Seminar on Algebraic Geometry and Ramification

Organizers

Fangzhou Jin, Peng Sun, Enlin Yang, Yigeng Zhao

Upcoming talks

Past talks

Frédéric Déglise (Ecole Normale Supérieure de Lyon): Chow-Künneth Decompositions
2023-08-17, 10:00-11:00
Room 108, Zhiyuan Building, Tongji University

Sebastian Bartling (Universität Duisburg-Essen): On the étale cohomology of the Fargues-Fontaine curve
2023-05-16, 05-18, 09:00-11:15
Room 1114, Sciences Building No.1, Peking University
Talk 1: In the first talk, I will introduce the Fargues-Fontaince curve (associated to a non-archimedean local field E and an algebraically closed, non-archimedean complete field F) in a way that makes the relationship between étale torsion local systems and torsion Galois representations of E transparent. Then I explain the three different incarnations of the curve and recall their basic properties. In the end, I will summarize some results of Laurent Fargues on the étale cohomology of torsion étale local systems on the curve.
Talk 2: In this talk, I start to consider constructible sheaves on the curve. Guided by the analogy between the curve (associated to an algebraically closed field F in char p) and a classical smooth projective curve over an algebraically closed field, Laurent Fargues made several conjectures about the étale cohomology of the curve. I will explain how to handle these conjecture completely in the prime to p torsion case and how the comparison of étale cohomology of the algebraic curve and the adic curve would imply the p torsion case. If time permits, I'll explain an approach for proving the last statement (which has an analogy for P1 and gives there a new proof of the comparison).

Heer Zhao (Universität Duisburg-Essen): Introduction to log geometry
2023-03-10, 03-17, 03-24, 03-31, 19:00-21:30
Tencent Meeting:976-6523-2427 Password:235711
Logarithmic geometry is the natural framework for dealing with degenerations and compactifications, and it has played and will keep playing important roles in many branches of algebraic geometry, arithmetic geometry and complex geometry. In this short course, after a quick review of commutative unitary monoids, we will first introduce the basic notions log structures, charts, frames, log smooth maps, log étale maps and log differentials. Then we will turn to two specific topics log products and log étale cohomology.

Yanbo Fang (Université Paul Sabatier): Critical Fubini-Study metrics over non-Archimedean fields
2023-02-13 15:30-16:30
Tencent Meeting:638-710-287 Password:235711
Critical Fubini-Study metrics on an ample line bundle are those minimizing (among its SL-orbits) the induced norm of the Chow form of the variety. Over complex numbers, such metrics are characterised by the constancy of its Bergman function (also known as balanced metrics in Kahler geometry). We give an analogous characterization over a non-Archimedean field in terms of a polymatroid polytope. (work based on arXiv:2212.02486)

Qin Xue (Westlake University): Frobenius-Witt differentials and cotangent bundle of regular schemes
2023-01-13 19:00-21:30
Tencent Meeting:736-523-618 Password:235711

Massimo Pippi (Universität Regensburg): Unipotent Deligne-Milnor conjecture
2023-01-11 16:00-17:00
Zoom Id:664 5939 7723 Password:172338
Let (C^n,0)->(C,0) be the germ of an holomorphic function with an isolated singularity. The Milnor formula computes the difference of the Euler characteristics of the generic and special fiber in terms of the Milnor number, i.e. the (complex) dimension of the Jacobian algebra. Let S an henselian DVR with residue characteristic p and let X be a flat, separated S-scheme of finite type, that is moreover regular. Let us suppose that the structural morphism has only an isolated singularity. The Deligne-Milnor formula, conjectured by P. Deligne, generalizes the Milnor formula to the positive and mixed characteristics case. It computes the difference of the Euler characteristics (l-adic, for l a prime number different from p) of the geometric fibers of X/S in terms of a number of algebraic nature (which is also called the Milnor number) and of a number of arithmetic nature (the Swan conductor of the generic fiber). The equicharacteristic case has been proved by P. Deligne (together with the case of relative dimension 0 and of ordinary quadratic singularities). In mixed characteristics, the conjecture is still open outside the case of relative dimension 1, which follows from theorems due to S. Bloch and F. Orgogozo. In this talk, we will discuss how to prove the conjecture under the additional assumption of unipotent action of the inertia group on the cohomology of the geometric generic fiber using derived and non commutative geometry, following ideas of B. Toen and G. Vezzosi. This provides us with new cases in the mixed characteristics situation. This talk is based on work in collaboration with D. Beraldo: arXiv:2211.11717.

