Toric degenerations and symplectic geometry of projective varieties

The purpose of this talk is to discuss some recent general results about symplectic geometry of smooth projective varieties using toric degenerations (motivated by commutative algebra). The main result is the following: Let X be a smooth n-dimensional complex projective variety embedded in a projective space and equipped with a Kahler structure induced from a Fubini-Study Kahler form. We show that for any \epsilon>0 the manifold X has an open subset U (in the usual topology) such that vol(X \ U) < \epsilon and moreover U is symplectomorphic to the algebraic torus (\C^*)^n equipped with a toric Kahler form. The proof is based on construction of a toric degeneration of X. As applications we obtain lower bounds on the Gromov width of X. Moreover, we show that X has a full symplectic ball packing by d balls of capacity 1 where d is the degree of X.