算术几何讨论班内容整理

德国数学在算术几何方面是一流的，可惜却没有吸引太多中国学生去攻读硕士和博士，因此对柏林大学算术几何讨论班的内容进行整理（来自主页http://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/index.html），作为对比。

2018 Summer

Abhyankar's conjecture

主题：

Let k be an algebraically closed field of characteristic p > 0. For every finite group G we will denote by p(G) the subgroup of G generated by its p-Sylow subgroups. We will say that a group G is a quasi-p-group if G = p(G).

In 1957 Abhyankar formulated the following conjecture.

Conjecture (Abhyankar’s Conjecture for affine curves). Let X be a smooth projective curve of genus g defined over k and S a non-empty set of closed points of X of cardinality r. A finite group G is the Galois group of some finite etale Galois cover of X − S if and only if G/p(G) admits a generating set of cardinality at most 2g + r − 1.

In particular the conjecture says that a finite group G is the Galois group of some Galois cover of A^1 if and only G is a quasi-p-group.

The conjecture is now completely proven by the work of Serre, Raynaud and Harbater. Serre has proven the conjecture when X −S = A^1 and G is assumed to be solvable. Later Raynaud succeeded to prove the entire conjecture over X −S = A^1 , using the case proven by Serre. Finally Harbater have proven the full conjecture.

工具：

The theory of schemes and formal schemes, the basic theory on the etale site, etale fundamental group and etale cohomology, the necessary facts on the ramification theory for curves, on the semi-stable reduction, on group’s cohomology and on rigid geometry (using Tate’s approach)

内容：

Belyi’s type theorems in positive characteristic, Hurwitz’s formula, Swan conductors, Grothendieck-Ogg-Shafarevich formula, Artin-Schreier theory, Brief introduction to rigid analytic space, Tate’s acyclity theorem, Rigid analytic GAGA, Proper mapping theorem, Raynaud’s generic fiber, Runge’s theorem in rigid geometry, Formal patching, Construction of coverings via rigid geometry, Extension and algebraization, The combinatorial step, Last case using semi-stable curves

de Rham epsilon factors à la Beilinson-Bloch-Esnault

主题：

考虑有限域上光滑射影曲线上l-adic可构造层F，可定义其L函数，是有理函数，Poincare对偶给出函数方程，方程中常数因子e(F)称为epsilon factor of F，其=Frob在F的cohomology的determinant这一一维线性空间上作用的特征值。

The constant e(F) is also known as the global epsilon factor. A formalism of local epsilon factors allows one to associated to a non-zero rational 1-form ω on X and a closed point x ∈ X an (invertible) l-adic number e_{ω,x}(F) which is equal to 1 for almost all closed points x, such that we have a product formula e(F) = \prod_{x} e_{w,x}(F). Furthermore, the quantity e_{ω,x}(F) is only to depend on the constructible sheaf F (and the form ω) near x, that is, the restriction to the formal completion of X at x.

If we assume that F is generically of rank 1, methods akin to class field theory and Tate’s thesis can be used to build a theory of local epsilon factors in this case. It was conjectured by Deligne [Del73] that a formalism of local factors should exist even in the higher rank case. This was proven by Laumon [Lau87] using global methods (and therefore in some sense indirectly). A purely local definition of e_{ω,x}(F) remains elusive and would be highly desirable for the tales it might tell about wild ramification.

The de Rham analogue: Replacing finite fields by the complex numbers C (or a field of characteristic zero) and l-adic sheaves by holonomic D-modules, the determinant line of de Rham cohomology remains an interesting invariant (e.g. due to its connection to period matrices). It’s been observed by Deligne (unpublished) and Beilinson–Bloch–Esnault [BBE02] that one can define lines (i.e. 1-dimensional vector spaces) E_{ω,x}(F) for a holonomic D-module F and a non-zero rational 1-form on X, such that we have the following analogue of formula det(X, F) = \otimes_{x} E_{ω,x}(F).

Furthermore, this isomorphism is canonical (up to signs which can be dealt with). The fascinating aspect to their construction is that it’s inherently local and surprisingly elementary. The main goal is to understand this theory of de Rham epsilon factors in depth. If time permits we will also discuss epsilon periods following Beilinson.

