Patrick Erik Bradley
Riemann Existence Theorems for Galois covers of Mumford curves by Mumford curves are stated and proven. As an application, all finite groups are realised as full automorphism groups of Mumford curves in characteristic zero.
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The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.
We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.
An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit squares. By varying the complex structure of the torus one obtains easily accessible examples of Teichmüller curves in the moduli space of Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves. A p-adic origami is defined as a covering of Mumford curves with at most one branch point, where the bottom curve has genus one. A classification o…
The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group.
A Motivic Riemann-Roch Theorem for Deligne-Mumford Stacks
本文为光滑分离 Deligne-Mumford 栈发展了在 Voevodsky 三角化动机范畴中可表示的动形上同调理论，证明其高阶 Chow 群与高阶 $K$-理论典范同构，从而将 Grothendieck-Riemann-Roch 定理推广至该范畴。
Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces
本文通过轨形Mukai配对导出新的轨形HRR公式，用于计算K3曲面上有限群作用下稳定层等变模空间的维数，并证明这些模空间是形变等价于点Hilbert概型的不可约辛流形。
We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.
Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus
研究证明在非双曲曲线的Teichmüller族中，不存在具有严格最大Higgs场的Shimura曲线，特别是Mumford型Shimura曲线不在该族中。
We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.
Mumford representation and Riemann Roch space of a divisor on a hyperelliptic curve
本文研究带 Weierstrass 无穷远点的超椭圆曲线，证明可由零次除子的 Mumford 表示直接构造相应 Riemann–Roch 空间的基，并用于得到 Goppa 码生成矩阵。
Existence of coarse moduli superspaces for Deligne--Mumford superstacks
本文证明在 $\mathbb{Z}[1/2]$ 上具有有限惯性的 Deligne--Mumford 超栈存在粗模超空间，推广 Keel--Mori 定理，并给出反例。
The Deligne-Riemann-Roch isomorphism
本文通过发展新的陈类范畴和相对交理论，确立了德利尼计划中关于精细格罗滕迪克-里曼-罗奇公式的几何部分，并展示了其在Knudsen-Mumford展开式和格里菲斯丛中的应用。
The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.
Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.
This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.
We establish a Hirzebruch-Riemann-Roch type theorem and Grothendieck-Riemann-Roch type theorem for matrix factorizations on quotient Deligne-Mumford stacks. For this we first construct a Hochschild-Kostant-Rosenberg type isomorphism explicit enough to yield a categorical Chern character formula. We next find an expression of the canonical pairing of Shklyarov under the isomorphism.
This paper proves a result on the existence of finite flat scheme covers of Deligne-Mumford stacks. This result is used to prove that a large class of smooth Deligne-Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to one concerning Brauer groups of schemes.
An integrable system is a dynamic system characterized by the existence of constants of motion and the existence of algebraic invariants, having an origin in algebraic geometry. In the 1970s, Mumford introduced a new completely integrable system defined on a smooth hyperelliptic curve. In the 2000s, Vanhaecke completed the description of the Munford integrable system by defining a Poisson structure on the phase space of the Mumford system. In this article we will study the singular Mumford system. The starting point is to determine when and why the Mumford system is singular. For this we will …
Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain …
We study the property of \emph{continuous Castelnuovo-Mumford regularity}, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in \cite{Kuronya:Mustopa:2020} by Küronya and Mustopa. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety's endomorphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that we established in \cite{Grieve:R-R:abVars}. In a compleme…
Mumford--Morita--Miller classes in generalised cohomology theories
本文研究 Riemann 曲面模空间上的 Mumford--Morita--Miller 类在广义复定向上同调理论中的定义与性质，并强调新现象。
Orbifold Hirzebruch-Riemann-Roch
该研究针对商Deligne-Mumford栈的轨形HRR定理展开，分析其与有限群表示论的联系，借助轨形Mukai配对得到了新的轨形HRR公式，丰富了代数几何领域的相关理论。
Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general bases. We will use these results to study equivariant K-theory, and also to define new filtrations on K-theory of algebraic stacks.
