Deformations of Generalized Kahler Structures and Bihermitian Structures

Understand

Let $(X, J)$ be a compact Kahler manifold with a non-zero holomorphic Poisson structure $\beta$.

• If the obstruction space for deformations of generalized complex structures on $(X, J)$ vanishes, we obtain a family of deformations of non-trivial bihermitian structures $(J, J^-_t, h_t)$ on $X$ by using $\beta$.
• In addition, if the class $[\beta\cdot \omega]$ does not vanish for a K\"ahler form $\omega$, then the complex structure $J_t^-$ is not equivalent to $J$ for small $t\neq 0$ under diffeomorphisms.
• Our method is based on the construction of generalized complex and Kahler structures developed in \cite{Go1} and \cite{Go2}.