内容：

geometric class field theory and epsilon-factors, Holonomic D-modules and irregular flat connections on curves, Good lattice pairs, Graded lines and determinants and the connection with algebraic K-theory, Infinite-dimensional vector spaces and determinantal theories, de Rham epsilon factors and the epsilon connection, Patel’s higher-dimensional theory of epsilon-factors, Betti epsilon factors, Beilinson’s unicity theorem, Epsilon periods.

2017 Winter

p-adic Simpson Correspondence

主题：

The theory of non-abelian Hodge theory starts from Hitchin. Hitchin in [Hit87] studied the self-dual Yang-Mills equations and obtained the famous Hitchin equations as follows.

Let X/C be a smooth projective curve over the complex numbers. E is a C∞- complex vector bundle of rank 2 on X together with a C∞-Hermitian metric. ∇ is a C∞-connection on E, and Φ ∈ End(E) ⊗ A(1.0) is an End(E)-valued (1.0) form satisfying F(∇) + [Φ, Φ ∗ ] = 0 ，∇00(Φ) = 0， where F(∇) is the curvature of ∇, Φ∗ is the complex conjugation of Φ, and ∇00 is the (0, 1)-component of the connection ∇.

Theorem [Kob14, Chapter I, Proposition 1.3.7] tells us that on a Riemann surface, a C∞-complex vector bundle with a C∞-connection has a unique holomorphic structure determined by the (0, 1)-part of the connection. So if (E, ∇, Φ) is a solution of the Hitchin equations, ∇00 will induce a holomorphic structure E∇00 on E. From the second equation, we see that Φ is holomorphic with respect E∇00. In particular, the pair (E∇00 , Φ) forms a Higgs bundle. By calculation, the connection ∇e := ∇ + Φ + Φ∗ is flat and induces another holomorphic structure on E (and the Chern classes of E vanish).

Hitchin shows in [Hit87] that the solutions of the Hitchin equations modulo gauge equivalent are in one-to-one correspondence with poly-stable Higgs bundles with vanishing Chern classes. In [Don87] after [Hit87], Donaldson showed that every irreducible flat connection is gauge equivalent to a connection of the form ∇+ Φ + Φ∗ where (E, ∇, Φ) is a solution of the Hitchin equations.

Therefore we get a bijection between poly-stable Higgs bundles with vanishing Chern classes and semi-simple local systems on X as follows:

semi-simple π_1(X)-representations ↔ bundles with flat connections ↔ solutions of Hitchin equations ↔ polystable Higgs -bundles

For Φ = 0, we recover the famous result by Narasimhan and Seshadri [NS65]: a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the Riemann surface.

Simpson in [Sim88, Sim92, Sim94, Sim97] generalized this correspondence to the general case where X is a smooth projective variety of arbitrary dimension, E is a C∞-vector bundle of rank r, and he refined the correspondence with additional information given by variation of Hodge structures. He showed the corresponding moduli spaces MdR(X/C), MB(X/C), MDol(X/C), which we mean rank r integrable connections, local systems, semi-stable Higgs bundles with vanishing Chern class respectively, are real analytically isomorphic. Furthermore, he give a comparison of Dolbeault cohomology and de Rham cohomology and show that the correspondence between semi-stable Higgs bundles and flat connections are compatible with extensions. Simpson called this correspondence non-abelian Hodge theory, and some other mathematicians called this Simpson correspondence.

The first step of p-adic analogue of Simpson correspondence is given by Faltings in [Fal05]. For curves or small affine space over p-adic field, Faltings constructed an equivalence between the category of Higgs bundles (the Chern classes may not vanish) and “generalised representations” using p-adic Hodge theory and almost etale coverings (the local computation looks similar to the calculation in [Ols05, Chapter 3,4]). This kind of representations include usual representations as a subcategory.

Faltings’ construction appears to be satisfactory only for curves and, even in this case, many fundamental questions remain open. The equivalence depends on the choice of an exponential function for the multiplicative group.

If we restrict the inverse functor as a functor from usual representations of geometric π_1 to Higgs bundles. It is difficult to characterise its image. All Higgs bundles in the image are semi-stable of slops zero, but people don’t know whether the image contains all those semi-stable Higgs bundles of slop zero. People also don’t know what kind of Higgs bundles comes from “genuine representations”.

