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.

**CM points have everywhere good reduction**

with J. Tsimerman

preprint (pdf, arXiv)**Periods in families and derivatives of period maps**

with J. Pila and J. Tsimerman

preprint (pdf, arXiv)**The linear Shafarevich conjecture for quasiprojective varieties and algebraicity of Shafarevich morphisms**

with Y. Brunebarbe and J. Tsimerman

preprint (pdf, arXiv)**Integral canonical models of exceptional Shimura varieties**

with A. N. Shankar and J. Tsimerman

preprint (pdf, arXiv)**A Hodge theoretic proof of Hwang's theorem on base manifolds of Lagrangian fibrations**

with C. Schnell

preprint (pdf, arXiv)**Appendix to "Teichmüller curves in hyperelliptic components of meromorphic strata"**, by M. Möller and S. Mullane

with S. Mullane

preprint (arXiv)**A short proof of a conjecture of Matsushita**

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

with J. Tsimerman

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

with Y. Brunebarbe and J. Tsimerman

*J. Reine Angew. Math.*, Volume 2023, No. 804 (2023) (pdf, arXiv, journal)**Definable structures on flat bundles**

with S. Mullane

*Bull. Lond. Math. Soc.*, Volume 55, Issue 5 (2023) (pdf, arXiv, journal)**Definability of mixed period maps**

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

*J. Eur. Math. Soc.*, to appear (pdf, arXiv, journal)**Finiteness for self-dual classes in integral variations of Hodge structure**

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

*Épijournal Géom. Algébrique*, Special volume in honour of Claire Voisin (2023) (pdf, arXiv, journal)**o-minimal GAGA and a conjecture of Griffiths**

with Y. Brunebarbe and J. Tsimerman

*Invent. Math.*, Volume 232, Issue 1 (2023) (pdf, arXiv, journal)**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 to “Tame topology of arithmetic quotients and algebraicity of Hodge loci”**

*J. Amer. Math. Soc.*, Volume 36, No. 4 (2023) (pdf, journal)**The Ax-Schanuel conjecture for variations of Hodge structures**

with J. Tsimerman

*Invent. Math.*, Volume 217, Issue 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**

**Hwang's theorem on the base of a Lagrangian fibration revisited**

Geometry of Hyperkähler varieties, Hangzhou, September 2022.

Abstract.Irreducible hyperkahler manifolds are higher dimensional analogs of K3 surfaces; their geometry is tightly controlled by the existence of a nowhere degenerate holomorphic 2-form. The only nontrivial fibration structure f:X -> B a hyperkahler manifold X admits is a fibration by Lagrangian tori, and for such Lagrangian fibrations the base B is conjectured to always be isomorphic to projective space. In 2008 Hwang proved that this is the case if B is assumed to be smooth by using the theory of varieties of minimal rational tangents on Fano manifolds. In this talk I will present a simpler proof of this result which leans more heavily on Hodge theory. Specifically, the main input is a basic functoriality result coming from Hodge modules. This is joint work with C. Schnell.**The geometric André--Grothendieck period conjecture**

Midrasha Mathematicae, December 2023.

Abstract.A period integral of a complex algebraic variety is the integral of an algebraic differential form along a topological cycle. These numbers are at the heart of Hodge theory. In this talk I will explain how to prove a version of the Ax--Schanuel conjecture for these period integrals in families, and how it provides a capstone to the advances in the transcendence theory of period maps made over the past decade. I will also discuss the relationship with the functional version of the Andre--Grothendieck period conjecture, which predicts that all algebraic relations between such periods integrals arise from geometry. This is joint work with J. Tsimerman.**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.

My course websites are maintained here.

- UIC Fall 2024: Complex manifolds II (Math 555).

- UIC Fall 2023: Introduction to Advanced Mathematics (Math 215) -- (11am, 1pm).
- UIC Fall 2022: Calculus III (Math 210) -- (4pm, 5pm).
- 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****Hank Morris****Anh Tran****Edward Varvak**