Please note that the papers below may differ from their published versions.
My current research program
Most of my current work focuses on arithmetic and geometric aspects of fundamental groups — that is, “anabelian geometry” — very broadly construed. I’m interested in connections to arithmetic, geometry, and topology.
papers on Arithmetic and topology of algebraic varieties
• Motives, mapping class groups, and monodromy (arXiv version, 2024)
\(\qquad\)+ Abstract
We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic varieties, we study mapping class group actions on character varieties and their algebro-geometric avatar -- isomonodromy differential equations -- from the point of view of both complex and arithmetic geometry. We then collect some open questions and conjectures on these topics. These notes are an extended version of my talk at the April 2024 Current Developments in Mathematics conference at Harvard.
• Big monodromy for higher Prym representations (accepted at Geometry and Topology, arXiv version, 2024, joint with Aaron Landesman and Will Sawin)
\(\qquad\)+ Abstract
Let \(\Sigma_{g'}\to \Sigma_g\) be a cover of an orientable surface of genus g by an orientable surface of genus g', branched at n points, with Galois group H. Such a cover induces a virtual action of the mapping class group \(\text{Mod}_{g,n+1}\) of a genus g surface with n+1 marked points on \(H^1(\Sigma_{g'},\mathbb{C})\). When g is large in terms of the group H, we calculate precisely the connected monodromy group of this action. The methods are Hodge-theoretic and rely on a "generic Torelli theorem with coefficients."
• Finite braid group orbits on \(SL_2\)-character varieties (arXiv version, 2023, joint with Yeuk Hay Joshua Lam and Aaron Landesman)
\(\qquad\)+ Abstract
Let X be a 2-sphere with n punctures. We classify all conjugacy classes of Zariski-dense representations \(\rho: \pi_1(X)\to SL_2(\mathbb{C})\) with finite orbit under the mapping class group of X, such that the local monodromy at one or more punctures has infinite order. We show that all such representations are "of pullback type" or arise via middle convolution from finite complex reflection groups. In particular, we classify all rank 2 local systems of geometric origin on the projective line with n generic punctures, and with local monodromy of infinite order about at least one puncture.
• Geometric local systems on the projective line minus four points (accepted at Compositio Mathematica, arXiv version, 2023, joint with Yeuk Hay Joshua Lam)
\(\qquad\)+ Abstract
Let \(J(m)\) be an \(m\times m\) Jordan block with eigenvalue 1. For \(\lambda\in \mathbb{C}\setminus \{0,1\}\), we explicitly construct all rank 2 local systems of geometric origin on \(\mathbb{P}^1\setminus\{0,1,λ,∞\}\), with local monodromy conjugate to \(J(2)\) at \(0,1,\lambda\) and conjugate to \(−J(2)\) at \(\infty\). The construction relies on Katz's middle convolution operation. We use our construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-Zuo).
• Prill’s problem (accepted to Compositio, Algebraic Geometry, arXiv version, 2022, joint with Aaron Landesman)
\(\qquad\)+ Abstract
We solve Prill's problem, originally posed by David Prill in the late 1970s and popularized in ACGH's "Geometry of Algebraic Curves." That is, for any curve \(Y\) of genus 2, we produce a finite étale degree 36 connected cover \(f: X\to Y\) where, for every point \( y \in Y, f^{-1}(y)\) moves in a pencil.
• An introduction to the algebraic geometry of the Putman-Wieland conjecture (European Journal of Mathematics, 2023, arXiv version, 2022, joint with Aaron Landesman)
\(\qquad\)+ Abstract
We give algebraic and geometric perspectives on our prior results toward the Putman-Wieland conjecture. This leads to interesting new constructions of families of "origami" curves whose Jacobians have high-dimensional isotrivial isogeny factors. We also explain how a hyperelliptic analogue of the Putman-Wieland conjecture fails, following work of Marković.
• Applications of the algebraic geometry of the Putman-Wieland conjecture (Proceedings of the LMS, 2023, arXiv version, 2022, joint with Aaron Landesman)
\(\qquad\)+ Abstract
We give three applications of our prior work toward the Putman-Wieland conjecture. First, we solve Prill's problem about special divisors on covers of curves. Next, we deduce a strengthening of a result of Marković-Tošić on virtual mapping class group actions on the homology of covers. Finally, we show that for \(g\geq 3\) and and \(\Sigma_{g',n'}\to \Sigma_{g, n}\) a finite \(H\)-cover of topological surfaces, the virtual action of the mapping class group of \(\Sigma_{g,n+1}\) on an \(H\)-isotypic component of \(H^1(\Sigma_{g'})\) has infinite image.
