Department of Mathematics, Statistics, and Computer Science

851 S. Morgan St.

Chicago, IL 60607

I am an associate professor of Mathematics at UIC.

UIC Algebraic Geometry Seminar

Midwest Algebraic Geometry Graduate Conference (MAGGC), UIC, August 10-11 2022.

**A short proof of a conjecture of Matsushita**

preprint (pdf, arXiv)**Functional transcendence of periods and the geometric André–Grothendieck period conjecture**

with J. Tsimerman

preprint (pdf, arXiv)**Definable structures on flat bundles**

with S. Mullane

submitted (pdf, arXiv)**Finiteness for self-dual classes in integral variations of Hodge structure**

with T. W. Grimm, C. Schnell, and J. Tsimerman

preprint (pdf, arXiv)**Quasiprojectivity of images of mixed period maps**

with Y. Brunebarbe and J. Tsimerman

submitted (pdf, arXiv)**o-minimal GAGA and a conjecture of Griffiths**

with Y. Brunebarbe and J. Tsimerman

*Invent. Math.*, to appear (pdf, arXiv)**Definability of mixed period maps**

with Y. Brunebarbe, B. Klingler, and J. Tsimerman

*J. Eur. Math. Soc.*, to appear (pdf, arXiv)**The global moduli theory of symplectic varieties**

with C. Lehn

*J. Reine Angew. Math.*, Volume 2022, No. 790 (2022) (pdf, arXiv, journal)**Algebraic approximation and the decomposition theorem for Kahler Calabi–Yau varieties**

with H. Guenancia and C. Lehn

*Invent. Math.*, Volume 228, Issue 3 (2022) (pdf, arXiv, journal)**A global Torelli theorem for singular symplectic varieties**

with C. Lehn

*J. Eur. Math. Soc.*, Volume 23, Issue 3 (2021) (pdf, arXiv, journal)**Tame topology of arithmetic quotients and algebraicity of Hodge loci**

with B. Klingler and J. Tsimerman

*J. Amer. Math. Soc.*, Volume 33, No. 4 (2020) (pdf, arXiv, journal)**Erratum**, submitted (pdf)**The Ax-Schanuel conjecture for variations of Hodge structures**

with J. Tsimerman

*Invent. Math.*, Volume 217, No. 1 (2019) (pdf, arXiv, journal)**The Mercat conjecture for stable rank 2 vector bundles on generic curves**

with G. Farkas

*Amer. J. Math.*, Volume 140, No. 5 (2018) (pdf, arXiv, journal)**The geometric torsion conjecture for abelian varieties with real multiplication**

with J. Tsimerman

*J. Differential Geom.*, Volume 109, No. 3 (2018) (pdf, arXiv, journal)**The Kodaira dimension of complex hyperbolic manifolds with cusps**

with J. Tsimerman

*Compos. Math.*, Volume 154, Issue 3 (2018) (pdf, arXiv, journal)**A classification of Lagrangian planes in holomorphic symplectic varieties**

*J. Inst. Math. Jussieu*, Volume 16, Issue 4 (2017) (pdf, arXiv, journal)**p-torsion monodromy representations of elliptic curves over geometric function fields**

with J. Tsimerman

*Ann. of Math.*184, No. 3 (2016) (pdf, arXiv, journal)**Lagrangian 4-planes in holomorphic symplectic varieties of K3^[4] type**

with A. Jorza

*Cent. Eur. J. Math.*, Volume 12, Issue 7 (2014) (pdf, arXiv, journal)

Computational appendix and Code**On the Frey-Mazur conjecture over low genus curves**

with J. Tsimerman

arXiv preprint (2013) (pdf, arXiv)**Higher rank stable pairs on K3 surfaces**

with A. Jorza

*Commun. Number Theory Phys.*, Volume 6, Number 4 (2012) (pdf, arXiv, journal)**Hodge polynomials of moduli spaces of stable pairs on K3 surfaces**

My thesis from Princeton University, under Rahul Pandharipande

June 2010 (pdf)

**Hodge theory and o-minimality**

Notes from the Felix Klein lecture series in Bonn, May 2019 (pdf)**Lectures on the Ax-Schanuel Conjecture**

with J. Tsimerman

*Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces*, CRM Short Courses, Springer (2020) (pdf, book)**Slides**

**The Matsushita alternative**

Hyperkähler varieties and related topics, September 2022.