After [Fal05], Abbes , Gros and Tsuji in [AGT16] explicitly explained the theory of Faltings. Liu Ruochuan and Zhu Xinwen in [LZ17] formalized the Simpson correspondence functor on a rigid analytic space via pulling back to pro-etale site [Sch12, Sch13]. (One can show that this is implied by the results of Faltings et al., but this is difficult). They used Scholze’s machinery to construct a Riemann-Hilbert correspondence for p-adic local systems on rigid analytic varieties using this Simpson correspondence functor. As a consequence, they obtained rigidity theorems for p-adic local systems on a connected rigid analytic variety. Finally, they gave an application of their results to Shimura varieties.

On the other hand, the p-adic Simpson correspondence can also be built from lifting a characteristic p correspondence. Ogus and Vologodsky in [OV07] established the nonabelian Hodge theorem in positive characteristic. They constructed a functor, which they called “inverse Cartier transform” from a category of certain nilpotent Higgs modules to a category of certain nilpotent flat modules on a W2(k)-liftable smooth variety. (Here k is some perfect field of positive characteristic)

Lan Guitang, Sheng Mao and Zuo Kang in [LSZ13] lifted the inverse Cartier transform to the truncated Witt ring Wn(Fq) and introduced the notion of Higgs-de Rham flows. They showed that there is a Higgs correspondence from crystalline representations to periodic Higgs-de Rham flows by passing form Wn(Fq) to W(Fq). The theory developed in [LSZ13] also turns out to be useful in the study of Higgs bundles with nontrivial Chern classes. This has been demonstrated in the recent work [Lan15] of A. Langer on an algebraic proof of the Bogomolov’s inequality for Higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 .

内容：

Classical Simpson Correspondence， Exponential Twisting ， Azumaya Picture， Fontaine Laffaille Modules ， Inverse Cartier transform over a truncated Witt ring，Higgs correspondence in mixed characteristic ， Strongly semistable Higgs modules， Rigidity theorem for Fontaine modules 2017 Summer

Langlands correspondence for function fields

主题：

Langlands Programme in the function field case, with a particular focus on the geometric nature of the constructions， presentation of the general ideas of Vincent Lafforgue’s proof of the Langlands’ correspondence for general reductive groups

内容：

Tate’s thesis for function fields： connection between adeles and bundles ， local zeta integrals and meromorphic continuation，Poisson summation formula ，Riemann-Roch and Serre duality for algebraic curves via harmonic analysis ， functional equation； Automorphic side： Hecke algebras， Satake isomorphism， L-functions， Automorphic representations， Statement of Langlands correspondence for function fields； The stack of bundles Bun_G, Shtukas for GL_2， Hecke stacks, Harder-Narasimhan polygon, cohomology of the stack of Shtukas， degenerated and pre-iterated Shtukas ，Basic Theorem， connection to counting bundles and formulae， function-sheaf dictionary， character sheaf / Hecke eigenproperty ，Geometric Class Field Theory ， Shtukas for reductive groups and general ideas of V. Lafforgue proof ， Drinfeld Modules ， Analytic moduli space of Drinfeld’s Modules and their cohomology

2016 Winter

Berkovich spaces, birational geometry and motivic zeta functions

主题：

understand the paper Poles of maximal order of motivic zeta functions by Nicaise-Xu.

Igusas p-adic zeta function Z(s) attached to an integer polynomial f in N variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation f = 0 modulo powers of a prime p. It is expressed as a p-adic integral, and Igusa proved that it is rational in p −s using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of Z(s) is at most N, the number of variables in f. In 1999, Wim Veys conjectured that the real part of every pole of order N equals minus the log canonical threshold of f (which is an invariant of the singularities of f).

Nicaise and Xu prove Veys’ conjecture by means of Berkovich skeletons and birational geometry. This seminar roughly covers the papers [MN13, NX15] and includes short introductions to Berkovich spaces, the minimal model program, p-adic and motivic zeta functions.

内容：

Minimal model program: log resolution, discrepancy, terminal, canonical, log terminal and log canonical pairs, dlt pairs, log canonical centers, log canonical threshold, ample, semiample, nef, big, effective and pseudoeffective divisors and the relations between their cones and with the cone of curves, Base-point-free theorem, Negativity Lemma, Cone theorem, Contraction theorem, three types of contractions , problem of termination of flips, log MMP.

