FR CNRS 2962

## [Nantes, Matpyl] Activité de printemps : Fibrés vectoriels sur les courbes.

Date début de l'évènement
Date de fin l'évènement
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Matpyl Spring activity on vector bundles overs curves

We will start on Monday at 9h. Registration 8h30!

This page is an overview of the topics that will be treated by the activity. Names behind the lectures indicate the distribution of the talks to be given by the participants.

There will be 6 lectures of 45 minutes a day, scheduled as follows

09h00 - 09h45 Lecture 1
09h45 - 10h15 Coffee break
10h15 - 11h00 Lecture 2
11h15 - 12h00 Lecture 3
12h00 - 14h00 Lunch
14h00 - 14h45 Lecture 4
15h00 - 15h45 Lecture 5
15h45 - 16h15 Coffee break
16h15 - 17h00 Lecture 6

Monday

AM : Salle Hypathia

Lecture 1 [Chloé Grégoire]

• Definition of a vector bundle over a smooth curve $X$ defined over an algebraically closed field $k$ (of any characteristic)
• Definition of (semi-)stability of a vector bundle
• Basic examples : if $X$ is the projective line, Grothendieck's theorem (with proof: [HL] Thm 1.3.1 ), Lazarsfeld's evaluation bundles
• Elementary properties: stable implies simple; the category of semi-stable vector bundles of fixed slope is abelian.

References [HL],[Se],[LeP]

Lecture 2 [Arvid Perego]

• The fundamental group $\pi_1(X)$ if $k=C$.
• Vector bundles : $E_{\rho}$ coming from representations $\rho$ of the fundamental group .
• Theorem of Weil (see e.g. [Se] page 46) (without proof)
• Theorem of Narasimhan-Seshadri (without proof). Corollary: $E$ and $F$ stable implies $E \otimes F$ semi-stable

References [Se]

Lecture 3 [Heinrich Hartmann]

• The Jordan-Hölder filtration of a semi-stable bundle
• The Harder-Narasimhan filtration of a bundle; existence and uniqueness the Harder-Narasimhan filtration is stable under base field extension (with proof) see [HL] Theorem 1.3.7

References: [LeP] and [HL]

PM : Salle Au Val

Lecture 4 [Yashonidhi Pandey]

• The algebraic fundamental group $\pi_1^{alg}(X)$
• Bundles $E_\rho$ coming from continuous representations $\rho$ of the algebraic fundamental group; these are the étale trivial bundles ([LaSt])
• The absolute/relative Frobenius morphism of the curve $X$ in positive characteristic
• Bundles coming from representations are fixed under the Frobenius ([LaSt] Satz 1.4).

References: for general facts on the algebraic fundamental group see e.g. chapter 9 by A. Mézard in [BLR], on the relative/absolute Frobenius see e.g [Ray] section 4.

Lecture 5/6 [Christian Lehn & Markus Zowislok]

• Definition of $(E, \nabla)$ bundle with connection $\nabla$ and $p$-curvature of $\nabla$ in char $p>0$
• Cartier's theorem on descent under Frobenius
• The generalized Verschiebung on vector bundles and some of its properties (without proof)
• Frobenius-destabilized vector bundles, definition of strongly semi-stability
• $E$ and $F$ strongly semi-stable implies $E \otimes F$ strongly semi-stable (with sketch of proof [RR]).

Reference: [K] section 5, [Ray] section 4, [LP], [RR]

Tuesday : Principal $G$-bundles

AM : Salle Eole

Lecture 1 [Manfred Lehn]

• Review of basics in representation theory of algebraic groups (in any characteristic)
• Definition of semi-simple/reductive algebraic groups, maximal torus, Borel subgroup, parabolic subgroup, root system, weight lattice, Weyl group

Reference: [Spr]

Lecture 2 [Tanja Becker]

• Definition of principal $G$-bundle $E_G$ over a smooth curve $X$ with $G$ reductive.
• Comparaison local triviality in étale topology and local trivialility in Zariski topology.
• Extension of structure group $G \rightarrow H$, associated fibre bundle $E_G(Y)$ for an action of $G$ on $Y$, important case: $Y$ is a linear representation of $G$
• Reduction of the structure group of $E_G$ to $H\subset G$.
• Automorphism group $Aut(E_{G})$ of a principal G-bundle.
• One has $Z(G)\subset Aut(E_G)$.
• Examples: $G=GL(r)$, $SL(r), SO(r), O(r)$.