Lei Zhang (Sun Yat-sen University Zhuhai Campus): A theorem on meromorphic descent and an application
2022-11-30 14:00-15:00
Tencent Meeting:938-410-977 Password:1130
Let R be a complete DVR with function field K, and let q: \mathfrak{Y} \to \mathfrak{X} be an adic faithfully flat map of adic noetherian formal schemes over Spf(R). Then by Grothendieck's general fpqc descent theory of quasi-coherent sheaves on schemes and a trick due to Ofer Gabber, one can show that the "meromorphic coherent sheaves" on \mathfrak{X} is equivalent to those on \mathfrak{Y} equipped with "meromorphic descent data". In particular, if q is a G-torsor for a finite discrete group G, then the category of "meromorphic coherent sheaves" on \mathfrak{X} is equivalent to that on \mathfrak{Y} equipped with a "meromorphic" G-action. In this talk we will show that this is still true when G is any discrete group. This generalizes Gieseker's construction of meromorhpic descent for the case when X is a stable curve over R with split degenerate special fiber and non-singular generic fiber and q: \mathfrak{Y} \to \mathfrak{X} is the formal completion of universal cover of the special fiber. Note that in this case q is a torsor under a flat schottky group G constructed by Mumford, and if X is of arithmetic genus g, then G is the free group on g generators. In the end, we will apply this technique to the specialization problem of the pro-etale fundamental group.

XiaoZong Wang (CAS): Smoothing of 1-cycles over finite fields
2022-11-11 15:00-16:00
Room 1114, Sciences Building No. 1, Peking University
It is known by Hironaka's work in 1968 that on a smooth projective variety defined over an infinite field, any algebraic 1-cycle is rationally equivalent to a smooth one. In this talk, I will show that the result is also true when the variety is defined over a finite field. In this setting, several results on the density of smooth divisors satisfying certain conditions are needed to construct the rationally equivalent smooth 1-cycle. I will also sketch how Poonen's closed point sieve works to get such density results.

Christophe Levrat (Télécom Paris): Computing the cohomology of constructible sheaves on curves
2022-10-05 16:00-17:00
Zoom Id:895 7231 2907 Password:666910
Given a curve X over an algebraically closed field, and an integer n invertible on X, the cohomology groups of constructible sheaves of Z/nZ-modules on X are finite. In this talk, I will give a completely explicit description of the cohomology complex of (complexes of) such sheaves in the case where X is smooth or nodal, as well as a description of the cup-products in the cohomology of finite locally constant sheaves on X when X is furthermore projective. I will give a detailed example to show that one can actually compute this cohomology complex using tools which - in a few very specific cases - are already implemented in current computer algebra systems.

Ziyang Gao (Leibniz Universität Hannover): Torsion points in families of abelian varieties
2022-09-28 16:30-17:30
Tencent Meeting:445-898-729 Password:072938
Given an abelian scheme defined over \IQbar and an irreducible subvariety X which dominates the base, the Relative Manin-Mumford Conjecture (inspired by S. Zhang and proposed by Zannier) predicts how torsion points in closed fibers lie on X. The conjecture says that if such torsion points are Zariski dense in X, then the dimension of X is at least the relative dimension of the abelian scheme, unless X is contained in a proper subgroup scheme. In this talk, I will present a proof of this conjecture. As a consequence this gives a new proof of the Uniform Manin-Mumford Conjecture for curves (recently proved by Kühne) without using equidistribution. This is joint work with Philipp Habegger.

