Benjamin Bakker

• 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)
• 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
  Forum Math. Sigma, to appear (pdf, arXiv)
• Appendix to "Teichmüller curves in hyperelliptic components of meromorphic strata", by M. Möller and S. Mullane
  with S. Mullane
  J. Inst. Math. Jussieu, to appear (arXiv, journal)
• Definability of mixed period maps
  with Y. Brunebarbe, B. Klingler, and J. Tsimerman
  J. Eur. Math. Soc., Volume 26, No. 6 (2024) (pdf, arXiv, journal)
• 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)
• 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

• The linear Shafarevich conjecture for quasiprojective varieties
  Algebraic Geometry MATRIX, December 2024.
  Abstract.
  Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex, meaning it admits a proper map to a Stein space. This was proven in the linear case---namely when X admits an almost faithful representation of its fundamental group---by Eyssidieux--Katzarkov--Pantev--Ramachandran using techniques from non-abelian Hodge theory. In joint work with Y. Brunebarbe and J. Tsimerman, we prove a version of the linear Shafarevich conjecture for quasiprojective varieties. The proof relies on a number of recent advances in non-abelian Hodge theory in the non-proper case, and we will focus on the role played by the twistor geometry of the stack of local systems.
• Integral canonical models and period maps
  Geometry over non-closed fields, Schloss Elmau, August 2024.
  Abstract.
  Work of Chai--Faltings, Milne, Moonen, Kisin, and Kottwitz constructs integral canonical models for abelian type Shimura varieties, but as the construction relies on the modular interpretation of the moduli space of abelian varieties, it does not apply to exceptional Shimura varieties. In joint work with A. Shankar and J. Tsimerman, we construct integral canonical models for all Shimura varieties at sufficiently large primes, as well as for the image of any period map arising from geometry. Our method passes through finite characteristic and relies on a partial generalization of the work of Ogus--Vologodsky. As applications, in the context of exceptional Shimura varieties we prove analogs of Tate semisimplicity in finite characteristic, CM lifting theorems for ordinary points, and the Tate isogeny theorem for ordinary points.
• 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.

My course websites are maintained here.

• UIC Spring 2025: Non-abelian Hodge theory (Math 571).
• UIC Spring 2025: Abstract algebra I (Math 330).

• 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