Berkovich spaces: Banach rings and the spectrum of a Banach ring and bounded morphisms, affinoid algebras, Berkovich analytic spaces and analytification, affinoid domains and special subsets,

Models: birational, divisorial and monomial points, snc models, ZariskiRiemann space, Berkovich skeleton, Weight function, Essential skeleton and relation to birational geometry, existence of flow

p-adic integration and the Igusa zeta function, Haar measure on a padic field and of p-adic integral, change of variables formula, proof of rationality of Z_f (s) and the explicit formula via resolution of singularities. Motivic integration and motivic zeta functions, arc spaces, the basics of motivic integration, transformation formula, Monodromy conjectures and proof of Veys’ conjecture.

2016 Summer

Motivic Galois groups and periods

主题：

A method for linearising the problem of studying topological spaces is to pass to (co)homology. The theory of motives is an attempt to linearise the study of algebraic varieties. Unlike in algebraic topology, we have many (co)homology theories for algebraic varieties together with canonical comparison isomorphisms: Betti, de Rham, ℓ-adic, crystalline, etc.. Moreover, these cohomology theories satisfy Kunneth-type formulas and often come with extra structure which can be encoded as a representation of a pro-algebraic group: Galois representation, mixed Hodge structure, etc. In the ideal world, for a field k, one wishes for a Tannakian category MM(k) of mixed motives over k and a functor M : Sm/k → MM(k) from the category of smooth schemes over k, such that cohomology theories of X with their additional structure and the higher Chow groups of X can be recovered from M(X). There has been many attempts to construct such a category.

In the pure case (restricting to smooth projective varieties), we have among others ① Grothendieck’s original proposal of numerical motives (abelian semi-simple category, for which the existence of fiber functors depends on the standard conjectures), and ② Andre motives built out of motivated cycles (avoiding the standard conjectures and constructing a semi-simple tannakian category with a pure motivic Galois group).

In the mixed case, Deligne and Beilinson observed that it might be easier to construct the derived category of MM(k) and then try to recover MM(k) as the heart of this category for the right t-structure. ① In the 90s a triangulated category DM(k) of motives were constructed by Voevodsky (along with similar constructions by Hanamura and Levine), based on his theory of A^1 -homotopy invariant sheaves with transfers. This category has natural realisation functors, and the higher Chow groups of X appear as extension groups. However, constructing the right t-structure on this category turned out to be at least as difficult as the standard conjectures [14]. ② Nori suggested another approch, constructing a tannakian category of mixed motives MM(k) and a mixed motivic Galois group based on his version of a weak tannakian formalism. The relation with algebraic cycles is unfortunately unclear.

③ In this seminar we will study yet another approach due to Ayoub. Like Nori’s, it is unconditional and produces a certain pro-algebraic group as candidate for the motivic Galois group. Unlike Nori’s, it builds on the work of Voevodsky and his successors. Ayoub first constructs a category DA(S) of etale motivic sheaves on a scheme S with rational coefficients (which is equivalent with Voevodsky’s triangulated category of motives with rational coefficients when S is the spectrum of a field). Then he develops a new variant of the tannakian formalism which works outside of the abelian category case, and applies it to the Betti realisation functor Bti∗ : DA(k) → D(Vec_Q) to obtain a Hopf algebra in D(Vec_Q), from which he constructs the motivic Galois group.

We will study the construction of DA(k) and the Betti realisation in details. A priori, these objects live in the world of monoidal triangulated categories; however, it is convenient to lift them to the more structured context of stable monoidal model categories. We will then present Ayoub’s construction of the motivic Galois group.

We will then proceed to study applications. First, we have a non-trivial reformulation of the conjecture of Grothendieck and Kontsevich-Zagier on transcendance properties of periods. Then, we have a motivic version of the theorem of the fixed part [10], which states that if a “motivic local system” (i.e., a representation of Ayoub’s motivic fundamental group!) over a variety X/k has the property that its underlying local system, after a base change k → C, on X_{C,an} is trivial, then it comes from the base field. Finally, and most importantly, we have the geometric version of the Kontsevich-Zagier conjecture [13, Theorem 1.6] - this last item is a “true” application, not a motivic or conjectural one!