• Canonical representations of surface groups (accepted at Annals of Mathematics, arXiv version, 2022, joint with Aaron Landesman)
\(\qquad\)+ Abstract
Let \(\Sigma_{g,n}\) be an orientable surface of genus \(g\) with \(n\) punctures. We study actions of the mapping class group \(\text{Mod}_{g,n}\) of \(\Sigma_{g,n}\) via Hodge-theoretic and arithmetic techniques. We show that if \(\rho: \pi_1(\Sigma_{g,n})\to \text{GL}_r(\mathbb{C})\) is a representation whose conjugacy class has finite orbit under \(\text{Mod}_{g,n}\), and \(r< \sqrt{g+1}\), then \(\rho\) has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz.
The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems.
• \(p\)-adic iterated integration on semi-stable curves (arXiv version, 2022, joint with Eric Katz)
\(\qquad\)+ Abstract
We reformulate the theory of p-adic iterated integrals on semistable curves using the unipotent log rigid fundamental group. This fundamental group carries Frobenius and monodromy operators whose basic properties are established. By identifying the Frobenius-invariant subgroup of the fundamental group with the fundamental group of the dual graph, we characterize Berkovich--Coleman integration, which is path-dependent, as integration along the Frobenius-invariant lift of a path in the dual graph. We demonstrate the basic properties for this integration theory. Vologodsky's path-independent integration theory which was previously described using a monodromy condition can now be identified as Berkovich--Coleman integration along a combinatorial canonical path arising from the theory of combinatorial iterated integration as developed by the first-named author and Cheng. We use this definition to lay the groundwork for computing iterated integrals.
• Geometric local systems on very general curves and isomonodromy (accepted at Journal of the AMS, arXiv version, 2022, joint with Aaron Landesman — see here for a shorter version that only proves the main results with the assumption of unipotent monodromy at infinity)
\(\qquad\)+ Abstract
We show that the minimum rank of a non-isotrivial local system of geometric origin, on a suitably general \(n\)-pointed curve of genus \(g\), is at least \(2\sqrt{g+1}\). We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformation, which additionally answers questions of Biswas, Heu, and Hurtubise.
• Surface bundles and the section conjecture (arXiv version, 2020, joint with Wanlin Li, Nick Salter, and Padma Srinivasan, published in Mathematische Annalen)
\(\qquad\)+ Abstract
We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph \(\Gamma\) of genus \(g>2\), and every field \(k\), the generic curve with reduction type \(\Gamma\) over \(k\) satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over \(p\)-adic fields and number fields via a Chebotarev argument.
We construct two Galois cohomology classes \(o_1\) and \(\widetilde{o_2}\), which obstruct the existence of \(\pi_1\)-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves \(\mathscr{M}_g\) studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs \(\pi_1\)-sections and hence rational points.
Among our geometric results are a new proof of the section conjecture for the generic curve of genus \(g>2\), and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).
• Level structures, arithmetic representations, and non-commutative Siegel linearization (arXiv version, 2021, joint with Borys Kadets, published in Journal für die reine und angewandte Mathematik - Crelle's Journal)
\(\qquad\)+ Abstract
Let \(\ell\) be a prime, \(k\) a finitely generated field of characteristic different from \(\ell\), and \(X\) a smooth geometrically connected curve over \(k\). Say a semisimple representation of \(\pi_1^{et}(X_{\bar k})\) is arithmetic if it extends to a finite index subgroup of \(\pi_1^{et}(X)\). We show that there exists an effective constant \(N=N(X,\ell)\) such that any semisimple arithmetic representation of \(\pi_1^{et}(X_{\bar k})\) into \(GL_n(\mathbb{Z}_\ell)\), which is trivial mod \(\ell^N\), is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel's linearization theorem and the \(\ell\)-adic form of Baker's theorem on linear forms in logarithms.
