Moduli Spaces of $J$-holomorphic Curves with General Jet Constraints

TL;DR

This paper proves the surjectivity of the tangent map of holomorphic jet evaluation on the universal moduli space of simple J-holomorphic curves, establishing smoothness and expected dimension for generic almost complex structures with jet constraints.

Contribution

It demonstrates the surjectivity of the tangent map of holomorphic jet evaluation and shows that the moduli space with jet constraints is smooth and of expected dimension for generic J.

Findings

Tangent map of holomorphic jet evaluation is surjective.

Moduli space with jet constraints is smooth for generic J.

Expected dimension of the moduli space is achieved.

Abstract

In this paper, we prove that the tagent map of the holomorphic $k$- jet evaluation $j^k_{hol}$ from the mapping space to holomorphic $k$-jet bundle, when restricted on the universal moduli space of simple J-holomorphic curves with one marked point, is surjective. From this we derive that for generic $J$, the moduli space of simple $J$-holomorphic curves in class $\beta\in H_2(M)$ with general jet constraints at marked points is a smooth manifold of expected dimension.