One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only im…
We generalize the result of Voronov (1988) to give an expression for the super Mumford form $μ$ on the moduli spaces of super Riemann surfaces with Ramond and Neveu-Schwarz punctures in the limit where the number of punctures is large compared to the genus. In the case of Neveu-Schwarz punctures we consider the super Mumford form over the component of the moduli space corresponding to an odd spin structure. The super Mumford form $μ$ can be used to create a measure whose integral computes scattering amplitudes of superstring theory. We express $μ$ in terms of local bases of $H^0(Σ, ω^j)$ for $…
We generalize the result of Voronov (1988) to give an expression for the super Mumford form $μ$ on the moduli spaces of super Riemann surfaces with Ramond and Neveu-Schwarz punctures. In the Ramond case we take the number of punctures to be large compared to the genus. We consider for the case of Neveu-Schwarz punctures the super Mumford form over the component of the moduli space corresponding to an odd spin structure. The super Mumford form $μ$ can be used to create a measure whose integral computes scattering amplitudes of superstring theory. We express $μ$ in terms of local bases of $H^0(X…
In the moduli space M_g of genus g Riemann surfaces, consider the locus RM_O of Riemann surfaces whose Jacobians have real multiplication by the order O in a totally real number field F of degree g. If g = 2 or 3, we compute the closure of RM_O in the Deligne-Mumford compactification of M_g and the closure of the locus of eigenforms over RM_O in the Deligne-Mumford compactification of the moduli space of holomorphic one-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O Boundary strata of RM_O are parameterized by configurations of el…
We study the existence of canonical Kähler metrics on the projectivisation of strictly Mumford semistable holomorphic vector bundles over a complex curve. We also provide an algebro-geometric characterization of these metrics.
In this note, we give a proof for a variant of the functorial Deligne-Riemann-Roch theorem in positive characteristic based on ideas appearing in Pink and Rössler's proof of the Adams-Riemann-Roch theorem in positive characteristic (see \cite{Pi}). The method of their proof appearing in \cite{Pi}, which is valid for any positive characteristic and which is completely different from the classical proof, will allow us to prove the functorial Deligne-Riemann-Roch theorem in a much easier and more direct way. Our proof is also partially compatible with Mumford's isomorphism.
Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves
本文以代数栈重述 Hilbert-Mumford 判据，给出保证稳定对象子栈存在分离粗模空间的几何条件，并应用于构造曲线上 parahoric 群概形下 torsor 的粗模空间。
The Super Mumford Form and Sato Grassmannian
本研究推广了Kontsevich等人的经典构造，利用超黎曼曲面模空间与佐藤格拉斯曼流形的关联，证明了特定线丛上平坦全纯联络的存在性及超共形Noether正规化引理。
The Neveu-Schwarz group and Schwarz's extended super Mumford form
该研究计算了Neveu-Schwarz作用在超tau函数上的形式，证明Schwarz的扩展超Mumford形式在超Heisenberg-Neveu-Schwarz作用下不变，支撑了超佐藤格拉斯曼流形内轨迹可作为超弦理论通用模空间的猜想，还构造了相关形式群。
Suppose $\mathcal{S}$ is a smooth, proper, and tame Deligne-Mumford stack. Toën's Grothendieck-Riemann-Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of Toën's Grothendieck-Riemann-Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.
A Hilbert--Mumford criterion for nilsolitons
本文提出幂零实李代数的代数准则作为对应李群存在nilsoliton的充要条件，推广了Nikolayevsky准则并证明修正版Taketomi–Tamaru猜想给出存在性障碍。
In this paper, we prove the functorial Riemann-Roch theorem in positive characteristic for a smooth and projective morphism with any relative dimension. In the case of relative dimension $1$, we have given an analogue with Deligne's functorial Riemann Roch theorem in previous author's paper. For any relative dimension, our result can deduce an analogue to the Knudsen-Mumford extension. The present result is a generalization, which mainly originated from the extended Deligne pairing by S. Zhang and the Adams Riemann Roch theorem in positive characteristic by R. Pink and D. Rössler.
Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights
针对超椭圆Mumford曲线，提出计算$p$-adic Schottky群的算法，并利用该均匀化计算$p$-adic Abelian积分和Schneider高度，通过SageMath数值示例验证。
A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing moduli and deformation parameters. By considering each generalized Tate curve as a family of degenerating Riemann surfaces, we give explicit formulas of the period isomorphism between its de Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. A remarkable fact is that similar formulas hold also for families of Mumfor…
A Hilbert--Mumford criterion for generalised Monge--Ampère equations
本文提出一个符合GIT精神的数值判据，用Paul意义下的配对稳定性刻画逆黑塞方程（含J-方程）解的存在性，并证明广义Chow线丛与混合Deligne配对的等距性。
We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we provide.
We consider two problems associated to the space of Riemann surfaces that are closely related to string theory. We first consider the problem of existence of heterotic-string and type-II-superstring field theory vertices in the product of spaces of bordered surfaces parameterizing left- and right-moving sectors of these theories. It turns out that this problem can be solved by proving the existence of a solution to the BV quantum master equation in moduli spaces of bordered spin-Riemann surfaces. We first prove that for arbitrary genus $g$, $n_{\text{NS}}$ Neveu-Schwarz boundary components, an…
A number of compactifications familiar in complex-analytic geometry, in particular, the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose nonempty intersections are Eilenberg-MacLane spaces. We exploit this fact to describe the (rational) homotopy type of these spaces and the natural maps between them in terms of the simplicial sets attached to certain categories. We thus generalize an old result of Charney-Lee on the Baily-Borel compactification of A_g and recover (and rephrase) a more recent one of E…
Given a vector bundle $F$ on a smooth Deligne-Mumford stack $\X$ and an invertible multiplicative characteristic class $\bc$, we define the orbifold Gromov-Witten invariants of $\X$ twisted by $F$ and $\bc$. We prove a "quantum Riemann-Roch theorem" which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A Quantum Lefschetz Hyperplane Theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of $\X$ and that of a complet…
We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a weak analog of the Keel-Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne-Mumford stack. We apply our result to prove that the moduli stacks of A_k and A_k^+-stable curves admit good moduli spaces. In forthcoming work, we will prove that these moduli spaces are projective and use them to construct the second flip in the log minimal model program for M_g.
Mumford-Tate groups of 1-motives and Weil pairing
本文研究1-动机的几何如何决定其动机Galois群的维数，通过周期矩阵描述Mumford-Tate群的作用，并在半椭圆情形下分类周期关系、计算群表示，为Grothendieck周期猜想提供新视角。
Boundary of equisymmetric loci of Riemann surfaces with abelian symmetry
本文研究了亏格$g\geq 2$黎曼曲面模空间中具有阿贝尔自同构群的等对称轨迹，确定了其边界最大维数层的拓扑类型，并用具有固定边数的树描述了超椭圆和循环$p$-边作用边界处的所有拓扑层。
We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definiti…
Let ${mathcal M}_g^n$ be the moduli space of n-pointed Riemann surfaces of genus g. Denote by ${\bar {\mathcal M}}_g^n$ the Deligne-Mumford compactification of ${mathcal M}_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of ${\bar {\mathcal M}}_g^n$ for any g and n such that n>2-2g.
Supersymmetric curves are the analogue of Riemann surfaces in super geometry. We establish some foundational results about complex Deligne-Mumford superstacks, and we then prove that the moduli superstack of supersymmetric curves is a smooth complex Deligne-Mumford superstack. We then show that the superstack of supersymmetric curves admits a coarse complex superspace, which, in this case, is just an ordinary complex space. In the second part of this paper we discuss the period map. We remark that the period domain is the moduli space of ordinary abelian varieties endowed with a symmetric thet…
Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of this curve and its definition field.
Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation
研究超椭圆曲线上代数几何码，通过芒福德表示刻画非特殊除子，构造LCP码并给出MDS准则，在特定条件下得到LCD码，应用于极大超椭圆曲线给出特定参数的MDS LCD码示例并推测其对所有$q \geq 4$存在。
A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Theta_s. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize speci…
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