Brauer groups of moduli of hyperelliptic curves via cohomological invariants

正特征下代数栈的上同调不变量与Brauer群

Cohomological invariants and Brauer groups of algebraic stacks in positive characteristic

引入特征p下代数栈的带模p^r系数上同调不变量理论，并据此完成了任意域上椭圆曲线栈的Brauer群与上同调不变量的计算。

A complete description of the cohomological invariants of even genus hyperelliptic curves

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic

Two torsion in the Brauer group of a hyperelliptic curve

We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C_1. In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.

Brauer group of punctual Quot scheme of points on a smooth projective surface

Let $X$ be a smooth projective surface over an algebraically closed field $k$ such that $char(k) \neq 2$. Let $X^{[d]}$ denote the punctual Hilbert scheme of zero dimensional quotients of degree $d$ and $X^{(d)}$ denote the symmetric product of $X$. For $\ell \neq 2$, we give a formula for the $\ell$-primary part of the Brauer group of $X^{[2]}$. We show that the Hilbert to Chow morphism induces an isomorphism of cohomological Brauer groups for $d=2$ and a similar result for $d \geq 3$. Let $Q(r,d)$ denote the punctual Quot-scheme parametrising zero dimensional quotients of $\mathcal{O}_X^{ \o…

Computing the Cohomological Brauer Group of a Toric Variety

The purpose of this article is to show how one might compute the étale cohomology groups $H^p_{\acute{e}t}(X,G_m)$ in degrees $p=0$, $1$ and $2$ of a toric variety $X$ with coefficients in the sheaf of units. The method is to reduce the computation down to the problem of diagonalizing a matrix with integral coefficients. The procedure outlined in this article has been fully implemented by the author as a program written in the ``C'' programming language.

稳定曲线模栈的Brauer群的消失

Vanishing of Brauer groups of moduli stacks of stable curves

本文证明了亏格$g\geq 2$的稳定曲线模栈的上同调Brauer群消失，并在特定范围内推广至带标记点版本，同时讨论了Deligne-Mumford栈的有限性结果。

The cohomological Brauer group of a toric variety

Toric varieties are a special class of rational varieties defined by equations of the form {\it monomial = monomial}. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety $X$ contains a cover by affine open sets described in terms of arrangements (called fans) of convex bodies in $\Bbb R^r$. The coordinate rings of each of these affine open sets is a graded ring generated over the ground field by monomials. As a consequence, toric varieties provide a good context in which cohomology can be calculated. The purpose of this article is to describe the seco…

Cohomological invariants of root stacks and admissible double coverings

We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the stack of admissible double coverings.

Brauer groups and quotient stacks

A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of $H^2(X,{\mathbb G}_m)$ is surjective.

Cohomological invariants of algebraic stacks

The purpose of this paper is to lay the foundations of a theory of invariants in étale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces.

Topological methods for complex-analytic Brauer groups

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-analytic surfaces.

Mackey-functor structure on the Brauer groups of a finite Galois covering of schemes

Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite étale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object "Brauer-Mackey functor". We realize this as a {\it cohomological Mackey functor} on the Galois category of finite étale coverings. For any finite étale covering of schemes, we can associate two homomorphisms for Brauer groups, namely the pull-back and the norm map. These homomorphisms make Brauer groups into a bivariant functor ($=$ Mackey functor) on the Galois…

Brauer groups of schemes associated to symmetric powers of smooth projective curves in arbitrary characteristics

In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves: we prove such results only for curves defined over the field of complex numbers.

