### Arithmetic Aspects of Fundamental Groups

My current program is an attempt to study the structure of the fundamental groups of algebraic varieties -- in particular, I am interested in phenomena that go beyond the pro-unipotent fundamental group. These structures include Galois actions, mixed Hodge structures, and their analogues in \(p\)-adic Hodge theory.

### Representation Theory of fundamental groups

•** Arithmetic representations of fundamental groups II: finiteness**

*(draft)*

#### \(\qquad\)+ Abstract

Let \(X\) be a smooth curve over a finitely generated field \(k\), and let \(\ell\) be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod \(\ell\) representations of the geometric fundamental group of \(X\). Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero.

For example, we show that if \(X\) is a normal, connected variety over \(\mathbb{C}\), the (typically infinite) set of representations of \(\pi_1(X^{\text{an}})\) into \(GL_n(\overline{\mathbb{Q}_\ell})\), which come from geometry, has no limit points. As a corollary, we deduce that if \(L\) is a finite extension of \(\mathbb{Q}_\ell\), then the set of representations of \(\pi_1(X^{\text{an}})\) into \(GL_n(L)\), which arise from geometry, is finite.

• * Arithmetic representations of fundamental groups I* (

*Inventiones Mathematicae (or here)*, 2018, arXiv version)

#### \(\qquad\)+ Abstract

Let \(X\) be a normal algebraic variety over a finitely generated field \(k\) of characteristic zero, and let \(\ell\) be a prime. Say that a continuous \(\ell\)-adic representation \(\rho\) of \(\pi_1^{\text{et}}(X_{\bar k})\) isarithmeticif there exists a finite extension \(k'\) of \(k\) and a representation \(\gamma\) of the etale fundamental group of \(\pi_1^{\text{et}}(X_{k'})\), with \(\rho\) a subquotient of \(\gamma|_{\pi_1^{\text{et}}(X_{\bar k})}\). We show that there exists an integer \(N=N(X, \ell)\) such that every nontrivial, semisimple arithmetic representation of \(\pi_1^{\text{et}}(X_{\bar k})\) is nontrivial mod \(\ell^N\). As a corollary, we prove that any nontrivial \(\ell\)-adic representation of \(\pi_1^{\text{et}}(X_{\bar k})\) whicharises from geometryis nontrivial mod \(\ell^N\).

This is a streamlined and completely rewritten account of the most interesting results in the paper below.

•** Arithmetic restrictions on geometric monodromy**

*(not intended for publication, arXiv version)*

#### \(\qquad\)+ Abstract

This is a leisurely account of the results in the paper above and some complements; some of what did not appear in the paper above will appear in its sequel,Arithmetic representations of fundamental groups II. This paper is not intended for publication.

### Hodge-theoretic aspects of fundamental groups

•** Canonical paths on algebraic varieties **(in preparation)

#### \(\qquad\)+ Abstract

This paper constructs a "canonical path" (an element of the pro-unipotent fundamental groupoid) connecting any two points on a normal complex variety. This gives (1) a canonical isomorphism between any two fibers of any unipotent local system, and (2) a monodromy-free notion of iterated integration, in analogy with the p-adic setting. This latter notion gives a geometric interpretation and construction of e.g. single-valued multiple polylogarithms, on any curve. Please email me if you'd like a (very rough) draft.

### Galois actions and \(p\)-adic Hodge theory of fundamental groups

•** Semisimplicity of the Frobenius action on \(\pi_1\)**

*(in preparation, joint with Alexander Betts)*

#### \(\qquad\)+ Abstract

Let \(X\) be a smooth variety over a \(p\)-adic field \(K\), with semi-stable reduction. This paper shows (1) that for any choice of Frobenius element of the Galois group of \(K\), Frobenius acts semi-simply on the \(\mathbb{Q}_\ell\)-pro-unipotent etale fundamental group of \(X\), and (2) an analogous statement for the crystalline Frobenius (acting on the log-crystalline pro-unipotent fundamental group). We give applications to Kim's unipotent Selmer varieties. The draft is very rough, but email me if you'd like to take a look.

•** \(p\)-adic iterated integrals on semi-stable curves** (in preparation, joint with Eric Katz)

#### \(\qquad\)+ Abstract

To appear.

## Classical Algebraic Geometry

I have pretty broad algebro-geometric interests: Lefschetz theorems, positivity and vanishing theorems for vector bundles in positive characteristic, the fine classification of algebraic varieties, algebraic dynamics, and birational geometry.

### Vanishing and positivity

• *Vanishing for Frobenius twists of ample vector bundles** *(to appear, *Tohoku Math Journal, *2018, arXiv version)

#### \(\qquad\)+ Abstract

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 *(

*Journal of Algebraic Geometry*, 2018, arXiv version)

#### \(\qquad\)+ Abstract

This is the edited version of my thesis, the goal of which was to prove a Lefschetz hyperplane theorem for any representable functor. 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 (taking \(Y=BG\) for \(G\) a finite group); 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,*2017, arXiv version)

#### \(\qquad\)+ Abstract

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 (arXiv version). Thus Sommese's conjecture is now a theorem as well.)

### Dynamics

• ** Dynamical Mordell-Lang and automorphisms of blowups **(joint with John Lesieutre, to appear,

*Algebraic Geometry, Foundation Compositio Mathematica*, arXiv version)

#### \(\qquad\)+ Abstract

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.

### The Grothendieck ring of varieties

• * Zeta functions of curves with no rational points *(

*Michigan Math Journal,*2015, arXiv version)

#### \(\qquad\)+ Abstract

Kapranov showed that his motivic zeta function (a geometric analogue of the zeta function of a variety over a finite field) 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.

• ** Symmetric powers do not stabilize **(

*Proceedings of the AMS,*2014, arXiv version)

#### \(\qquad\)+ Abstract

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.