• Representations of surface groups with universally finite mapping class group orbit (arXiv version, joint with Brian Lawrence, Mathematical Research Letters, 2023)
\(\qquad\)+ Abstract
Let \(\Sigma_{g,n}\) be the orientable genus \(g\) surface with \(n\) punctures, such that \(2−2g−n < 0 \). Let \[\rho:\pi_1(\Sigma_{g,n})\to GL_m(\mathbb{C})\] be a representation. Suppose that for each finite covering map \(f: \Sigma_{g′,n′}\to \Sigma_{g,n}\), the orbit of (the isomorphism class of) \(f_∗(\rho)\) under the mapping class group \(MCG(\Sigma_{g′,n′})\) of \(\Sigma_{g′,n′}\) is finite. Then we show that \(\rho\) has finite image. The result is motivated by the Grothendieck-Katz \(p\)-curvature conjecture, and gives a reformulation of the \(p\)-curvature conjecture in terms of isomonodromy.
• Arithmetic representations of fundamental groups II: finiteness (arXiv version, 2021, Duke Mathematics Journal version here)
\(\qquad\)+ 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 \(\pi_1^{\text{et}}(X_{\bar k})\) is arithmetic if 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 \(\pi_1^{\text{et}}(X_{\bar k})\) which arises from geometry is 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.
• Integral points on algebraic subvarieties of period domains (manuscripta mathematica, arXiv version here, 2019, joint with Ariyan Javanpeykar)
\(\qquad\)+ Abstract
We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of \(S\)-integral points implies finiteness (resp. non-Zariski-density) of points over all \(\mathbb{Z}\)-finitely-generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, as well as the recent resolution of Griffiths's conjecture by Bakker-Brunebarbe-Tsimerman. We give straightforward applications to Shimura varieties, locally symmetric varieties, and the moduli space of smooth hypersurfaces in projective space. Using similar arguments and results of Viehweg-Zuo, we obtain similar arithmetic finiteness (resp. non-Zariski-density) statements for complete subvarieties of the moduli of canonically polarized varieties.
• Tamely ramified morphisms of curves and Belyi’s theorem in positive characteristic (arXiv version, 2020, joint with Kiran Kedlaya and Jakub Witaszek, published in IMRN)
\(\qquad\)+ Abstract
We show that every smooth projective curve over a finite field \(k\) admits a finite tame morphism to the projective line over \(k\). Furthermore, we construct a curve with no such map when \(k\) is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.
• Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class (EPIGA, arXiv version, 2020, joint with Dean Bisogno, Wanlin Li, and Padma Srinivasan)
\(\qquad\)+ Abstract
Let \(\ell\) be a prime and \(G\) a pro-\(\ell\) group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when \(\ell=2\). We apply this to the pro-\(\ell\) étale fundamental groups of smooth curves to obtain Galois-cohomological analogues, and discuss their relationship to work of Hain and Matsumoto in the case the curve is proper. We analyze many of the fundamental properties of these classes and use them to give an example of a non-hyperelliptic curve whose Ceresa class has torsion image under the \(\ell\)-adic Abel-Jacobi map.
• Semisimplicity and weight-monodromy for fundamental groups (arXiv version, joint with Alexander Betts, 2019, appeared in Proceedings of the Simons Symposium on p-adic Hodge Theory)
\(\qquad\)+ Abstract
Let \(X\) be a smooth, geometrically connected variety over a \(p\)-adic local field \(K\). We show that the pro-unipotent fundamental group of \(X\) (in both the etale and crystalline settings) satisfies the weight-monodromy conjecture, following Vologodsky. We deduce (in the etale setting) that Frobenii act semisimply on the Lie algebra of the pro-unipotent fundamental group of \(X\), and (in the crystalline setting) that the same is true for a \(K\)-linear power of the crystalline Frobenius. We give applications to the representability and geometry of the Selmer varieties appearing in the Chabauty-Kim program, even in cases of bad reduction.
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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.
Miscellaneous
• The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces (arXiv version, 2022, joint with Philip Engel and Raju Krishnamoorthy)
\(\qquad\)+ Abstract
Let A be a simple abelian surface over an algebraically closed field k. Let \(S\subset A(k)\) be the set of torsion points of A contained in the image of a map from a genus 2 curve C to A, sending a Weierstrass point of C to the origin. The purpose of this note is to show that if k has characteristic zero, then S is finite --- this is in contrast to the situation where k is the algebraic closure of a finite field, where \(S=A(k)\), as shown by Bogomolov and Tschinkel. We deduce that if \(k=\overline{\mathbb{Q}}\), the Kummer surface associated to A has infinitely many k-points not contained in a rational curve arising from a genus 2 curve in A, again in contrast to the situation over the algebraic closure of a finite field.
Vanishing and positivity
• Vanishing for Frobenius twists of ample vector bundles (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, Algebraic Geometry, Foundation Compositio Mathematica, 2016, 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.