publications and preprints

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Arithmetic Aspects of Fundamental Groups

My current program is an attempt to study the Galois action on fundamental groups of algebraic varieties -- in particular, I am interested in phenomena that go beyond the pro-unipotent fundamental group

  • Arithmetic restrictions on geometric monodromy -- (submitted, updated 12/2016) -- This paper studies which representations of fundamental groups can come from geometry.  The main theorem is: Let X be a normal complex algebraic variety, and p a prime. Then there exists an integer \(N=N(X, p)\) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial mod \(p^N\).  I think this is a pretty interesting result. The proof is "anabelian" -- it involves a study of the action of the Galois group of a finitely generated field on the fundamental group of X.
  • The spectrum of Frobenius on the fundamental group (in preparation)
  • Integral p-adic Hodge theory for the fundamental group (in preparation)

Positivity and Lefschetz theorems

Broadly speaking, Lefschetz hyperplane theorems compare the geometry of a projective variety to that of a hyperplane section.  My thesis work was to give a proof of a Lefschetz hyperplane theorem for any smooth representable functor.   Some of my related interests are: positivity and vanishing theorems for vector bundles in positive characteristic, and the fine classification of algebraic varieties.

  • Vanishing for Frobenius Twists of Ample Vector Bundles -- This note proves several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, I prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.
  • Non-abelian Lefschetz hyperplane theorems (submitted) -- This is the edited version of my thesis.  Given a smooth projective variety \(X\) and an ample divisor \(D\) in \(X\), we study the problem of extending a map \(D\to Y\) to a map \(X\to Y\).  This is mainly interesting when \(Y\) is a moduli space. The Lefschetz hyperplane theorem for the etale fundamental group is a corollary of the main result; we get many more Lefschetz theorems as well. For example, we find Lefschetz theorems for extending families of curves from an ample divisor, extending sections to certain maps, extending maps to varieties (DM stacks) with nef cotangent bundle, and so on. The main results are in characteristic zero, but the proofs pass through positive characteristic.
  • Manifolds containing an ample \(\mathbb{P}^1\)-bundle (Manuscripta Mathematica, ArXiv version here) -- Sommese gives a conjectural classification of manifolds containing a projective bundle as an ample divisor; I prove his conjecture in many new cases, and I prove it in general assuming a natural conjectural characterization of projective space, along the lines of the famous characterization due to Andreatta and Wisniewski.  (Update 11/2016: This conjectural characterization is now a theorem, due to Jie Liu.  Thus Sommese's conjecture is now a theorem as well.)
 

Dynamics of algebraic varieties

How does the automorphism group of a variety change under birational modifications?  This question is extremely mysterious -- for example, we don't know if the component group of the automorphism group of a rational surface is always finitely generated.  Nonetheless, one can say some things, interestingly, using p-adic methods.

  • Dynamical Mordell-Lang and automorphisms of blowups (joint with John Lesieutre, submitted) -- We prove a non-reduced analogue of the dynamical Mordell-Lang conjecture, via \(p\)-adic methods.  We use this to place restrictions on the dynamics of the exceptional divisor of a blowup under the action of a regular automorphism.  As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of the automorphism group, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism. Here is a link to a video of John talking about this paper.

The Grothendieck Ring of Varieties

The Weil conjectures may be interpreted geometrically as saying (among other things) that the generating function for the number of effective 0-cycles on a variety over a finite field is rational.  Kapranov asked a "motivic" version of this question -- is the generating function for symmetric powers of a variety (as a power series over a ring whose elements are "linear combinations of varieties") rational?  The answer is "no" in general, but one may ask interesting weaker questions about this power series (called the Kapranov motivic zeta function), by analogy to the Weil conjectures.

  • Symmetric powers do not stabilize (Proceedings of the AMS, ArXiv version here) -- My advisor, Ravi Vakil, conjectured that the classes of the symmetric powers of a variety "stabilize" in the Grothendieck ring, in a certain sense.  I show that the answer to this question is "no" if one asks this stabilization question in a slightly different way, and conditional on various (very difficult) questions about the Grothendieck ring of varieties, that the answer to the original question is "no" as well.  On the other hand, I show that these stabilization questions are controlled by the Hodge theory of the variety, and give an analogue of the famous "Newton above Hodge" theorem in the Grothendieck ring.  Here is a video of me talking about this paper.
  • Zeta functions of curves with no rational points (Michigan Math Journal, ArXiv version here) -- Kapranov showed that his motivic zeta function is rational for smooth curves which have a rational point.  By analyzing the classes of certain Severi-Brauer schemes in the Grothendieck ring of varieties, I show that the same thing is true without the assumption of the existence of a rational point.