HU Arithmetic Geometry Seminar

Humboldt Arithmetic Geometry Seminar

Humboldt University Berlin, Summer 2025

Time: Tuesdays 15:15 - 16:45

Venue: Room 3.006, Rudower Chaussee 25, 12489 Berlin

— See also the Algebraic Geometry Seminars at HU and FU —


NOTE:	This semester the seminar will take place from 15:15 - 16:45 !!!
20.05.2025	Marco Maculan (Jussieu, Paris)
Title:	Affine vs. Stein in rigid geometry
Abstract:	What is the relation between coherent cohomology on a complex variety and that of the associated analytic space? The natural map between them is certainly not surjective for cardinality reasons. It is not even injective in general: this is a consequence of the existence of a nonaffine algebraic variety which is Stein. In a joint work with J. Poineau, we show that over non-Archimedean field the situation is, pun intended, far more rigid.
10.06.2025	Abhishek Oswal (Freiburg)
Title:	p-adic hyperbolicity of the moduli space of abelian varieties
Abstract:	By a theorem of Borel, any holomorphic map from a complex algebraic variety to the moduli space of abelian varieties (and more generally to an arithmetic variety) is in fact algebraic. A key input is to prove that any holomorphic map from a product of punctured disks to such an arithmetic variety does not have any essential singularities. In this talk, I'll discuss a p-adic analogue of these results. This is joint work with Ananth Shankar and Xinwen Zhu (with an appendix by Anand Patel).
THURSDAY 19.06.2025	Philip Engel (Chicago, Illinois)
Title:	Boundedness theorems for abelian fibrations
Abstract:	I will report on forthcoming work, joint with Filipazzi, Greer, Mauri, and Svaldi, on boundedness results for abelian fibrations. We will discuss a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded. Conditional on the generalized semiampleness/hyperkahler SYZ conjecture, this bounds the number of deformation classes of hyperkahler varieties in a fixed dimension, with second Betti number at least 5.
24.06.2025	Denis-Charles Cisinski (Regensburg)
Title:	l-adic completion of motivic sheaves
Abstract:	Rigidity theorems imply that l-adic sheaves are motivic in nature, so that l-adic realization of Voevodsky's motivic sheaves simply amounts to l-adic completion. We will explain the structural properties of this process, with positive and negative results: the analogy between rings of integers and smooth curves breaks here, since motivic sheaves behave quite differently in equal characteristics and in mixed characteristics. This yields new cohomology theories that measure the lack of functoriality of integral models in situations of good reduction. We will also discuss a conjectural property of l-adic completion of Weil-étale sheaves over Witt vectors of a finite field that is interesting in its own right: it can be shown to be a consequence of the Tate conjecture combined with a conjecture of Beilinson, and it has spectacular consequences, namely independence of l of suitable derived categories of l-adic sheaves of geometric origin over varieties over a finite field.

NOTE:

This semester the seminar will take place from 15:15 - 16:45 !!!

20.05.2025

Marco Maculan (Jussieu, Paris)

Title:

Affine vs. Stein in rigid geometry

Abstract:

What is the relation between coherent cohomology on a complex variety and that of the associated analytic space? The natural map between them is certainly not surjective for cardinality reasons. It is not even injective in general: this is a consequence of the existence of a nonaffine algebraic variety which is Stein. In a joint work with J. Poineau, we show that over non-Archimedean field the situation is, pun intended, far more rigid.

10.06.2025

Abhishek Oswal (Freiburg)

Title:

p-adic hyperbolicity of the moduli space of abelian varieties

Abstract:

By a theorem of Borel, any holomorphic map from a complex algebraic variety to the moduli space of abelian varieties (and more generally to an arithmetic variety) is in fact algebraic. A key input is to prove that any holomorphic map from a product of punctured disks to such an arithmetic variety does not have any essential singularities. In this talk, I'll discuss a p-adic analogue of these results. This is joint work with Ananth Shankar and Xinwen Zhu (with an appendix by Anand Patel).

THURSDAY 19.06.2025

Philip Engel (Chicago, Illinois)

Title:

Boundedness theorems for abelian fibrations

Abstract:

I will report on forthcoming work, joint with Filipazzi, Greer, Mauri, and Svaldi, on boundedness results for abelian fibrations. We will discuss a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded. Conditional on the generalized semiampleness/hyperkahler SYZ conjecture, this bounds the number of deformation classes of hyperkahler varieties in a fixed dimension, with second Betti number at least 5.

24.06.2025

Denis-Charles Cisinski (Regensburg)

Title:

l-adic completion of motivic sheaves

Abstract:

Rigidity theorems imply that l-adic sheaves are motivic in nature, so that l-adic realization of Voevodsky's motivic sheaves simply amounts to l-adic completion. We will explain the structural properties of this process, with positive and negative results: the analogy between rings of integers and smooth curves breaks here, since motivic sheaves behave quite differently in equal characteristics and in mixed characteristics. This yields new cohomology theories that measure the lack of functoriality of integral models in situations of good reduction. We will also discuss a conjectural property of l-adic completion of Weil-étale sheaves over Witt vectors of a finite field that is interesting in its own right: it can be shown to be a consequence of the Tate conjecture combined with a conjecture of Beilinson, and it has spectacular consequences, namely independence of l of suitable derived categories of l-adic sheaves of geometric origin over varieties over a finite field.