Humboldt University Berlin, Summer 2023
Time: Tuesdays 13:15 - 14:45
Venue: Room 3.006, Rudower Chaussee 25, 12489 Berlin
— See also the Algebraic Geometry Seminars at HU and FU —
| 25.04. | Ruijie Yang (MPI Bonn) |
| Title: | The Riemann-Schoktty problem and Hodge theory |
| Abstract: | It is a classical problem, going back to Riemann, to decide which principally polarized abelian varieties come from the Jacobians of curves. In 2008, Casalaina-Martin proposed a question towards the Riemann-Schottky problem in terms of the codimension of multiplicity locus of the theta divisor, which includes Debarre’s conjecture as a special case. In this talk, I will talk about a partial affirmative answer to this question using a newly developed theory “Higher multiplier ideals”, which builds on Sabbah-Schnell’s theory of complex Hodge modules and Beilinson-Bernstein’s language of twisted D-modules. It is based on joint work with Christian Schnell. |
| 02.05. | Marco Maculan (IMJ Paris) |
| Title: | Counting rational points on varieties with large fundamental group |
| Abstract: | A nonsingular projective curve of genus at least 2 on a number field admits only finitely many rational points. Elliptic curves might have infinitely many rational points (as the projective line does), but “way less” than the projective line. In a joint work with Y. Brunebarbe, inspired by a recent result of Ellenberg-Lawrence-Venkatesh, we prove an analogous statement in higher dimension: projective varieties with large fundamental group in the sense of Kollár-Campana have “way less” rational points than Fano varieties. |
| 09.05. | Marco Flores (HU Berlin) |
| Title: | A cohomological approach to formal Fourier-Jacobi series |
| Abstract: | Siegel modular forms admit various expansions, one of the most important being the Fourier-Jacobi expansion. Algebraically, these expansions take the form of a series whose coefficients are Jacobi forms satisfying a certain symmetry condition. One poses the following modularity question: does every formal series of this shape arise from a Siegel modular form? Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014. In this talk I will consider this question over the ring of integers, and reformulate it as a matter of cohomological vanishing on a toroidal compactification of the moduli space A_g of principally polarized abelian varieties. I will present a proof that said vanishing is equivalent to our modularity question, and explore its relationship with the singularities of the minimal compactification of A_g. |
| 16.05. | Emil Jacobsen (U Zürich / Stockholm University) |
| Title: | On (generic) motivic fundamental groups |
| Abstract: | I will introduce the (generic) motivic fundamental group of a smooth variety, and explain its relation to the usual (generic) fundamental group. The main result can also be phrased as follows: generic local systems of motivic origin are stable under extension in the category of generic local systems. I will also present a (generic) motivic version of a classical theorem of Hain, on Malcev completions of monodromy representations. At the end, I'll explain some of the group/representation theoretic tools that go into these result. If there's time, I can explain some analogous Hodge theoretic results, as well as some future directions. |
| 23.05. | Alexandre Minets (U Edinburgh) |
| Title: | A proof of the P=W conjecture |
| Abstract: | Let \( C \) be a smooth projective curve. The non-abelian Hodge theory of Simpson is a homeomorphism between the character variety \( M_B \) of \( C \) and the moduli of (semi)stable Higgs bundles \( M_D \) on \( C \). Since this homeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on \( H^*(M_B) \) gets sent to the perverse filtration on \( H^*(M_D) \), associated to the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on \( H^*(M_D) \). Based on joint work with T. Hausel, A. Mellit, O. Schiffmann. |
| 30.05. | Josh Lam (HU Berlin) |
| Title: | Properties of non-abelian Hodge theory mod p: Periodicity |
| Abstract: | This is part of the reading group on flat connections in positive and mixed characteristic. For details see here. |
| 06.06. | Haohao Liu (IMJ Paris) |
| Title: | Fourier-Mukai transform on complex tori |
| Abstract: | Classical Fourier transform occupies a major part of the analysis. An analog on abelian varieties is introduced by S. Mukai in 1981, which is now known as Fourier-Mukai transform. Similar to the Fourier inversion formula, Mukai proved a duality theorem for his transform. This result reveals the phenomenon that, the derived category of coherent modules of two non-isomorphic projective varieties can be equivalent. In this talk, I will present the work of O. Ben-Bassat, J. Block and T. Pantev about the analytic version of Fourier-Mukai transform on complex tori. |
| 13.06. | Gabriel Ribeiro (École Polytechnique Paris) |
| Title: | Cartier duality, character sheaves, and generic vanishing |
| Abstract: | Since Green and Lazarsfeld's seminal work, generic vanishing theorems have influenced many fields ranging from birational geometry to analytic number theory. In this talk, I will present a new generic vanishing theorem for holonomic D-modules, that may shed new light on both of these worlds. Following ideas of Laumon, I'll explain how "character sheaves" play the role of topologically trivial line bundles and construct their moduli space based on a stacky version of Cartier duality. |
| 20.06. | Michael McBreen (U Hong Kong) |
| Title: | Introduction to Microlocal Sheaves |
| Abstract: | This is the first of a series of lectures about microlocal sheaves. Details can be found here. |
| 27.06. | Per Salberger (U Gothenburg) |
| Title: | On n-torsion in class groups of number fields |
| Abstract: | It is well known that the class group of a number field is of size bounded above by roughly the square root of its discriminant. But one expects by conjectures of Cohen-Lenstra that the the n-torsion part of this group should be much smaller and there have recently been several papers on this by prominent mathematicians. We present in our talk some of these results and a partial improvement of an estimate of Bhargava, Shankar, Taniguchi, Thorne, Zimmerman and Zhao. |
| 04.07. | Hsueh-Yung Lin (National Taiwan University) |
| Title: | Motivic invariants of birational maps and Cremona groups |
| Abstract: | (Joint with E. Shinder) In characteristic zero, birational maps of projective varieties factorize through a sequence of blow-ups and blow-downs along smooth centers. We study to which extent these factorization centers are unique, and construct invariants of motivic nature which account for the non-uniqueness of centers. For surfaces over a perfect field, we prove (with E. Shinder and S. Zimmermann) the uniqueness of centers in the strongest possible sense. In higher dimension, we construct examples showing that the centers fail to be unique. Relying on the non-uniqueness, we provide new explanations of the non-simplicity of Cremona groups. |
| 11.07. | Jan Hendrik Bruinier (TU Darmstadt) |
| Title: | Special cycles on toroidal compactifications of orthogonal Shimura varieties |
| Abstract: | A famous theorem of Gross-Kohnen-Zagier states that the generating series of Heegner divisors on a modular curve is a weight 3/2 modular form with values in the first Chow group. An analogous result for special divisors on orthogonal Shimura varieties was proved by Borcherds, and for higher codimension special cycles by Zhang, Raum and myself. We report on joint work with Shaul Zemel generalizing the modularity result to special divisors on toroidal compactifications of orthogonal Shimura varieties. |