On the Brauer group of the product of a torus and a semisimple algebraic group

Let T be a torus (not assumed to be split) over a field F, and denote by $_n{H^{2}_{et}(X,Gm)}$ the subgroup of elements of exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism $_n{H^2_{et}(T,Gm)}\to_n{H^2_{et}(X\times T,Gm)}$ is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characterist…

Cohomological obstruction theory for Brauer classes and the period-index problem

Let U be a connected scheme of finite cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that alpha is a class in H^2(U_et,Gm)_{tors}. For each positive integer m, the K-theory of alpha-twisted sheaves is used to identify obstructions to alpha being representable by an Azumaya algebra of rank m^2. The etale index of alpha, denoted eti(alpha), is the least positive integer such that all the obstructions vanish. Let per(alpha) be the order of alpha in H^2(U_{et},Gm)_{tors}. Methods from stable homotopy theory give an upper bound on the e…

Cohomology of the hyperelliptic Torelli group

Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of genus g. This is the subgroup of the mapping class group of S_g consisting of elements that act trivially on H_1(S_g;Z) and that commute with some fixed hyperelliptic involution of S_g. We prove that the cohomological dimension of SI(S_g) is g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely generated when g > 1. In particular, SI(S_3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B_n at t = -1 has cohomological dim…

Kummer varieties and their Brauer groups

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with larg…

The Cohomological Excess of Certain Moduli Spaces of Curves of Genus $g$

The open subvariety $\overline{M}_g^{\leq k}$ of $\overline{M}_g$ parametrizes stable curves of genus $g$ having at most $k$ rational components. By the work of Looijenga, one expects that the cohomological excess of $\overline{M}_g^{\leq k}$ is at most $g-1+k$. In this paper we show that when $k=0$, the conjectured upper bound is sharp by showing that there is a constructible sheaf on $\overline{H}_g^{\leq k}$ (the hyperelliptic locus) which has non-vanishing cohomology in degree $3g-2$.

Cohomology of the moduli space of non-hyperelliptic genus four curves

We compute the intersection Betti numbers of the GIT model of the moduli space of Brill-Noether-Petri general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett-Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities.

线性代数群的2次上同调不变量

Degree 2 cohomological invariants of linear algebraic groups

本文利用单纯概形上的 étale 上同调，刻画任意域上光滑连通线性代数群取值于 Brauer 群的2次不变量，并涵盖非完善域上的非约化群。

Algebraic families of hyperelliptic curves violating the Hasse principle

In $2000$, Colliot-Thélène and Poonen showed how to construct algebraic families of genus one curves violating the Hasse principle. Poonen explicitly constructed an algebraic family of genus one cubic curves violating the Hasse principle using the general method developed by Colliot-Thélène and himself. The main result in this paper generalizes the result of Colliot-Thélène and Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for $n > 5$ and $n \not\equiv 0 \pmod{4}$, we show that there is an algebraic family of hyperelliptic curves of genus $n$ that is counterexamples to…

The algebraic Brauer group of a pinched variety

Added lemma provided by Michel Brion. Other (minor) changes. Submitted version. Let k be any field, let X' be a projective and geometrically integral k-scheme and let Y' be a finite closed subscheme of X'. If f: Y'-> Y is a schematically dominant morphism between finite k-schemes and X is obtained by pinching X' along Y' via f, we describe the kernel (and, in certain cases, the cokernel) of the induced pullback map Br_{1}X -> Br_{1}X' between the corresponding algebraic (cohomological) Brauer groups of X and X'_solely_ in terms of the Brauer groups of the residue fields of Y and Y' and the Ami…

Explicit descent on elliptic curves and splitting Brauer classes

We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting $μ_N$-gerbes, we use crucial input from the arithmetic of modular curves and explicit $N$-descent on elliptic curves.

The Brauer group of analytic K3 surfaces

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family.

Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves

We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results ar…

亏格3曲线、阿贝尔三重折叠和平面曲线模空间的Brauer群

The Brauer groups of moduli of genus three curves, abelian threefolds and plane curves

本文通过计算低阶上同调不变量，确定了特征不为2的域上亏格3曲线模栈、特定条件下平面曲线模栈及3维主极化阿贝尔簇模栈的Brauer群的$\ell$-准原挠。

次数为4的del Pezzo曲面与亏格模4余1的超椭圆曲线的代数族违反Hasse原则

Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4

本文构造了无限多个显式单参数代数族，包含次数为4的del Pezzo曲面和亏格模4余1的超椭圆曲线，证明其每个K成员都违反Hasse原则，并由Brauer-Manin障碍解释。

