Holomorphic forms and non-tautological cycles on moduli spaces of curves

Abstract We prove, for infinitely many values of g and n , the existence of non-tautological algebraic cohomology classes on the moduli space $$\mathcal {M}_{g,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> of smooth, genus- g , n -pointed curves. In particular, when $$n=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , our results show that there exist non-tautological algebraic cohomology classes on $$\mathcal {M}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> for $$g=12$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:math> and all $$g \ge 16$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> . These results generalize the work of Graber–Pandharipande and van Zelm, who proved that the classes of particular loci of bielliptic curves are non-tautological and thereby exhibited the only previously-known non-tautological class on any $$\mathcal {M}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> : the bielliptic cycle on $$\mathcal {M}_{12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>12</mml:mn> </mml:msub> </mml:math> . We extend their work by using the existence of holomorphic forms on certain moduli spaces $$\overline{\mathcal {M}}_{g,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new double-cover loci are non-tautological.