References: [So], [Ra]

Lecture 3 [Heinrich Hartmann]

• Description of $G$-bundles $E_G$ as tannakian functor
• Topological classification of $G$-bundle
• Various definitions of semi-stability of $E_G$: degree of $T^{\vert}$, characters of parabolic subgroups
$P\subset G$
• For $G= GL(r)$ one recovers semi-stability for vector bundles.

References: [Ra], [So1], [N]

PM : Salle Au Val

Lecture 4 [Olivier Serman]

• Semi-stable $G$-bundles with fixed topological type form a bounded family (in any characteristic)

Reference: [Beh] (8.2.6), [HN]

Lecture 5 [Jochen Heinloth]

Existence and uniqueness of canonical reduction of $E_G$ (with sketch of proof...)

Lecture 6 [Jochen Heinloth]

• Behrend's conjecture ([B] conjecture 7.6)
• Its implications (rationality of canonical reduction), low-height representations, and a counter-example to Behrend's conjecture for $G = G_2$ and $p=2$.

References: [B], [IMP], [He2]

Wednesday: moduli spaces

AM : Salle Hypathia

Lecture 1 [Alessandra Sarti]

• functors of points, universal family, scheme (co-) representing a functor, existence of Grothendieck Quot scheme
• notions de quotients (catégorique, bon, géométrique)

References: [HL] Section 2.2, section 4; [Do], [LeP]

Lecture 2 [Alessandra Sarti]

• Introduction to GIT. Les critères utiles de semi-stabilité

References: [LeP]

Lecture 3 [Samuel Boissière]

• Construction GIT de M(r,d)

References: [LeP]

PM Salle Hypathia

Lecture 4 [Samuel Boissière]

• Construction GIT de M_G + propriétés

References: [BLS]

Lecture 5 [Olivier Serman]

• Semi-stable reduction theorem(s)
• Show that $M_X(G)$ is proper in characteristic zero or if characteristic $p$ large: do first the vector bundle case, then go to $G$-bundles.

References: [Lan], [HL] section 2B for the vector bundle case; [BP], [F], [He1] for $G$-bundles

Lecture 6 [Etienne Mann]

• Principal $G$-bundles over elliptic curves
• Sketch of proof that the moduli space $M_X(G)$ is isomorphic to a weighted projective space over an elliptic curve $X$

[FMW], [Las]

Thursday: Conformal blocks and Verlinde formula

AM : Salle des séminaires

Lecture 1 [Timo Schürg]

• Introduction to moduli stacks of $G$-bundles.
• Definition of algebraic stack, the stack of $G$-bundles is algebraic, smooth. Its dimension.

[Go], [So1]

Lectures 2/3 [Manfred Lehn]

• Uniformization of $G$-bundles, loop spaces

References: [So1], [BL], [LS2], [F2]

PM : Salle Hypathia

Lecture 4 [Manfred Lehn]

• Representations of affine Lie algebras,
• space of (co)-vacua (conformal block)
• Virasoro algebras, Sugawara construction.

[So2], [SU]

Lecture 5 [Christoph Sorger]

• Infinite Grassmannians, line bundles over the stack of $G$-bundles

References: [So1],[So3],[BL], [LS2], [F2]

Lecture 6 [Christian Lehn]

• Isomorphism between space of conformal blocks and space of generalized theta functions

[BL], [LS2]

Friday: projective connections, WZW and Hitchin

AM : Salle Hypathia

Lectures 1/2/3 [Christian Pauly]

• Constructions of projective connections on the spaces of covacua by WZW and by Hitchin, comparison

PM : Salle Au Val

Lectures 4/5/6 [Christoph Sorger]

• Fusions rings and the Verlinde formula

[Be], [So2]

Saturday: Level-Rank duality

AM : Salle Hypathia

Lecture 1/2 [Rémy Oudompheng]

• Level-Rank (or strange) duality of theta-functions

Lecture 3 [everybody]

• Open questions, conjectures, ideas, what to do next ...

