Humboldt University Berlin, Summer 2024

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 —

30.04.2024 | Alessandro Ghigi (Pavia) |

Title: | Projective structures on Riemann surfaces and metrics on the moduli space of curves |

Abstract: | I will describe some recent results on projective structures on Riemann surfaces. After recalling some basic definitions I will explain a correspondence between varying projective structures over the moduli space of curves and (1,1)-forms over it. I will describe explicitely the correspondence in two examples: the projective structure coming from uniformization and a projective structure coming from Hodge theory. Finally I will also describe a new projective structure obtain from the line bundle \( 2 \Theta\). |

07.05.2024 | Victor Gonzalez Alonso (Hannover) |

Title: | Infinitesimal rigidity of certain modular morphisms |

Abstract: | The Torelli morphism maps (the isomorphism class of) a smooth complex projective curve to its polarized jacobian variety. It has been recently proved by Farb that this is the only non-constant holomorphic map from the moduli space of curves to that of principally polarized abelian varieties, and Serván has recently proved a similar result for the Prym morphism. These result can be interpreted by saying that a certain moduli space of morphisms consists of just one point, and it is natural to ask whether this point is reduced. In this talk I will present a joint work with Giulio Codogni and Sara Torelli, where we show that this is indeed the case (in the setting of moduli stacks): These morphisms do not admit non-trivial infinitesimal deformations. The proof uses the Fujita decomposition of the Hodge bundle of a family of curves, and can be applied to other morphisms involving moduli of smooth curves. |

21.05.2024 | Emiliano Ambrosi (Strasbourg) |

Title: | Reduction modulo p of the Noether problem |

Abstract: | Let k be an algebraically closed field of characteristic \( p\ge 0 \) and V a faithful k-rational representation of an \(l\)-group G. Noether's problem asks whether V/G is (stably) birational to a point. If \( l = p \), then Kuniyoshi proved that this is true, while for \( l\neq p \) Saltman constructed \(l\)-groups for which V/G is not stably rational. Hence, the geometry of V/G depends heavily on the characteristic of the field. We show that for all the groups G constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and p. More precisely, we show that there does not exist a complete valuation ring R of mixed characteristic (0,p) and a smooth proper R-scheme \( X \to \mathrm{Spec}(R) \) whose special fiber and generic fiber are both stably birational to V/G. The proof combines the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a joint work with Domenico Valloni. |

28.05.2024 | Paul Ziegler (Darmstadt) |

Title: | p-adic integration on Artin stacks |

Abstract: | After giving an introduction to the technique of p-adic integration, I will explain how this technique can be extended to Artin stacks, and give an application to BPS invariants. This is joint work with M. Groechenig and D. Wyss. |

11.06.2024 | Marco Flores (HU Berlin) |

Title: | Modularity of formal Fourier-Jacobi series from a cohomological point of view |

Abstract: |
In the Kudla program, generating series constructed by means of arithmetic intersection numbers are conjectured to appear as Fourier coefficients of modular forms. One instance of this program was the proof in 2015 by Bruinier and Raum of Kudla's modularity conjecture over \(\mathbb{C}\) for special cycles on a Shimura variety of orthogonal type, obtained by showing that certain formal series of Jacobi forms are in fact modular forms. This result was reformulated and proven over \(\mathbb{Z}\) by Kramer using the Faltings-Chai theory of toroidal compactifications of \(\mathcal{A}_g\). In this talk, we contextualize the above result using cohomology of line bundles on such a toroidal compactification \(\overline{\mathcal{A}}_g\), which we believe is the more natural point of view. We then obtain cohomological vanishing results characterizing the modularity of formal Fourier-Jacobi series, taking advantage of the natural resolution morphism from \(\overline{\mathcal{A}}_g\) to the minimal compactification \(\mathcal{A}^*_g\). |

25.06.2024 | Gari Yamel Peralta Alvarez (HU Berlin) |

Title: | Toric arithmetic varieties and adelic intersection numbers |

Abstract: | A result of Burgos, Philippon and Sombra states that the height of a toric arithmetic variety with respect to a line bundle equipped with a torus-invariant continuous semipositive metric is given by the integral of a convex function over a polytope. In this talk, I will discuss a generalization of this formula for line bundles with torus-invariant singular metrics. The techniques used to obtain this result are based on Yuan and Zhang's theory of adelic line bundles. Moreover, this formula agrees with the generalized adelic intersection numbers of Burgos and Kramer. |

02.07.2024 | Remy van Dobben de Bruyn (Utrecht) |

Title: | Constructible sheaves and exodromy |

Abstract: | Locally constant sheaves are most easily understood as representations of the fundamental group, via the monodromy correspondence. In algebraic geometry, it is often preferable to use the larger class of constructible sheaves, as these are stable under (higher) pushforward. In 2018, Barwick, Glasman, and Haine proved an _exodromy correspondence_ for constructible étale sheaves, using ideas from higher topos theory and profinite stratified homotopy theory. In this talk, I will present a more direct geometric proof of the étale and pro-étale exodromy theorems, based on joint work with Sebastian Wolf. |

16.07.2024 | Jürg Kramer (HU Berlin) |

Title: | Arithmetic intersections of line bundles with singular metrics |

Abstract: | In our talk, we will present an extension of arithmetic intersection theory of adelic divisors on quasi-projective varieties introduced by Yuan-Zhang to the case where these divisors are not necessarily arithmetically nef. The key tool to realize this extension is the concept of relative finite energy established by T. Darvas et al.. In particular, our theory will allow to compute heights on mixed Shimura varieties, e.g. the arithmetic self-intersection number of the line bundle of Siegel-Jacobi forms on the universal abelian variety. This is joint work with José Burgos Gil. |