内容：

model category, Bousfield localisation and local homotopy theory, homotopical algebra, Spectra and motives， suspension and loop adjunction， Betti realisation and the motivic Hopf algebra， strongly dualizable object ， The D 1 -localisation functor ， Approximation of singular chains， Motivic Galois group and period torsor， Motivic fundamental group and the theorem of the fixed part， Relative motivic Galois groups， Geometric Kontsevich-Zagier conjecture

2015 Winter

Supersingular K3 surfaces are unirational, after C. Liedtke

主题：

A K3 surface X over an algebraically closed field k is a connected smooth proper surface which satisfies ω_X := Ω^2_{X/k} ∼= O_X and H^1(X, O_X) = 0. For the Hodge numbers we have h 0,2 = h 2,0 = 1 and h 1,1 = 20. In characteristic zero the Hodge Chern class injects the Neron-Severi group into H^1(X, Ω^1_X) and its rank is therefore bounded by 20. In positive characteristic, the crystalline Chern class injects the Neron-Severi group into the slope 1 part of crystalline cohomology and it can happen its image generates the whole W(k)-module H^2_{crys}(X/W). In this case the rank of the Neron-Severi group of X is 22 and we say that it is (Shioda) supersingular. It was conjectured by Artin et al. that supersingular K3 surfaces are unirational (since all the examples they had were so). The aim of this seminar is to understand the proof of the following recent result of Liedtke:

Theorem ([Lie15b]). Supersingular K3 surfaces in positive characteristic ≥ 5 are unirational.

Lieblich also announced a proof of this result, see [Lie14], but we will follow the approach taken in [Lie15b]. Notice that for a smooth unirational variety in characteristic 0 one always has H^i(X, OX) = 0, for all i ≥ 1. Therefore a K3 surface in charateristic 0 can never be unirational since we have H^2 (X, O_X) ∼= H^2 (X, ω_X) = H^0 (X, O_X). Whereas the unirationality in positive characteristic only implies H^i(X, WOX)/torsion = 0 for all i ≥ 1. Actually we will see in the seminar that for a supersingular K3 surface, H^2 (X, WOX) is an infinitely generated W(k)-torsion module.

内容：

K3 surfaces, Witt vectors, construction of crystalline cohomology, de Rham-Witt complex, formal groups and their Dieudonne modules, ´etale-connected exact sequence, p-divisible group, formal Brauer group, sheaf of p-typical curves, F-crystals, slope decomposition and associated Hodge and Newton polygons, Rudakov-Shafarevich vanishing, Illusie’s refinement of the Igusa-Artin-Mazur inequality, Theorem of Rudakov-Shafarevich-Zink stating that supersingular K3 surfaces in characteristic p ≥ 5 have potential good reduction, Supersingular K3 crystals, The moduli space of N-marked K3 surfaces, Ogus’ crystalline Torelli theorem for supersingular K3 surfaces, N-rigidified supersingular K3 crystals and moving torsors, Supersingular K3 surfaces are unirational (p>=5)

2015 Summer

"Rational sections and Serre's conjecture" following de Jong-He-Starr

主题：

Recall the following conjecture of Serre.

Conjecture. Let K be a perfect field of cohomological dimension ≤ 2, G be a semisimple and simply connected algebraic group over K . Then H^1(K,G) = 0.

The goal of this seminar is to prove a version of this conjecture following [JHS] when K = k(S), where S is a surface over an algebraically closed field k. (Note that K = k(S) is not perfect, so technically speaking, it is a special case of a stronger form of Serre’s conjecture.)

For simplicity we may first assume that G is defined over k (we have to wait until the last talk to discuss the general situation). Now suppose we are given an element x ∈ H^1 (K,G), then x corresponds to a G-torsor π : P → Spec(K) in the étale topology . Thus x = 0 ⇔ P is a trivial G-torsor ⇔ π admits a section. We may shrink S a little bit to get a model π_S : P_S → S of π which is still an étale G-torsor. Then π admits a section ⇔ π_S admits a rational section, the latter means that there is an dense open U ⊆ S and a k-morphism s_U :U → P_S such that π_S ◦ s_U = id_U . Now the algebraic problem is converted into a problem which is of strong geometric flavor – finding rational sections. Thus we can use geometric tools such as moduli spaces and Hodge theory to attack the conjecture. This is what A. J. de Jong, X. He and J. M. Starr did in their paper

Theorem 2. Let X → C be a morphism of smooth projective varieties over an algebraically closed field k of characteristic 0 with C a curve. Let L be an invertible sheaf on X which is ample on each fibre of X →C. Assume the geometric generic fibre of X →C is rationally simply connected by chains of free lines and contains a very twisting scroll. In this case for e sufficiently large there exists a canonically defined irreducible component Z_e ⊆ Sections^e (X/C/k) so that

the restriction of αL :Sections^e (X/C/k) −→ Pic^e (C/k) to Z^e has rationally connected fibres