Brauer group of a moduli space of parabolic vector bundles over a curve

Let ${\mathcal P}{\mathcal M}^α_s$ be a moduli space of stable parabolic vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X) \geq 2$. If $g(X) = 2$, then we assume that $n > 2$. Let $m$ denote the greatest common divisor of $d$, $n$ and the dimensions of all the successive quotients of the quasi-parabolic filtrations. We prove that the cohomological Brauer group ${\rm Br}({\mathcal P}{\mathcal M}^α_s)$ is isomorphic to the cyclic group ${\mathbb Z}/ m{\mathbb Z}$. We also show that ${\rm Br}({\mathcal P}{\mathcal M}^…

Brauer Spaces of Spectral Algebraic Stacks

We study the question of whether the Brauer group is isomorphic to the cohomological one in spectral algebraic geometry. For this, we prove the compact generation of the derived category of twisted sheaves for quasi-compact spectral algebraic stacks with quasi-affine diagonal, which admit a quasi-finite presentation; in particular, we obtain the compact generation of the unbounded derived category of quasi-coherent sheaves and the existence of compact perfect complexes with prescribed support for such stacks. We also study the relationship between derived and spectral algebraic stacks, so that…

A cohomological obstruction to weak approximation for homogeneous spaces

Let X be a homogeneous space, X = G/H, where G is a connected linear algebraic group over a number field k, and H is a k-subgroup of G (not necessarily connected). Let S be a finite set of places of k. We compute the Brauer-Manin obstruction to weak approximation for X in S in terms of Galois cohomology.

There are enough Azumaya algebras on surfaces

Using Maruyama's theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for separated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces.

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological Brauer-Grothendieck group of $X$ and $Br_0(X)$ the image of $Br(k)$ in $Br(X)$. We write $Br_1(X)$ for the subgroup of elements in $Br(X)$ that become trivial after replacing $k$ by its algebraic closure. We prove that $Br(X)/Br_0(X)$ is finite if $X$ is a $K3$ surface. When $X$ is (a principal homogeneous space of) an abelian variety over $k$ then we prove that $Br(X)/Br_1…

Rationally simply connected varieties and pseudo algebraically closed fields

The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension $1$. Every transcendence degree $1$ extension of such a field has cohomological degree $2$. We prove Serre's "Conjecture II" for such fields of …

代数堆上的Brauer-Manin 障碍

The Brauer-Manin obstruction on algebraic stacks

本文研究代数堆上的Brauer-Manin集和配对，扩展了Colliot-Thélène和Sansuc的降理论，证明了局部由变分线性群商的堆的Brauer群的 torsion 性，并展示了Brauer-Manin障碍与其它上同调障碍的一致性。

A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension

A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends …

The relative units-Picard complex and the Brauer group of a product

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors. Under certain hypotheses, we obtain a five-term exact sequence involving the preceding groups which enables us to solve the indicated problem in the case of, e.g., (certain types of) ruled varieties over a field of characteristic zero.

通过gerbes得到的箭图模空间奇点解消的Brauer群

Brauer groups of resolved quiver moduli via gerbes

研究半稳定箭图表示模空间奇点解消的Brauer群，通过扩展箭图 - 曲线字典，将关于曲线上向量丛模的证明转换到箭图表示模情形，给出平凡性新证明及相关消失结果的新代数几何证明。

2-descent 对 Bloch-Kato Selmer 组和超椭圆曲线上的有理点 I

2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves I

本文提出了一种通过 Galois 协同论方法确定超椭圆曲线 Jacobian 的 Bloch-Kato Selmer 组秩的方法，并利用该方法回答了关于有理点的问题。

广义 del Pezzo 环面的 torsor 与 Brauer 群

Torsors of Generalized del Pezzo Tori and Brauer Groups

利用上同调 Mackey 函子，给出广义 del Pezzo 簇上环面 torsor 的显式 Brauer 群分类，推广了 Blunk 对六次 del Pezzo 曲面的 Brauer 理论描述。

