On coisotropic deformations of holomorphic submanifolds

TL;DR

This paper studies the deformation theory of holomorphic Poisson manifolds and coisotropic submanifolds, establishing unobstructedness of certain deformations and relating them to homotopy Lie algebroids.

Contribution

It introduces a differential graded Lie algebra framework for Poisson and coisotropic deformations, extending classical stability and moduli space results.

Findings

Infinitesimal first order deformations are unobstructed under mild conditions.

Provides a homotopy equivalence between the constructed dg Lie algebra and the homotopy Lie algebroid.

Generalizes classical theorems like Kodaira stability and McLean-Voisin for these deformations.

Abstract

We describe the differential graded Lie algebras governing Poisson deformations of a holomorphic Poisson manifold and coisotropic embedded deformations of a coisotropic holomorphic submanifold. In both cases, under some mild additional assumption, we show that the infinitesimal first order deformations induced by the anchor map are unobstructed. Applications include the analog of Kodaira stability theorem for coisotropic deformation and a generalization of McLean-Voisin's theorem about the local moduli space of lagrangian submanifold. Finally it is shown that our construction is homotopy equivalent to the homotopy Lie algebroid, in the cases where this is defined.