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SUMMARY:Noether-Lefschetz cycles on the moduli space of abelian varieties 
 - Aitor Iribar Lopez\, ETH Zurich
DTSTART:20250521T131500Z
DTEND:20250521T141500Z
UID:TALK232051@talks.cam.ac.uk
CONTACT:Dhruv Ranganathan
DESCRIPTION:We study $A_g$\, the moduli space of principally polarized abe
 lian varieties of dimension $g$. The tautological ring\, generated by the 
 chern classes of the Hodge bundle\, was fully determined by Gerard van der
  Geer in 1999\, but the question of which geometrically defined cycles bel
 ong to this subring remains open. In 2024\, Canning\, Oprea and Pandharipa
 nde showed that $[A_1 \\times A_5]$ is not tautological in $A_6$\, and lat
 er I showed that $[A_1 \\times A_{g-1}]$ is not tautological for $g=12$ or
  $g \\geq 16$ even.\n\nThe cycle $[A_1\\times A_{g-1}]$ is one of the Noet
 her-Lefschetz cycles on the moduli spaces. With Greer and Lian\, we conjec
 ture that these cycles are related to modular forms of weight $2g$.\n\nA n
 ew technique\, which was not available in 1999 is the existence of a proje
 ction operator by Canning\, Oprea\, Molcho and Pandharipande onto the taut
 ological ring. This leads to interesting conjectures about Gromov-Witten i
 nvariants on a moving elliptic curve\, which now have been proven in colla
 boration with Pandharipande and Tseng\, and are also connected to the fail
 ure of the Gorenstein conjecture on the moduli space of curves of compact 
 type\n\nThe talk will be an overview of the intersection theory of $A_g$ a
 nd the moduli space of curves\, and I will explain briefly the ideas behin
 d some proofs.
LOCATION:CMS MR13
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