Xiaoyu Su (Tsinghua University): Parabolic Hitchin System and Mirror Symmetry
2022-08-08 10:00-11:00
Tencent Meeting:909-888-873 Password:235711
A very important class of hyperkaehler mirror partners are moduli spaces of SL_r and PGL_r-Higgs bundles. Hausel and Thaddeus proved that the moduli spaces of SL_r and PGL_r-Higgs bundles on a smooth projective curve are mirror partners in the sense of Strominger-Yau-Zaslow [SYZ96] (i.e. generic fibers are dual abelian varieties). Inspired by the SYZ philosophy, Hausel-Thaddeus in [HT01, HT03] conjectured that the moduli spaces of SL_r and PGL_r-Higgs bundles over both smooth and parabolic curves are topological mirror partners (i.e. have equal stringy Hodge numbers) and proved this for r=2,3. On the other hand, a very recent paper by Groechenig-Wyss-Ziegler [GWZ20] uses p-adic integration to prove Hausel and Thaddeus' topological mirror symmetry conjecture holds over a smooth projective curve.
In this talk we will talk about our recent work on the topological mirror symmetry conjecture for moduli of Higgs bundles over parabolic curves. We will first review the concept of mirror symmetry and in particular in the since of Strominger-Yau-Zaslow and Hausel-Thaddeus. Then we will talk about our way to check the SYZ and topological mirror symmetry partners property for moduli of Higgs bundles over a parabolic curve. This is a joint work with Xueqing Wen and Bin Wang.

Sabrina Pauli (University of Duisburg-Essen): The Bézoutian and the A1-degree
2022-07-25 16:00-17:00
Zoom Id:831 2409 6013 Password:791738
Morel's A1-degree is the analogue of the Brouwer degree in A1-homotopy theory. It is the key input in A1-enumerative geometry, that is enumerative geometry over an arbitrary field where answers are valued in the Grothendieck-Witt ring of the field. In the talk I will give a brief introduction to the A1-degree and its applications and present formulas to compute it including a formula using the multivariate Bézoutian. This is based on joint work with Thomas Brazelton and Stephen McKean.

Heng Xie (Sun Yat-Sen University): Cohomology of spinor varieties via blow-ups
2022-05-20 14:00-15:00
Tencent Meeting:534-704-254 Password:235711
I will talk about spinor varieties, which are certain closed subschemes inside Grassmann bundles defined by a non-degenerate quadratic form. We consider the blow-up of a spinor variety along a spinor sub-variety, which helps us to compute the oriented cohomology theory (e.g. K-theory and Chow groups) of spinor varieties, and Witt groups in the non-oriented setting.

Yigeng Zhao (Westlake University): A relative twist formula of epsilon factors
2022-05-13 14:00-15:00
Tencent Meeting:494-628-807 Password:235711
For an l-adic sheaf on a smooth variety, its epsilon factor is the constant term in the functional equation of its L-function. In this talk, we review the product and twist formulas of epsilon factors in the light of the developments in geometric ramification theory, and suggest a generalization in the relative case. This is a joint work in progress with Enlin Yang.

Xucheng Zhang (University of Duisburg-Essen): Existence of Good Moduli Space of Vector Bundles
2022-03-16 17:00-18:00
Zoom Id:630 7871 3736 Password:188978
We prove that, among all open substacks of moduli stacks of rank 2 bundles over a curve, the open substack of semi-stable bundles is the unique maximal one that admits a separated good moduli space. One example in rank 3 case shows that a similar phenomenon does not hold in higher rank case and also suggests a general formulation.

Xiping Zhang (Tongji University): Local Euler Obstructions of Projective Reflective Varieties
2022-02-24 10:00-11:00
Tencent Meeting:225-158-631 Password:235711
The local Euler Obstructions are very important local invariants in the study of stratified singular spaces, but are very hard to compute in general. In this talk I will discuss projective reflective varieties. These are stratified projective varieties such that the dual varieties of the strata have certain dimension constraints. I will show that for such varieties these local invariants are completely determined by their Chern classes, followed by some explicitly computed examples.

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.