Toric degenerations and symplectic geometry of smooth projective varieties

Abstract

Let $X$ be an $n$-dimensional smooth complex projective variety embedded in $\mathbb{C}\mathbb{P}^{N}$. We construct a smooth family $\mathcal{X}$ over $\mathbb{C}$ with an embedding in $\mathbb{C}\mathbb{P}^{N} \times \mathbb{C}$ whose generic fiber is $X$ and the special fiber is the torus $(\mathbb{C}^*)^n$ sitting in $\mathbb{C}\mathbb{P}^{N}$ via a monomial embedding. We use this to show that if $ω$ is an integral Kähler form on $X$ then for any $ε> 0$ there is an open subset $U_ε\subset X$ such that $vol(X \setminus U_ε) < ε$ and $U_ε$ is symplectomorphic to $(\mathbb{C}^*)^n$ equipped with a (rational) toric Kähler form. As an application we obtain lower bounds for the Gromov width of $(X, ω)$ in terms of its associated Newton-Okounkov bodies. We also show that if $ω$ lies in the class $c_1(L)$ of a very ample line bundle $L$ then $(X, ω)$ has a full symplectic packing with $d$ equal balls where $d$ is the degree of $(X, L)$.