Tautological and non-tautological cycles on the moduli space of Abelian varieties

Abstract The tautological Chow ring of the moduli space $\mathcal{A}_{g}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> of principally polarized abelian varieties of dimension $g$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from $\mathcal{A}_{g}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> to the moduli space $\mathcal{M}_{g}^{\operatorname{ct}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>g</mml:mi> <mml:mo>ct</mml:mo> </mml:msubsup> </mml:math> of genus $g$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> of curves of compact type, we prove that the product class $[\mathcal{A}_{1}\times \mathcal{A}_{5}]\in \mathsf{CH}^{5}( \mathcal{A}_{6})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>]</mml:mo> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>CH</mml:mi> <mml:mn>5</mml:mn> </mml:msup> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:math> is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>6</mml:mn> <mml:mo>ct</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:math> in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>6</mml:mn> <mml:mo>ct</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:math> has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_{1}\times \mathcal{A}_{5}]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:math> . More generally, the Torelli pullback of the difference between $[\mathcal{A}_{1}\times \mathcal{A}_{g-1}]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> </mml:math> and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^{*}(\mathcal{M}_{g}^{\operatorname{ct}})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>g</mml:mi> <mml:mo>ct</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:math> . The product map $\mathcal{A}_{1}\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_{g}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub>