Seminar on Algebraic Geometry and Ramification

Organizers

Fangzhou Jin, Peng Sun, Enlin Yang, Yigeng Zhao

Upcoming talks

Past talks

Yisheng Tian (SUSTech): Some arithmetic properties of linear groups over p-adic function fields
2021-12-27 15:00-16:00
Zoom ID:899 2860 6536 Password:984423
In this talk, we first recall several classical results about the arithmetic of linear groups over number fields. Namely, we talk briefly about the Hasse principal, weak approximation and the Borel-Serre theorem. Subsequently, we introduce some arithmetic dualities associated to reductive groups and obstructions to weak approximation for certain reductive groups over p-adic function fields. These works motivate us to study weak approximation for semi-simple simply connected groups and related Galois cohomology theoretic results on the finiteness (trivialness) of the Tate-Shafarevich set.  If we still have some time, we will also talk about the obstruction to the Hasse principal for torsors under tori and obstructions to weak approximation for homogenous spaces under reductive groups.

Yichen Qin (École Polytechnique): L-functions of Kloosterman sheaves
2021-12-27 16:00-17:00
Zoom ID:899 2860 6536 Password:984423
Kloosterman sums are finite field analogues of Bessel functions. They appear as traces of Frobenius on some \ell-adic local systems Kl_{n+1} on G_m, called Kloosterman sheaves. In a recent work, Fresán-Sabbah-Yu have constructed some motives attached to symmetric powers of Kloosterman sheaves Sym^k Kl_{n+1}, and showed that for n=1, their L-functions have meromorphic extensions to the complex plane and satisfy functional equations. For small values of k, these L-functions are given by modular forms that one can explicitly write down. In this talk, I will present some new results about the L-functions of Sym^k Kl_3. In particular, I will specify some modular forms and explain why these motives are modular in these cases.

Ran Azouri (University of Duisburg-Essen): Quadratic conductor formulas
2021-12-07 16:00-17:00
Zoom ID:821 8253 0518 Password:071242
Motivic methods allow replacing the ring of integers by the Grothendieck-Witt ring of a field to get refined versions to formulas in algebraic geometry. We will review Milnor's number formula for complex degenerations and its analogues in algebraic geometry. We will then report on a work by Levine, Pepin Lehalleur and Srinivas, computing the motivic Euler characteristic for projective hypersurfaces and obtaining a refined conductor formula in quadratic forms. We will describe how to compute the motivic Euler characteristic of the motivic nearby cycles functor to obtain a quadratic conductor formula for a more general degeneration with several isolated singularities, refining Milnor's formula.

Zhizhong Huang(Institute of Science and Technology Austria): The Hilbert Irreducibility Theorem for Kummer varieties
2021-11-29,16:00-17:00
Zoom ID:858 9779 5553 Password:007574
An algebraic variety X defined over a number field k is called Hilbertian if the set of rational points X(k) is not thin. In other words, rational points on any Zariski dense open subset of X do not lift into any collection of finitely many dominant finite morphisms of degree>1. Clearly the variety X being Hilbertian is stronger than the set X(k) being Zariski dense. This notion originates from constructing finite extensions with prescribed Galois group and goes back to Hilbert and Noether. A conjecture of Corvaja and Zannier predicts that simply connected varieties are Hilbertian. In this talk we report our result which confirms this conjecture for a family of Kummer varieties associated to the jacobians of hyperelliptic curves of genus at least two. This is based on joint work in progress with Damián Gvirtz (UCL).

Xiaoyu Su(Tsinghua University): A Noether-Lefschetz Theorem for Spectral Varieties with Applications
2021-10-25,10:00-11:00
Zoom ID:831 3317 0890 Password:078177
In this talk we will talk about our recent results about the Picard group of generic (very general) Vafa-Witten spectral varieties. We will first review the concept of spectral varieties, Higgs sheaves and their relation with the Noether-Lefschetz type Theorems. Then we introduce our main results and an application.

