discs with injective points.

Derived an adjunction inequality for singularities of holomorphic discs.

Proved that the number of non-immersed holomorphic discs is even for generic coordinate classes.

Abstract

We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an adjunction inequality for singularities is derived. Moreover, we show that for a coordinate class of a monotone Lagrangian split torus generically the number of non-immersed holomorphic discs is even.