scholarly journals Geometric Langlands in prime characteristic

2017 ◽  
Vol 153 (2) ◽  
pp. 395-452 ◽  
Author(s):  
Tsao-Hsien Chen ◽  
Xinwen Zhu
Keyword(s):  
Differential Operators ◽  
Derived Category ◽  
Closed Field ◽  
Projective Curve ◽  
Local Systems ◽  
Smooth Projective Curve ◽  
Simple Algebraic Group ◽  
Coherent Sheaves ◽  
Geometric Langlands ◽  
Moduli Stack

Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve{G}$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$ of genus at least two. Denote by $\operatorname{Bun}_{G}$ the moduli stack of $G$-bundles on $C$ and $\operatorname{LocSys}_{\breve{G}}$ the moduli stack of $\breve{G}$-local systems on $C$. Let $D_{\operatorname{Bun}_{G}}$ be the sheaf of crystalline differential operators on $\operatorname{Bun}_{G}$. In this paper we construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}(\operatorname{LocSys}_{\breve{G}}^{0}))$ of quasi-coherent sheaves on some open subset $\operatorname{LocSys}_{\breve{G}}^{0}\subset \operatorname{LocSys}_{\breve{G}}$ and bounded derived category $D^{b}(D_{\operatorname{Bun}_{G}}^{0}\text{-}\text{mod})$ of modules over some localization $D_{\operatorname{Bun}_{G}}^{0}$ of $D_{\operatorname{Bun}_{G}}$. This generalizes the work of Bezrukavnikov and Braverman in the $\operatorname{GL}_{n}$ case.

On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn

2006 ◽  
Vol 58 (5) ◽  
pp. 1000-1025 ◽  
Author(s):  
Ajneet Dhillon
Keyword(s):  
Vector Bundles ◽  
Hodge Structure ◽  
Betti Numbers ◽  
Mixed Hodge Structure ◽  
Projective Curve ◽  
Ground Field ◽  
Smooth Projective Curve ◽  
Poincaré Polynomial ◽  
Moduli Stack ◽  
Moduli Of Vector Bundles

AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

scholarly journals ON THE VOEVODSKY MOTIVE OF THE MODULI STACK OF VECTOR BUNDLES ON A CURVE

2020 ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Vector Bundles ◽  
Intersection Theory ◽  
Line Bundles ◽  
Projective Curve ◽  
Classifying Space ◽  
Smooth Projective Curve ◽  
Quot Schemes ◽  
Homotopy Colimit ◽  
Moduli Stack ◽  
Fixed Rank

Abstract We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

2019 ◽  
Vol 99 (2) ◽  
pp. 195-202
Author(s):  
LINGGUANG LI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

scholarly journals Spherical Hall Algebras of Weighted Projective Curves

2018 ◽  
Vol 2020 (15) ◽  
pp. 4721-4775
Author(s):  
Jyun-Ao Lin
Keyword(s):  
Vector Bundles ◽  
Characteristic Functions ◽  
Projective Curve ◽  
Hall Algebra ◽  
Smooth Projective Curve ◽  
Coherent Sheaves ◽  
Hall Algebras ◽  
Arbitrary Genus ◽  
Projective Curves ◽  
Fixed Rank

Abstract In this article, we deal with the structure of the spherical Hall algebra $\mathbf{U}$ of coherent sheaves with parabolic structures on a smooth projective curve $X$ of arbitrary genus $g$. We provide a shuffle-like presentation of the bundle part $\mathbf{U}^&gt;$ and show the existence of generic spherical Hall algebra of genus $g$. We also prove that the algebra $\mathbf{U}$ contains the characteristic functions on all the Harder–Narasimhan strata. These results together imply Schiffmann’s theorem on the existence of Kac polynomials for parabolic vector bundles of fixed rank and multi-degree over $X$. On the other hand, the shuffle structure we obtain is new and we make links to the representations of quantum affine algebras of type $A$.

scholarly journals Sheaves of categories with local actions of Hochschild cochains

2019 ◽  
Vol 155 (08) ◽  
pp. 1521-1567 ◽  
Author(s):  
Dario Beraldo
Keyword(s):  
Differential Operators ◽  
Algebraic Stacks ◽  
Finiteness Conditions ◽  
Coherent Sheaves ◽  
Singular Support ◽  
Mild Conditions ◽  
Geometric Langlands ◽  
Category Of Modules ◽  
Langlands Programme

The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$ , affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ , where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$ . In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$ . Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$ -modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$ , its descent properties and the notion of $\mathbb{H}$ -affineness. We then prove the $\mathbb{H}$ -affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$ -category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$ -category of modules for $\mathbb{H}({\mathcal{Y}})$ , the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$ , then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$ , where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$ . As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$ , thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$ .

scholarly journals Harder–Narasimhan reduction of a principal bundle

2004 ◽  
Vol 174 ◽  
pp. 201-223 ◽  
Author(s):  
Indranil Biswas ◽  
Yogish I. Holla
Keyword(s):  
Algebraic Group ◽  
Principal Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Mild Assumption

AbstractLet E be a principal G–bundle over a smooth projective curve over an algebraically closed field k, where G is a reductive linear algebraic group over k. We construct a canonical reduction of E. The uniqueness of canonical reduction is proved under the assumption that the characteristic of k is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of k is positive.

Integral orbits over function fields

2014 ◽  
Vol 10 (08) ◽  
pp. 2187-2204
Author(s):  
Hsiu-Lien Huang ◽  
Chia-Liang Sun ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Forward Orbit ◽  
Siegel Theorem

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.

scholarly journals Formal orbifolds and orbifold bundles in positive characteristic

2019 ◽  
Vol 30 (12) ◽  
pp. 1950067
Author(s):  
Manish Kumar ◽  
A. J. Parameswaran
Keyword(s):  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Positive Characteristic ◽  
Closed Field ◽  
Fundamental Groups ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental groups approximate the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statements about vector bundles which are true in characteristic zero are also proved. Some of these are positive characteristic avatars of notions which appear in the second author’s work [A. J. Parmeswaran, Parabolic coverings I: Case of curves, J. Ramanujam Math. Soc. 25(3) (2010) 233–251.] in characteristic zero.

scholarly journals Geometric Waldspurger periods

2008 ◽  
Vol 144 (2) ◽  
pp. 377-438 ◽  
Author(s):  
Sergey Lysenko
Keyword(s):  
Derived Categories ◽  
Hecke Operators ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Geometric Langlands Program ◽  
Geometric Langlands ◽  
Langlands Functoriality

AbstractLet X be a smooth projective curve. We consider the dual reductive pair $H=\mathrm {G\mathbb {O}}_{2m}$, $G=\mathrm {G\mathbb {S}p}_{2n}$ over X, where H splits on an étale two-sheeted covering $\pi :\tilde X\to X$. Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, $H=\pi _*{\mathbb {G}_m}$ with $\tilde X$ connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LH→LG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from $\mathrm {Bun}_{\mathrm {G\mathbb {O}}_4}$ to $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$ (one of them makes sense for ${\mathcal D}$-modules only).

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