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<p>We prove a formula for the cycle class of the supersingular locus in the Chow ring with rational coefficients of the moduli space of principally polarized abelian varieties of dimension <i>g</i> in characteristic <i>p</i>. This formula determines this class as a monomial in the Chern classes of the Hodge bundle up to a factor that is a polynomial in <i>p</i>. This factor is known for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(g\le 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We also determine the factor for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(g=4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The cycle class of the supersingular locus of principally polarized abelian varieties

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月13日

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  • Gerard van der Geer,
  • Shushi Harashita

摘要

We prove a formula for the cycle class of the supersingular locus in the Chow ring with rational coefficients of the moduli space of principally polarized abelian varieties of dimension g in characteristic p. This formula determines this class as a monomial in the Chern classes of the Hodge bundle up to a factor that is a polynomial in p. This factor is known for \(g\le 3\) g ≤ 3 . We also determine the factor for \(g=4\) g = 4 .