In the theorem the space Sections^e (X/C/k) parametrizes sections σ :C → X of degree e (with respect to L) and the map αL assigns to a section σ the point of Pic^e(C/k) corresponding to the invertible sheaf σ^∗L . This theorem plus the main result of [GHS] lead to the following:

Theorem 1. Let f : X → S be a faithfully flat morphism between smooth projective varieties over an algebraically closed field k of characteristic 0, where S is a connected surface. Suppose that L is an f -ample invertible sheaf and the geometric generic fibre of f satisfies the following two conditions 1. the space X_{η¯} is rationally simply connected by chains of free lines; 2. the space X_{η¯} has very twisting scroll, then there exists a rational section of f .

Amazingly one is able to deduce from 1, which is only proved over fields of characteristic 0, the following theorem in arbitrary characteristic:

Theorem 3. Let k be an algebraically closed field of any characteristic. Let S be a quasiprojective surface over k. Let X → S be a projective morphism. Let η¯ be the spectrum of the algebraic closure of the function field of k(S). If X_{η¯} is of the form G/P for some linear algberaic group G and parabolic subgroup P and Pic(X) → Pic(X_{η¯}) is surjective, then X → S has a rational section.

With the above theorem plus a little more work and a result of [CGP] one canfinally conclude the following theorem:

Theorem 4. (Serre’s conjecture II for function fields) Let k be an algebraically closed field. Let k ⊆ K be a finitely generated field extension of transcendence degree 2. For every connected, simply connected, semisimple algebraic group G_K over K , every G_K -torsor over K is trivial.

内容：

Brauer-Severi varieties， Azumaya algebras，rational sections， algebraic stacks，Stack of curves，Stack of sections， Stack of Kontsevich stable maps，free lines， very twisting scrolls

2014 Winter

Chow groups of zero cycles over p-adic fields

Let K be a p-adic field, i.e. a finite extension of Q——p and V a smooth geometrically connected variety over K. Denote by CH_0(V ) the Chow group of zero-cycles on V and by A_0(V ) ⊂ CH_0(V ) the subgroup of degree zero-cycles. It was conjectured by Colliot-Thelene that the group A_0(V ) is the direct sum of a finite group and a p^'-divisible group. Here a group D is called p^' -divisible if the multiplication map D → D, d→ nd is surjective for any natural number n which is prime to p. In this seminar we want to understand the proof of this conjecture given by Saito-Sato in the case of quasi-semistable reduction:

Theorem 1 ([SS10, Thm 9.7]). Let K be a p-adic field with ring of integers O_K and residue field k. Let X be a regular connected scheme which is flat and projective over O_K and assume that (X ⊗_{O_K} k)_red is a divisor with simple normal crossings. Then A_0(X ⊗_{O_K} K) is the direct sum of a finite group and a p'-divisible group.

A key ingredient in the proof is the following theorem:

Theorem 2 ([SS10, Thm 1.16]). Let R be an excellent henselian DVR with fraction field K and residue field k, which we assume to be either finite or separably closed. Let X be a regular scheme, flat and projective of pure relative dimension d over R and assume that (X ⊗_{R} k)_red is a divisor with simple normal crossings. Then for any natural number n ≥ 1 which is prime to the characteristic of k the cycle map ρ_X : CH^1(X)/n → H^{2d}_{et} (X, {µ_n}^⊗d ) is an isomorphism.

Now to prove Theorem 1 one can use Theorem 2 and Bertini arguments to reduce to the case where V = X ⊗_{O_K} K is a curve and then theorem follows from a classical result of Mattuck on the structure of the group of K-rational points of an abelian variety over K.

Most of the seminar will be devoted to understand the techniques used in the proof of Theorem 2. In fact as remarked by Bloch [Blo13, Rmk A.4] one can use Gabber’s refinement of de Jong’s alteration theorem to obtain Theorem 2 also in the case without the quasisemistable assumption on X. In case the residue field k is separably closed Bloch also gives a shorter proof of this theorem. Since we are mainly interested in the case of finite residue field we essentially follow [SS10], but we try to incorporate a simplification of Bloch. As an additional reference, one can mention [CT11].

