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On a symplectic generalization of a Hirzebruch problem

Published 1 Mar 2024 in math.SG and math.AG | (2403.00949v2)

Abstract: 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 $T2$-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.

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