Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916570252443648 |
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| author | Denisi, Francesco Antonio |
| author_facet | Denisi, Francesco Antonio |
| contents | Let $X$ be a projective irreducible holomorphic symplectic manifold. We associate with any big $\mathbf{R}$-divisor $D$ on $X$ a convex polygon $Δ_E^{\mathrm{num}}(D)$ of dimension 2, whose Euclidean volume is $\mathrm{vol}_{\mathbf{R}^2}(Δ_E^{\mathrm{num}}(D))=q_X(P(D))/2$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03295 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds Denisi, Francesco Antonio Algebraic Geometry 14J42, 14M25 Let $X$ be a projective irreducible holomorphic symplectic manifold. We associate with any big $\mathbf{R}$-divisor $D$ on $X$ a convex polygon $Δ_E^{\mathrm{num}}(D)$ of dimension 2, whose Euclidean volume is $\mathrm{vol}_{\mathbf{R}^2}(Δ_E^{\mathrm{num}}(D))=q_X(P(D))/2$, 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. |
| title | Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds |
| topic | Algebraic Geometry 14J42, 14M25 |
| url | https://arxiv.org/abs/2311.03295 |