无限CW复形、布劳尔群与幻影上同调

Infinite CW-complexes, Brauer groups and phantom cohomology

本研究推广Serre的有限CW复形结果，证明紧空间的布劳尔群与上同调布劳尔群一致，同时用同伦论方法重证并拓展了Eilenberg–MacLane空间、无限维F_p向量空间的分类空间不满足该相等性的结论，还给出幻影上同调的同调表述公式。

Cohomology of moduli spaces of curves of genus three via point counts

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of…

Non-abelian reciprocity laws and higher Brauer-Manin obstructions

We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…

超椭圆轨迹上的万有 Picard 层的 Chow 环

The Chow ring of the universal Picard stack over the hyperelliptic locus

本文确定了超椭圆轨迹上万有 Picard 层的有理 Chow 环及其生成元与关系，并计算了整 Picard 群，证明了其 Brauer 类阶由 $d-g$ 奇偶性决定。

Locally free twisted sheaves of infinite rank

We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of "very positive" infinite rank vector bundles on projective variet…

循环群与循环群半直积扩张的上同调核

Cohomological Kernels for Cyclic by Cyclic Semi-Direct Product Extensions

本文针对 Galois 闭包为循环群半直积的域扩张，利用 Positelski 的工具建立六项正合序列以刻画 Galois 上同调核，并涵盖相对 Brauer 群情形。

On the Brauer Group of Real Algebraic Surfaces

Let X be a real projective algebraic manifold, s numerates connected components of X(R) and _2Br(X) the subgroup of elements of order 2 of the cohomological Brauer group Br(X). We study the natural homomorphism ξ: _2Br(X) \to (Z/2)^s and prove that ξis epimorphic if H^3(X(C)/G;Z/2) \to H^3(X(R);Z/2) is injective. Here G=Gal(C/R). For an algebraic surface X with H^3(X(C)/G;Z/2)=0 and X(R)\not=\emptyset, we give a formula for dim _2Br(X). As a corollary, for a real Enriques surfaces Y, the ξis epimorphic and dim _2Br(Y)=2s-1 if both liftings of the antiholomorphic involution of Y to the universa…

群论中的Johnson类与一个非双曲曲线的 torsion Ceresa 类

Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class

本文通过群论方法构造Johnson/Morita余数，应用于曲线的pro-l étale基本群，讨论其与Hain和Matsumoto工作的关系，并给出一个非双曲曲线的Ceresa类在l-adic Abel-Jacobi映射下具有torsion图像的例子。

A higher dimensional generalization of Lichtenbaum duality in terms of the Albanese map

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the Néron-Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparison result of cohomological- and Azumaya-Brauer groups. We will also mention the local-global problem of the Albanese-cokernel; the abelian group on the "local side" turns out to be a finite group.

Twisted sheaves and the period-index problem

We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber's theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong's period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over …

Linear manifolds in the moduli space of one-forms

We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to strata. For non-generic strata similar results can be shown by a case-by-case inspection. We also propose to study a notion of 'linear manifold' that comprises Teichmueller curves, Hilbert modular s…

J-invariant of linear algebraic groups

Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behaviour of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as we…

来自 Theta 特征的曲线模空间上的上同调类

Cohomology classes on moduli of curves from Theta Characteristics

本文利用 theta 特征的 Galois–Stiefel–Whitney 类，证明全实域上曲线与主极化阿贝尔簇模栈在指定次数存在非平凡上同调类，并计算亏格 3 情形的 Brauer 群拉回。

函数域非分歧扩张上的Azumaya代数

Azumaya algebras over unramified extensions of function fields

本文利用Milnor-Quillen $K$-群解释函数域的étale上同调挠部分，证明了在范数映射的温和条件下，光滑簇上存在可由Azumaya代数表示的上同调Brauer类，从而部分回答了Grothendieck的问题。

模栈与向量丛扭曲模空间的Torelli定理

Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves

本文证明了向量丛模空间的两个Torelli定理：从固定行列式模栈恢复曲线，及从扭曲模空间恢复曲线和Brauer类，方法涉及特殊线性线性化、Hitchin构造和Kummer覆盖等。