Unobstructed K-deformations of generalized complex structures and bi-Hermitian structures

We introduce K-deformations of generalized complex structures on a compact Kähler manifold M=(X,J) with an effective anti-canonical divisor and show that obstructions to K-deformations of generalized complex structures on M always vanish. Applying unobstructed K-deformations and the stability theorem for generalized Kähler structures, we construct deformations of bi-Hermitian structures in the form (J,Jt−,ht) on a compact Kähler surface with a non-zero holomorphic Poisson structure. Then we prove that a compact Kähler surface S admits a non-trivial bi-Hermitian structure with the torsion condition and the same orientation if and only if S has a non-zero holomorphic Poisson structure. We also obtain bi-Hermitian structures (J,J−,h) on del Pezzo surfaces, degenerate del Pezzo surfaces and some ruled surfaces for which the complex structure J is not equivalent to J− under diffeomorphisms.