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<p>Motivated by a problem of Hirzebruch, we study 8-dimensional, closed, connected, 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1224_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(T^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-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. We apply our results to obtain new consequences for Hirzebruch’s problem in the algebraic setting. 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.</p>

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On a symplectic generalization of a Hirzebruch problem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月1日

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  • Leonor Godinho,
  • Nicholas Lindsay,
  • Silvia Sabatini

摘要

Motivated by a problem of Hirzebruch, we study 8-dimensional, closed, connected, 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 \(T^2\) 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. We apply our results to obtain new consequences for Hirzebruch’s problem in the algebraic setting. 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.