内容：

Chow group of zero-cycles of a smooth projective and geometrically connected variety over a finite field, a local field, a number fields and the complex numbers， extraordinary inverse image functor， Etale Homology， the cycle map， Kato homology， Vanishing Theorem， Bertini Theorem over a DVR， Moving Lemma， The structure of the group of rational points of abelian varieties over a p-adic field

2014 Summer

Pro-étale cohomology after Bhatt-Scholze

主题：

The etale cohomology theory which was initially suggested by Grothendieck in 1960s plays a very important role in Modern algebraic geometry. In particular, when X is an algebraic variety over an algebraically closed field k of characteristic p \not=l, the l-adic cohomology which is derived from the general étale cohomology theory provides a nice Weil-cohomology theory. However, since the higher étale cohomology groups of a locally constant sheaf over a normal variety are always torsion, in order to get a Weil-cohomology theory with coefficients in a field of characteristic 0, one has to define the l-adic cohomology groups in an "indirect" manner, i.e. one has to define H^i(X_{et},Ql ) by a projective limit of H^i (X_{et},Z/ l^nZ) (and then tensor with Q_l ) instead of "directly" taking cohomology for the constant étale sheaf Q_l.

For some reasons (mainly for the study of perverse sheaves), one has to work with complexes of l-adic constructible sheaves and do operations at the level of derived categories instead of working with the single l-adic sheaf Q_l and considering only cohomology groups. Again these notions are defined in an "indirect" manner, i.e. one has to deal with projective systems instead of single objects.

These notions, although a bit "indirect", work very well: H^i (X_{et},Q_l ) is a Weil-cohomology theory, the category of l-adic lisse sheaves (which is the full subcategory of the category of constructible l-adic sheaves consisting of objects whose n-components are locally constant for each n) is equivalent to the category of continous l-adic representations of π^et_1(X,x), D^b_c (X,Q_l ) ⊆ D^b (X,Q_l ) is a sub-triangulated category and is preserved by Grothendieck’s six operations (under suitable finiteness conditions).

The goal of this seminar is to understand the new "direct" approach to the étale cohomology theory which is developed by Bhatt and Scholze in a recent paper [BS].

内容：

Grothendieck topology and pretopology， weakly étale and ind-étale morphisms， pro-étale site/ topos, w-local spectral spaces，w-local ring，w-strictly local， w-contractible ring， PRO-ÉTALE TOPOS IS LOCALLY W-CONTRACTIBLE，comparison of etale and pro-etale topology， functoriality for locally closed immersions （ the proétale site has the advantage that a closed immersion induces a continous morphism between the sites while the usual étale site does not ），perfect complex, compact objects， derived completions，Grothendieck's six operations， smooth base change theorem，pro-etale fundamental group

2013 Summer

"p-adic periods and derived de Rham cohomology" after A. Beilinson

主题：

If X is a variety over a p-adic field K we have the de Rham cohomology of X (we need de Jong’s alteration theorem to construct nice proper hypercoverings). As a natural analog of Betti cohomology we have p-adic etale cohomology H^i_{et}(X¯, Q_p). Now we have an isomorphism (∗) H^i_{dR}(X) ⊗_K B_{dR} \cong H^i_{et}(X¯,Qp) ⊗_{Qp} B_{dR}. B_{dR} is a complete discrete valuation field with ring of integers B^{+}_{dR} and residue field C_p, which is equipped with a filtration and a G_K- action, such that the graded C_p-algebra B_{HT} with its G_K-action equals gr_{Fil}B_{dR}.

In case X is proper and smooth the isomorphism (*) was first proved by Faltings using his theory of almost etale extensions. Before that the existence of the Hodge-Tate decomposition was proved in special cases by Tate, Raynaud and Bloch-Kato. In the seminar we will follow Beilinson’s approach in [Bei12] who proves the isomorphism (∗) in all generality. He constructs a crystalline period map in a second article [Bei13].

Notice that Beilinson defines an isomorphism already in the derived category using the language of E∞ -algebras. It seems that this is a nice way to express the compatibility with cup products on the level of complexes. In the seminar we will ignore this extra information and simply work with complexes instead of E∞-algebras.

内容：

p-adic Hodge theory， Fontaine’s period ring， Λ-pro-infinitesimal thickening of V， Illusie’s derived de Rham complex， cotangent complex， homotopy limit and the completion functor， de Jong’s alteration theorems， h-topology， Log geometry and the log derived de Rham complex， Cohomological Descent and de Rham cohomology， p-adic Poincar´e Lemma， p-adic Poincar´e Lemma

2013 Winter

Deligne’s Finiteness Theorem