References :

• [AB] M.F. Atiyah, R. Bott: The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A, 308 (1982), 523-615
• [BP] V. Balaji, A.J. Parameswaran: Semistable principal bundles II. Positive characteristics. Transform. Groups 8 (2003), 3-36
• [Be] A. Beauville: Conformal blocks, fusion rules and the Verlinde formula, In Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (1996)
• [BL] A. Beauville, Y. Laszlo: Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385-419
• [BLS] A. Beauville, Y. Laszlo, C. Sorger: The Picard group of the moduli of $G$-bundles on a curve, Compositio Math. 112 (1998), 183-216
• [B] K. Behrend: Semi-stability of reductive group schemes over curves, Math. Ann. 301 (1995), 281-305
• [BH] I. Biswas, Y.I. Holla: Harder-Narasimhan reduction of a principal bundle, Nagoya Math. J (2004), 201-223
• [BLR] J.-B. Bost, F. Loeser, M. Raynaud: Courbes semi-stables et groupe fondamental en géométrie algébrique, Progress in Mathematics, Vol. 187, Birkhäuser Verlag
• [Do] I. Dolgachev: Lectures on Invariant Theory, London Mathematical Society Lecture Note Series 296, Cambridge University Press
• [F1] G. Faltings: Projective connections and G-bundles, J. Alg. Geometry 2, No. 3 (1993), 507-568
• [F2] G. Faltings: Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003),41-68
• [FMW] R. Friedman, J. Morgan, E. Witten: Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5 (1998), 97-118
• [Go] T. Gomez: Algebraic stacks, math.AG/9911199
• [He1] J. Heinloth: Semistable reduction for $G$-bundles over curves, J. Alg. Geom. 17 (2008), 167-183
• [He2] J. Heinloth: Bounds for Behrend's conjecture on the canonical reduction, arXiv:0712.0692
• [HN] Y. I. Holla, M. S. Narasimhan: A generalisation of Nagata's theorem on ruled surfaces, Compositio Math. 127 (2001), 321-332
• [HL] D. Huybrechts, M. Lehn: The Geometry of moduli spaces of sheaves, Aspects of Mathematics, E31 (1997)
• [IMP] S. Ilangovan, V.B. Mehta, A.J. Parameswaran: Semistability and semisimplicity in representations of low height in positive characteristic, A tribute to C.S.Seshadri
• [K] N. Katz: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Inst. Hautes Etudes Sci. Publ. Math. 39 (1970), 175-232
• [LaSt] H. Lange, U. Stuhler: Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Zeitschrift 156 (1977), 73-83
• [LS] Y. Laszlo, C. Sorger: The line bundles on the moduli of parabolic $G$-bundles over curves and their sections, Ann. Sci. Ecole Norm. Sup. 4 (1997), 499-525
• [Lan] S.G. Langton: Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2) 101 (1975), 88-110
• [Las] Y. Laszlo: About $G$-bundles over elliptic curves, Ann. Inst. Fourier 48 (1998), 413-424
• [LP] Y. Laszlo, C. Pauly: On the Hitchin morphism in positive characteristic, IMRN 3 (2001), 129-143
• [LeP] J. Le Potier: Lectures on vector bundles, Cambridge Studies in Advanced Mathematics 54, Cambridge University Press, 1997
• [N] M.V. Nori, The fundamental group scheme, Proc. Indian Acad. Sci. (Math. Sci.) 91 (1982), 78-122
• [RR] S. Ramanan, A. Ramanathan: Some remarks on the instability flag, Tohoku Math. Journal 36 (1984), 269-291
• [Ra] A. Ramanathan: Moduli for principal bundles over algebraic curves I, II, Proc. Indian Ac. Sci. Sci. 106 (1996), 301-328 and 421-449
• [Ray] M. Raynaud: Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France, Vol. 110 (1982), 103-125
• [Se] C.S. Seshadri: Fibrés vectoriels sur les courbes algébriques, Astérisque 96, 1982 (rédigé par J.-M. Drézet)
• [SU] Y. Shimizu, K. Ueno: Advances in Moduli Theory, Translations of Mathematical Monographs 206, AMS (2002)
• [So1] C. Sorger: Lectures on moduli of principal $G$-bundles over algebraic curves, School on Algebraic Geometry (Trieste, ICTP 1999), 1-57
• [So2] C. Sorger: La formule de Verlinde, Séminaire Bourbaki, 1994/95, Exp. No. 794, Astérisque 237 (1996), 87-114
• [So3] C. Sorger: On Moduli for $G$-bundles for exceptional G, Ann. Sci. Ecole. Norm. Sup, 32, 1999, 127-133
• [Spr] T.A. Springer: Linear Algebraic Groups, Progress in Mathematics, Birkhäuser (1983)

Organisers: Manfred Lehn, Christian Pauly, Christoph Sorger