Abstract.Matsushita conjectured that a Lagrangian fibration of an irreducible hyperkähler manifold is either isotrivial or of maximal variation. In this talk I will show how to prove this conjecture by adapting previous work of Voisin and van Geemen. I will also deduce some applications to the density of torsion points of sections of Lagrangian fibrations.**Period integrals of algebraic varieties**

Non-abelian Hodge theory, Saint-Jacut de la Mer, June 2022.

Abstract.Period integrals on complex algebraic varieties are the integrals of algebraic differential forms along topological cycles. They are at the heart of Hodge theory. In this talk I will survey some recent results on the behavior of the functions obtained by taking period integrals in algebraic families. In particular I will discuss the proof of an Ax–Schanuel type theorem on the transcendence of these functions and show it is equivalent to a geometric version of the André–Grothendieck period conjecture. This is joint work with J. Tsimerman.**Compact hyperkähler varieties: basic results**

Derived Categories, Moduli Spaces, and Hyperkähler Varieties, Ann Arbor, August 2022.

Abstract.Compact hyperkähler manifolds enjoy a number of nice properties, many of which are connected to the Hodge structure on their weight 2 cohomology. Surprisingly, much of this theory extends to the case of singular compact hyperkähler varieties, which arise naturally even in the study of hyperkähler manifolds and are interesting in and of themselves. The goal of these lectures is to introduce the basic objects and survey some important recent developments. Topics will include: basic definitions and examples, Hodge theory and deformation theory, birational geometry and the global Torelli theorem, and the Beauville-Bogomolov decomposition theorem.**Algebraic approximation of Calabi–Yau varieties and the decomposition theorem**

Geometry and TACoS, November 2020.

Abstract.Calabi–Yau manifolds are built out of simple pieces by the Beauville–Bogomolov decomposition theorem: any Calabi–Yau Kähler manifold up to an etale cover is a product of complex tori, irreducible holomorphic symplectic manifolds, and strict Calabi–Yau manifolds (which have no holomorphic forms except a holomorphic volume form). Work of Druel–Guenancia–Greb–Horing–Kebekus–Peternell over the last decade has culminated in a generalization of this result to projective Calabi–Yau varieties with the kinds of singularities that arise in the MMP, and the proofs critically use algebraic methods. In this talk I will describe joint work with H. Guenancia and C. Lehn extending the decomposition theorem to nonprojective varieties via deformation theory. A crucial step in the proof is the resolution of the K-trivial case of a conjecture of Peternell asserting that any minimal Kähler variety can be approximated by algebraic varieties.

My course websites are maintained here.

- UIC Spring 2022: Hyperkähler manifolds (Math 571).
- UIC Fall 2021: Second Course in Abstract Algebra I (Math 516).
- UIC Fall 2021: Definable Complex Analytic Geometry (Math 571).
- UIC Spring 2021: Linear Algebra (Math 320) -- (10am, 1pm).
- UGA Spring 2020: Moduli spaces (Math 8330).
- UGA Fall 2019: Calculus II for Science and Engineering (Math 2260)
- UGA Fall 2018: Differential equations (Math 2700).
- UGA Fall 2018: (Counter)examples in char. p geometry (Math 8330).
- UGA Spring 2018: Commutative algebra (Math 8020).
- UGA 2017-2018: Topics in Hodge theory (VRG) (Math 8850).
- UGA Fall 2017: Differential equations (Math 2700).
- UGA Spring 2017: Modern algebra and geometry I (Math 4000/6000).
- UGA Spring 2017: Abelian varieties (Math 8330).
- HU Summer 2016: Berkovich spaces.
- HU Winter 2015: Literature Seminar.
- HU Summer 2015: Faltings theorem.
- HU Winter 2014: Abelian varieties and Fourier-Mukai transforms.
- NYU Spring 2013: Number Theory (MATH-GA 2210.001).
- NYU Fall 2012: Theory of Numbers (MATH-UA 248.001).
- NYU Fall 2011: Algebra I (MATH-GA 2130.001).
- NYU Spring 2011: Topology II (MATH-GA 2320.001).
- NYU Fall 2010: Calculus I (V63.0121.026.FA10).

**Ben Tighe****Zhehao Li**