2015/08/31 by Kiumars Kaveh
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Cohomology #Embedding #Geometry and complex manifolds #Projective variety #Symplectic geometry #Toric variety #Torus #Variety (cybernetics) #math.AG #math.SG #msc:14D06 #msc:53D05
paper · pdf · doi:10.1112/jlms.12173
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
arxiv created 2016/03/21 · openalex created_date 2016/06/24 · openalex publication_date 2018/09/16 · arxiv updated 2018/09/26 · openalex updated_date 2026/08/05
Let X be an n-dimensional smooth complex projective variety embedded in C P N . We construct a smooth family X over C with an embedding in C P N × C such that its generic fiber is X and its special fiber is the torus ( C ∗ ) n sitting in C P N via a monomial embedding. We use this to show that if ω is a Kähler form on X with rational cohomology class, then for any ε > 0 there is an open subset U ε ⊂ X such that vol ( X ∖ U ε ) < ε and U ε is symplectomorphic to ( C ∗ ) n equipped with a (rational) toric Kähler form. As applications we obtain lower bounds for the Gromov width of ( X , ω ) , as well as construction of symplectic packings for it, in terms of associated Newton–Okounkov bodies.