Deformations of Generalized Kahler Structures and Bihermitian Structures

Abstract

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 Kähler form $\omega$, then the complex structure $J_t^{-}$ is not equivalent to $J$ for small $t\ne 0$ under diffeomorphisms. Our method is based on the construction of generalized complex and Kahler structures developed in \cite{Go1} and \cite{Go2}. As applications, we obtain such deformations of bihermitian structures on del Pezzo surfaces, the Hirtzebruch surfaces $F_2,F_3$ and degenerate del Pezzo surfaces. Further we show that del Pezzo surfaces $S_n(5\le n\le 8)$, $F_2$ and degenerate del Pezzo surfaces admit bihermitian structures for which $(X,J_t^{-})$ is not biholomorphic to $(X,J)$ for small $t\ne 0$.

39 pages

References in corpus (1)

• Anti-self-dual bihermitian structures on Inoue surfaces