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Let X be a Deligne-Mumford stack equipped with a perfect obstruction theory \(\phi :\mathbb {E} \to \mathbb {L}_X\) . By Kiem and Li (J. Am. Math. Soc. 26(4), 1025–1050 (2013)), if the obstruction sheaf \(Ob_X=h^1(\mathbb {E} ^\vee )\) admits a cosection \(\sigma :Ob_X\to {\mathcal O}_X\) , the virtual fundamental class \([X]^{\mathrm {vir}} \) of X is localized to a class \([X]^{\mathrm {vir}} _\sigma \) supported in the zero locus \(X(\sigma )\) of \(\sigma ^\vee \) . In many natural examples, \(X(\sigma )\) is an interesting space on its own and we may ask if there is a natural induced perfect obstruction theory of \(X(\sigma )\) whose virtual fundamental class equals \([X]^{\mathrm {vir}} _\sigma \) . In this expository article, we discuss two possible approaches, by global complex Kuranishi chart and by derived algebraic geometry.

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Virtual Fundamental Classes of the Vanishing Loci of Cosections

  • 会议论文

  • 出版模式：

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  • 发表日期：

    2026年

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  • Young-Hoon Kiem,
  • Hyeonjun Park

摘要

Let X be a Deligne-Mumford stack equipped with a perfect obstruction theory \(\phi :\mathbb {E} \to \mathbb {L}_X\) . By Kiem and Li (J. Am. Math. Soc. 26(4), 1025–1050 (2013)), if the obstruction sheaf \(Ob_X=h^1(\mathbb {E} ^\vee )\) admits a cosection \(\sigma :Ob_X\to {\mathcal O}_X\) , the virtual fundamental class \([X]^{\mathrm {vir}} \) of X is localized to a class \([X]^{\mathrm {vir}} _\sigma \) supported in the zero locus \(X(\sigma )\) of \(\sigma ^\vee \) . In many natural examples, \(X(\sigma )\) is an interesting space on its own and we may ask if there is a natural induced perfect obstruction theory of \(X(\sigma )\) whose virtual fundamental class equals \([X]^{\mathrm {vir}} _\sigma \) . In this expository article, we discuss two possible approaches, by global complex Kuranishi chart and by derived algebraic geometry.