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 $({\mathbb{C}}^{\ast }{)}^n$ sitting in $\mathbb{C}{\mathbb{P}}^N$ via a monomial embedding. We use this to show that if $\omega$ is an integral Kähler 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 $({\mathbb{C}}^{\ast }{)}^n$ equipped with a (rational) toric Kähler 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 $(X,L)$.

Revised in many places, applications to symplectic packing problem and lower bounds on Gromov width in terms of Newton-Okounkov bodies added, 26 pages, 1 figure

References in corpus (1)

• Simplices in Newton-Okounkov bodies and the Gromov width of coadjoint orbits

Cited by in corpus (4)

• Multipoint Okounkov bodies
• The geometric structure of symplectic contraction
• The Gromov width of Bott-Samelson varieties
• Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces