Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds

2023/11/06 by Denisi, Francesco Antonio
#14J42 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.03295

Abstract

Let X be a projective irreducible holomorphic symplectic manifold. We associate with any big R-divisor D on X a convex polygon Δ_E^num(D) of dimension 2, whose Euclidean volume is vol_R²(Δ_E^num(D))=q_X(P(D))/2, where E is any prime divisor on X, q_X is the Beauville-Bogomolov-Fujiki form, and P(D) is the positive part of the divisorial Zariski decomposition of D. We systematically study these polygons and observe that they behave like the Newton-Okounkov bodies of big divisors on smooth complex projective surfaces, with respect to a general admissible flag.

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