Simultaneous deformations and Poisson geometry

TL;DR

This paper develops an explicit L-infinity algebra framework for simultaneously deforming geometric structures like coisotropic submanifolds, twisted Poisson structures, and complex structures within generalized complex geometry.

Contribution

It introduces a novel L-infinity algebra approach based on derived brackets for analyzing simultaneous deformations of geometric structures.

Findings

Deformations governed by an explicitly constructed L-infinity algebra.

Applications include coisotropic submanifolds, twisted Poisson, and complex structures.

Method provides new insights not accessible via operad theory.

Abstract

We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an L-infinity algebra, which we construct explicitly. Our machinery is based on Th. Voronov's derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications can not be, to our knowledge, obtained by other methods such as operad theory.