# Deformations of Generalized Kahler Structures and Bihermitian Structures - Mathematics > Differential Geometry

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Abstract: Let $X, J$ be a compact Kahler manifold with a non-zero holomorphic Poissonstructure $\beta$. If the obstruction space for deformations of generalizedcomplex structures on $X, J$ vanishes, we obtain a family of deformations ofnon-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\-ahlerform $\omega$, then the complex structure $J t^-$ is not equivalent to $J$ forsmall $t eq 0$ under diffeomorphisms.Our method is based on the construction of generalized complex and Kahlerstructures developed in \cite{Go1} and \cite{Go2}.As applications, we obtain such deformations of bihermitian structures on delPezzo surfaces, the Hirtzebruch surfaces $F 2, F 3$ and degenerate del Pezzosurfaces. Further we show that del Pezzo surfaces $S n 5\leq n\leq 8$, $F 2$and degenerate del Pezzo surfaces admit bihermitian structures for which $X,J^- t$ is not biholomorphic to $X, J$ for small $t eq 0$.

Author: ** Ryushi Goto**

Source: https://arxiv.org/