Branes on Poisson varieties
Marco Gualtieri
TL;DR
This work treats a holomorphic Poisson structure ($I$,$\sigma_I$) as a generalized complex structure and develops a framework linking generalized connections, branes, and Poisson modules to generalized Kahler geometry. It introduces a canonical generalized Bismut connection and shows how branes and Poisson-module structures give rise to equivalences between categories across different holomorphic Poisson data, enabling explicit construction of generalized Kahler metrics on Poisson manifolds with positive Poisson line bundles. A deformation mechanism via flows of Poisson vector fields yields families of generalized Kahler structures that extend Hitchin’s bi-Hermitian constructions to higher dimensions, particularly on Fano/Poisson varieties. The paper also sketches connections to non-commutative algebraic geometry through groupoid of trivializations and elliptic loci, hinting at a broader geometric-algebraic landscape involving generalized geometry and noncommutative deformations.
Abstract
We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kaehler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is non-holomorphic in nature. Finally we show an equivalence between certain configurations of branes on Poisson varieties and generalized Kaehler structures, and use this to construct explicitly new families of generalized Kaehler structures on compact holomorphic Poisson manifolds equipped with positive Poisson line bundles (e.g. Fano manifolds). We end with some speculations concerning the connection to non-commutative algebraic geometry.