# Toric degenerations and symplectic geometry of projective varieties

# Toric degenerations and symplectic geometry of projective varieties

I will explain some recent general results about symplectic geometry of projective varieties using toric degenerations (motivated by commutative algebra and the theory of Newton-Okounkov bodies). The main result is the following: Let X be a smooth n-dimensional complex projective variety equipped with an integral 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)-vol(U)< epsilon, and moreover U is symplectomorphic to the algebraic torus (C*)^n equipped with a "toric" Kahler form. The proof is based on the construction of a toric degeneration of X. As applications we obtain lower bounds on the Gromov width of X. We also get a full symplectic ball packing of X by d balls of capacity 1 where d is the degree of X.