Unobstructed deformations of generalized complex structures induced by $C^\infty$ logarithmic symplectic structures and logarithmic Poisson structures

Abstract

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstructed deformations which are parametrized by an open set of the second de Rham cohomology group of the complement of type changing loci if the type changing loci are smooth. Complex surfaces with smooth effective anti-canonical divisors admit unobstructed deformations of generalized complex structures such as del pezzo surfaces and Hirzebruch surfaces. We also give some calculations of Poisson cohomology groups on these surfaces. Generalized complex structures ${\cal J}_m$ on the connected sum $(2k-1)\Bbb C P^2\# (10k-1)\ol {\Bbb C P^2}$ as in \cite{Cavalcanti_Gualtieri_2006}, \cite{Goto_Hayano} are induced by $C^\infty$ logarithmic symplectic structures modulo the action of $b$-fields and it turns out that {generalized complex structure}s ${\cal J}_m$ have unobstructed deformations of dimension $12k+2m-3$.}