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<p>Formulas of Harer–Zagier and Harder imply that the orbifold Euler characteristic of the moduli stacks of smooth curves or principally polarized abelian varieties have a sign given by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((-1)^{\dim {\mathscr {M}}_g }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>dim</mo> <msub> <mi mathvariant="script">M</mi> <mi>g</mi> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((-1)^{\dim {\mathscr {A}}_g }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>dim</mo> <msub> <mi mathvariant="script">A</mi> <mi>g</mi> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation>. This is generalized as follows: Given a perverse sheaf on a product of moduli stacks of either type, it is proved that the Euler characteristic is nonnegative when the base field has characteristic zero. This is shown to be false in positive characteristic.</p>

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Nonnegativity of signed Euler characteristics of moduli of curves and abelian varieties

  • 期刊论文

  • 发表日期：

    2026年2月13日

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  • Donu Arapura,
  • Deepam Patel

摘要

Formulas of Harer–Zagier and Harder imply that the orbifold Euler characteristic of the moduli stacks of smooth curves or principally polarized abelian varieties have a sign given by \((-1)^{\dim {\mathscr {M}}_g }\) ( - 1 ) dim M g or \((-1)^{\dim {\mathscr {A}}_g }\) ( - 1 ) dim A g . This is generalized as follows: Given a perverse sheaf on a product of moduli stacks of either type, it is proved that the Euler characteristic is nonnegative when the base field has characteristic zero. This is shown to be false in positive characteristic.