Shimura Varieties, Kummer Varieties, and Rational Curves
Bo-Hae Im, Michael Larsen, and Sailun Zhan

TL;DR
This paper investigates the presence of rational curves on Kummer varieties associated with products of elliptic curves and certain abelian varieties, revealing conditions under which these varieties lack rational curves.
Contribution
It establishes that for a very general product of seven or more elliptic curves, the Kummer variety contains no rational curves, and extends this to certain abelian varieties parametrized by Shimura varieties.
Findings
Rational curves on Kummer varieties of large products of elliptic curves project trivially onto factors.
Very general abelian varieties in certain Shimura families have Kummer varieties without rational curves.
The results connect the geometry of Kummer varieties with the structure of Shimura varieties.
Abstract
For a very general product of seven or more elliptic curves, every rational curve on the Kummer variety of projects trivially onto the Kummer variety of at least one of its factors. As a consequence, a very general member of certain families of abelian varieties parametrized by connected Shimura varieties of unitary type has the property that its Kummer variety has no rational curves.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Advanced Combinatorial Mathematics
