# Oberseminar Summer 2024

Die Vorträge finden jeweils donnerstags um 16:45 Uhr im **Raum WSC-N-U-3.05** (im Mathematikgebäude ) statt. Directions from the train station.

Der Tee findet ab 16:15 statt.

Alle Interessenten sind herzlich eingeladen!

The seminar takes place on Thursday, starting at 4:45pm. The duration of each talk is about 60 minutes.

Everybody who’s interested is welcome to join.

11.4.2024 | Yukako Kezuka (Paris) | Non-vanishing results for some CM elliptic curves |

18.4.2024 | Irene Spelta (Barcelona) | Special subvarieties of the Torelli locus: examples |

25.4.2024 | Alex Küronya (Frankfurt) | Structures on cohomology and P=W for abelian varieties |

16.5.2024 | Mihai Pavel (Bukarest) | Variation of moduli of sheaves in higher dimensions |

23.5.2024 | Stefano Marseglia (University of French Polynesia) | Cohen-Macaulay type of endomorphism rings of abelian varieties over finite fields |

6.6.2024 | – | reserviert |

13.6.2024 | Quentin Posva (Düsseldorf) | On the singularities of 1-foliations |

20.6.2024 | V. Srinivas (SUNY Buffalo) | An obstruction to lifting to characteristic 0 using the etale fundamental group |

27.6.2024 | – | Symposium Düsseldorf/Essen/Wuppertal |

4.7.2024 | Mario Kummer (TU Dresden) | The arithmetic writhe |

11.7.2024 | Alexander Petrov (IAS) | Secondary characteristic classes in arithmetic geometry |

18.7.2024 | Michael Thaddeus (Columbia) | Moduli of flags of sheaves on curves |

## Abstracts

### Yukako Kezuka: Non-vanishing results for some CM elliptic curves

We prove non-vanishing results for the values at $s=1$ of the complex L-series of quadratic twists of the elliptic curves with complex multiplication introduced by B. Gross in his thesis. From this, we obtain the finiteness of their Tate–Shafarevich group. For a prime $\mathfrak P$ lying above the prime 2, we also prove a $\mathfrak P$-converse theorem in the rank 0 case for the higher-dimensional abelian varieties obtained by restriction of scalars.

### Irene Spelta: Special subvarieties of the Torelli locus: examples

The Torelli morphism $j \colon \mathcal M_g \to \mathcal A_g$ associates to an algebraic curve $C$ its Jacobian $J_C$ (as a principally polarized abelian variety). Very classically, it is injective on geometric points. In this talk, we will discuss the interplay between the geometry of $\mathcal M_g$ and of $\mathcal A_g$. In particular, we will consider certain special subvarieties. We will explain how this is linked to a conjecture by Coleman and Oort, and we will analyse explicit examples.

### Alex Küronya: Structures on cohomology and P=W for abelian varieties

The purpose of this talk is to discuss the significance of various extra structures on the (co)homology of algebraic varieties, leading to a discussion of the recent P=W conjecture and its verification for abelian varieties.

### Mihai Pavel: Variation of moduli of sheaves in higher dimensions

We introduce a notion of semistability for coherent sheaves over a projective manifold, which is well-behaved from the perspective of variation in higher dimensions. Specifically, we obtain (new) moduli spaces of semistable pure sheaves, along with a locally finite wall-chamber structure that governs their variation with the change of semistability. This involves showing uniform boundedness statements for semistable pure sheaves.

### Stefano Marseglia: Cohen-Macaulay type of endomorphism rings of abelian varieties over finite fields

Let $A$ be an abelian variety over a finite field $\mathbb F_q$ with commutative $\mathbb F_q$-endomorphism ring $S$. We will study the (Cohen-Macaulay) type of $S$ and see some of its applications. More precisely, when the type is 1 or 2, we will see: that we can easily compute the isomorphism classes of the abelian varieties isogenous to $A$ with the same endomorphism ring $S$; that the group of $\mathbb F_q$-rational points of $A$ is uniquely determined by $S$; a necessary condition for $A$ to not be isomorphic to its dual variety.

### Quentin Posva: On the singularities of 1-foliations

In positive characteristic geometry, purely inseparable maps can be described in terms of sheaves of tangent vectors called 1-foliations. Their global properties have been used by several authors to produce pathological examples, but their local structure has not been extensively studied outside of a few cases. In this talk, I will present new results about singularities 1-foliations on surfaces, including a full resolution theorem.

### V. Srinivas: An obstruction to lifting to characteristic 0 using the etale fundamental group

The etale fundamental group of a smooth proper variety over an algebraically closed field of positive characteristic is a profinite group, now known to be finitely presented. In this talk, I will discuss a certain group theoretic property of this profinite group which needs to be satisfied in case the variety has a smooth proper lifting to characteristic 0; I will also discuss examples to show this is not always valid, thus providing a new obstruction to the existence of a lifting. This is a report on joint work with H. Esnault and J. Stix.

### Mario Kummer: The arithmetic writhe

We present an $\mathbb{A}^1$-count of secant lines to a space curve from a joint work with Daniele Agostini and explain how it can be thought of as an arithmetic knot invariant that we call the arithmetic writhe. We show that two smooth rational space curves of degree four are $\mathbb{A}^1$-isotopic if and only if they have the same arithmetic writhe.

### Alexander Petrov: Secondary characteristic classes in arithmetic geometry

To a complex vector bundle on a smooth manifold we can attach Chern classes living in singular cohomology in even degrees. If the bundle admits a flat connection, then its Chern classes are torsion, but moreover the particular choice of a flat connection gives rise to secondary characteristic classes living in odd degree cohomology. Following a joint work in progress with Lue Pan, I will discuss an analog of this construction in arithmetic geometry, where the role of a vector bundle with a flat connection is played by an etale local system on a scheme, and the secondary characteristic classes now take values in absolute etale cohomology. These classes can be partially computed by the methods of p-adic Hodge theory, and in some ways behave differently from their differential-geometric counterparts.

### Michael Thaddeus: Moduli of flags of sheaves on curves

The moduli problem of vector bundles on a fixed smooth curve has long been a favorite object of study in algebraic geometry. Such bundles are parametrized by a smooth Artin stack, locally of finite type, which contains only one smooth projective variety, namely the moduli space of stable bundles. Many variations on this theme have also been studied. In this seminar, a particularly interesting such variation will be described and explored, that of vector bundles equipped with a flag of coherent subsheaves. In contrast to the classical case, this problem admits many stability parameters; indeed, the case of rank n bundles admits 2n stability conditions. It is, roughly speaking, like studying bundles with parabolic structure at the generic point. The birational geometry of the moduli spaces of flags sets up correspondences between spaces of vector bundles of different ranks, raising the possibility of studying the enumerative geometry of these classical spaces by induction on the rank.