Keerthi Madapusi Pera
Department of Mathematics, SC 235,
One Oxford St,
Cambridge, MA 02138, U.S.A.
Email: keerthi [at] math [dot] harvard [dot] edu
I just graduated from the the department of mathematics at the University of Chicago, and am now a Benjamin Peirce Fellow at Harvard. My advisor was Mark Kisin, who is now also at Harvard. I grew up in
Click here if you would like to do a Senior Thesis with me.
Thesis: Toroidal Compactifications of Integral Canonical Models of Shimura varieties of Hodge type.
The results of my thesis have been superseded by those in the paper 'Toroidal compactifications...' below. If you would still like a copy of it, please let me know.
Publications and pre-prints:
We show that the classical Kuga-Satake construction gives rise, away from characteristic 2, to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain normal integral model for a Shimura variety of orthogonal type. This allows us to attach to every polarized K3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety. Using a result of Kisin, we can then prove the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic. We also show that the moduli stack of primitively polarized K3 surfaces of fixed degree 2d is quasi-projective and, when d is not divisible by p^2, is geometrically irreducible in characteristic p. We indicate how the same method applies to prove the Tate conjecture for co-dimension 2 cycles on cubic fourfolds.
We construct regular integral canonical models for Shimura varieties attached to Spin groups at (possibly ramified) primes $p>2$ where the level is not divisible by $p$. We exhibit these models as schemes of 'relative PEL type' over integral canonical models of larger Spin Shimura varieties with good reduction at $p$. Work of Vasiu-Zink then shows that the classical Kuga-Satake construction extends over the integral model and that the integral models we construct are canonical in a very precise sense. We also construct good compactifications for our integral models. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla's program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.
We construct smooth projective toroidal compactifications for the integral canonical models of Shimura varieties of Hodge type constructed by Kisin and Vasiu at primes where the level is hyperspecial. This construction is a consequence of the main result of the paper, which shows, without any unramifiedness conditions on the Shimura datum, that the Zariski closure of a Shimura sub-variety of Hodge type in a Chai-Faltings compactification always intersects the boundary in a relative Cartier divisor. This result also provides a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties (over number fields) whose Mumford-Tate group is anisotropic modulo center. We also construct integral models of the minimal (Satake-Baily-Borel) compactification for Shimura varieties of Hodge type.
Some (very incomplete) notes of mine:
The first two were written back when I was an innocent first year student, who believed that the only way to learn something was to write it all up in gory detail. I still find parts of them useful, so I’ve put them up for general consumption. Caveat: citation in these notes is quite poor, but obviously stuff has made it into them from all over the place. And nothing in them is original (except maybe some of the organization).
You might have known me under the last name Madapusi Sampath (or simply Madapusi). I'm in the process of getting my name officially to Madapusi Pera, a combination of my last name and my wife's.