On a symplectic generalization of a Hirzebruch problem
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916273639653376 |
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| author | Godinho, Leonor Lindsay, Nicholas Sabatini, Silvia |
| author_facet | Godinho, Leonor Lindsay, Nicholas Sabatini, Silvia |
| contents | Motivated by a problem of Hirzebruch, we study $8$-dimensional, closed, symplectic manifolds having a Hamiltonian torus action with isolated fixed points and second Betti number equal to $1$. Such manifolds are automatically positive monotone. Our main result concerns those endowed with a Hamiltonian $T^2$-action and fourth Betti number equal to $2$. We classify their isotropy data, (equivariant) cohomology rings and (equivariant) Chern classes, and prove that they agree with those of certain explicit Fano $4$-folds with torus actions. Moreover, under more general assumptions, we prove several finiteness results concerning Betti and Chern numbers of $8$-dimensional, positive monotone symplectic manifolds with a Hamiltonian torus action. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00949 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a symplectic generalization of a Hirzebruch problem Godinho, Leonor Lindsay, Nicholas Sabatini, Silvia Symplectic Geometry Algebraic Geometry 53D20, 57M60, 37B05 Motivated by a problem of Hirzebruch, we study $8$-dimensional, closed, symplectic manifolds having a Hamiltonian torus action with isolated fixed points and second Betti number equal to $1$. Such manifolds are automatically positive monotone. Our main result concerns those endowed with a Hamiltonian $T^2$-action and fourth Betti number equal to $2$. We classify their isotropy data, (equivariant) cohomology rings and (equivariant) Chern classes, and prove that they agree with those of certain explicit Fano $4$-folds with torus actions. Moreover, under more general assumptions, we prove several finiteness results concerning Betti and Chern numbers of $8$-dimensional, positive monotone symplectic manifolds with a Hamiltonian torus action. |
| title | On a symplectic generalization of a Hirzebruch problem |
| topic | Symplectic Geometry Algebraic Geometry 53D20, 57M60, 37B05 |
| url | https://arxiv.org/abs/2403.00949 |