Keerthi Madapusi Pera

Department of Mathematics, SC 235,
Harvard University,
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 India and first came to the US to get a BS in mathematics at Yale. You can find my CV here.

Click here if you would like to do a Senior Thesis with me.

Research Interests:

• Integral models of Shimura varieties and their compactifications.
• Hodge cycles on abelian varieties.
• Integral p-adic Hodge theory.
• Logarithmic Dieudonne theory.

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The results of my thesis have been superseded by those in the paper 'Toroidal compactifications...'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:

1. The Tate conjecture for K3 surfaces in odd characteristic. Abstract

   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.

   PDF (April 22, 2013)
2. Integral canonical models for Spin Shimura varieties. Abstract

   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.

   PDF (April 18, 2013)
3. Toroidal compactifications of integral models of Shimura varieties of Hodge type. Abstract

   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.

   PDF (Feb 19, 2013)

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).

· Basic Commutative Algebra

· Basic Algebraic Geometry

· Perfect complexes: The linear algebraic content of semi-continuity and base change

· Log p-divisible groups (after Kato)

Other news

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.