Jinhyun Park(KAIST): On motivic cohomology of singular algebraic schemes
2021-10-25 13:00-14:00
Zoom ID:820 6945 5808 Password:352729
Motivic cohomology is a hypothetical cohomology theory for algebraic schemes, including algebraic varieties, that can be seen as the counterpart in the domain of algebraic geometry, to the singular cohomology of topology. It has been constructed for smooth varieties, but for the singular ones the situation was not clear. In this talk, I will sketch some recent attempts of mine to provide an algebraic-cycle-based model for the motivic cohomology of singular algebraic schemes, using formal schemes. As this is very complicated, in the talk I will concentrate more on the concrete case of the fat points, where the situation is simpler, but not still trivial.

Ning Guo(Euler International Mathematical Institute): Grothendieck-Serre over valuation rings
2021-09-30 15:00-16:00
The Grothendieck-Serre conjecture predicts that for a reductive group scheme G over a regular local ring R, there is no nontrivial G-torsor over R trivializes over Frac(R). In this talk, we consider the case when R is instead assumed to be a valuation ring. This result is predicted by the original Grothendieck-Serre and the resolution of singularities. By using flasque resolution of tori, we prove for groups of multiplicative type. Subsequently, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat-Tits theory to conclude. In this talk, time permitting, we focus on techniques in algebraization (such as Gabber-Ramero triples) and their applications to lifting torsors and group schemes from certain closed subschemes.

Adeel Khan (Academia Sinica): Derived microlocalization and singular supports of étale sheaves in mixed characteristic
2021-08-18 10:00-11:00
Beilinson introduced an analogue of Kashiwara and Schapira's singular support for étale sheaves on a smooth scheme of equal characteristic. Recently, Saito has proposed an analogue of singular support for regular schemes in mixed characteristic, but it is only known to exist in some special examples. I will explain how to define a derived version of Verdier's specialization functor, and use it to give a candidate construction of the singular support in general.

Fangzhou Jin (Tongji University): A Gersten complex on real schemes
2021-07-15 09:00-10:00

Nanjun Yang (Tsinghua University): Split Milnor-Witt Motives
2021-07-15 10:15-11:15

Fei Ren (University of Wuppertal): Higher Chow groups and coherent duality
2021-05-10

Haoyu Hu (Nanjing University): Semi-continuity of conductors and ramification bound of nearby cycles
2021-03-15 15:00-16:00
In this talk, we firstly discuss a lower semi-continuity property for conductors of étale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumon's lower semi-continuity property of Swan conductors and is also an l-adic analogue of André's semi-continuity result of Poincaré-Katz ranks for meromorphic connections on complex relative curve. After that, we give a ramification bound for the nearby cycle complex of an étale sheaf ramifications along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier in 2019 and answers a conjecture of Leal in a geometric situation. If time permits, we discuss the common new ingredient behind the two aspects above, which is a decreasing property of the conductor divisor defined in terms of Abbes and Saito's ramification theory after pull-backs.

Hiroki Kato (Universite Paris-Saclay): Etale cohomology of rigid analytic varieties via nearby cycles over general bases
2021-02-25 16:00-17:00
One of the most fundamental results in the study of étale cohomology of rigid analytic varieties is the comparison with the nearby cycle cohomology, which gives a canonical isomorphism between the cohomology of an algebraizable rigid analytic variety and the cohomology of the nearby cycle. I will discuss a generalization of this comparison result to the relative case: For an algebraizable morphism, the compactly supported higher direct image sheaves are identified, up to replacing the target by a blowup, with a generalization of the nearby cycle cohomology, which is given by the theory of nearby cycles over general bases. This result can be used to show the existence of a tubular neighborhood that doesn't change the cohomology for algebraizable families.

Will Sawin (Columbia university):Stalks of perverse sheaves in characteristic p
2021-02-25, 10:00-11:00
Perverse sheaves are objects that efficiently encapsulate geometric information in multiple areas of algebraic geometry, number theory, representation theory, and topology. A key invariant of perverse sheaves is the characteristic cycle, which can be used to calculate the Euler characteristic or the rank of the vanishing cycles at a particular point. Massey showed that the characteristic cycle can be used to bound the stalk of the perverse sheaf at a particular point. We generalize Massey's formula from characteristic 0 to characteristic p. This relies on the recent construction of the characteristic cycle in characteristic p. It has multiple applications to number theory over the ring of polynomials in one variable over finite fields, since many natural arithmetic functions in that setting arise from the stalks of perverse sheaves - most famously, automorphic forms.