Symplectic mapping class groups of elliptic ruled surfaces

An elliptic ruled surface is a 4-manifold satisfying the condition that it has a holomorphic fibration over an elliptic curve with fibers that are projective lines. Every elliptic ruled surface is algebraic, and, in particular, a Kaehler surface. In this talk I would like to discuss the symplectomorphism group of elliptic ruled surfaces. More precisely, we will show that every symplectomorphism that is smoothly isotopic to the identity is isotopic to the identity within the symplectomorphism group.