Toric degenerations and symplectic geometry of smooth projective varieties

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 $\omega$ is an integral K\"ahler form on $X$ then for any $\epsilon > 0$ there is an open subset $U_\epsilon \subset X$ such that $vol(X \setminus U_\epsilon) < \epsilon$ and $U_\epsilon$ is symplectomorphic to $(\mathbb{C}^*)^n$ equipped with a (rational) toric K\"ahler form. As an application we obtain lower bounds for the Gromov width of $(X, \omega)$ in terms of its associated Newton-Okounkov bodies. We also show that if $\omega$ lies in the class $c_1(L)$ of a very ample line bundle $L$ then $(X, \omega)$ has a full symplectic packing with $d$ equal balls where $d$ is the degree of $(X, L)$.