For any genus g ≥ 2 we give an example of a family of smooth complex projective curves of genus g... more For any genus g ≥ 2 we give an example of a family of smooth complex projective curves of genus g such that the image of the monodromy representation of the Hitchin connection on the sheaf of generalized SL(2)-theta functions of level l = 1, 2, 4 and 8 contains an element of infinite order.
In this paper we develop a theory of Grothendieck's six operations for adic constructible she... more In this paper we develop a theory of Grothendieck's six operations for adic constructible sheaves on Artin stacks continuing the study of the finite coefficients case in math.AG/0512097.
We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of r... more We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than d, and for the size of the sum of 2 Betti numbers for the real form of complex manifolds of complex degree less than d.
In this paper we develop a theory of Grothendieck's six operations of lisse-\'etale const... more In this paper we develop a theory of Grothendieck's six operations of lisse-\'etale constructible sheaves on Artin stacks which are locally of finite type over suitable regular basis of dimension at most 1.
Let X be an ordinary smooth curve defined over an algebraically closed field of characteristic 2.... more Let X be an ordinary smooth curve defined over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map F on the moduli space M_X of rank 2 vector bundles with fixed trivial determinant. If the genus of X is 2, the moduli space M_X is isomorphic to projective space of dimension 3 (as over the complex numbers). In this case we explicitly give the equations of F, which enables us to determine, for example, its base locus (one point) and its image (different from M_X).
Let X be a smooth projective curve over a field of characteristic p>0. We show that the Hitchi... more Let X be a smooth projective curve over a field of characteristic p>0. We show that the Hitchin morphism, which associates to a Higgs bundle its characteristic polynomial, has a non-trivial deformation over the affine line. This deformation is constructed by considering the moduli stack of t-connections on vector bundles on X and an analogue of the p-curvature, and by observing that the associated characteristic polynomial is, in a suitable sense, a p-th power.
For any genus g > 1 we give an example of a family of smooth complex projective curves of genu... more For any genus g > 1 we give an example of a family of smooth complex projective curves of genus g such that the image of the monodromy representation of the Hitchin connection on the sheaf of generalized SL(2)-theta functions of level l different from 1,2,4 and 8 contains an element of infinite order.
This paper is concerned with the moduli space of principal G-bundles on an algebraic curve, for G... more This paper is concerned with the moduli space of principal G-bundles on an algebraic curve, for G a complex semi-simple group. While the case G = SL r , which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it became clear that these spaces play an important role in Quantum Field Theory. In particular, if L is a holomorphic line bundle on the moduli space MG , the space H
Uperieure S Ormale N Ecole about G-bundles over Elliptic Curves about G-bundles over Elliptic Curves about G -bundles over Elliptic Curves
A para^ tre aux Annales de l'Institut Fourier R esum e : Soit G un groupe alg ebrique complex... more A para^ tre aux Annales de l'Institut Fourier R esum e : Soit G un groupe alg ebrique complexe simple et simplement connexe, T un tore maximal et W le groupe de Weyl. On d emontre que l'espace de modules grossier M G param etrant les classes de S-equivalence de G-br es semistables sur une courbe elliptique X est isomorphe a ?(T) Z X]=W. D'apr es un r esultat de Looijenga, ceci prouve que M G est un espace projectif anistotrope. Abstract: Let G be a complex algebraic group, simple and simply connected, T a maximal torus and W the Weyl group. One shows that the coarse moduli space M G (X) parameterizing S-equivalence classes of semistable G-bundles over an elliptic curve X is isomorphic to ?(T) Z X]=W. By a result of Looijenga, this shows that M G (X) is a weighted projective space.
In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in &... more In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in "The six operations for sheaves on Artin stacks I: Finite Coefficients" and "The six operations for sheaves on Artin stacks II: Adic Coefficients" (math.AG/0512097 and math.AG/0603680)
Let $X$ be a smooth, complete and connected curve and $G$ be a simple and simply connected algebr... more Let $X$ be a smooth, complete and connected curve and $G$ be a simple and simply connected algebraic group over $\comp$. We calculate the Picard group of the moduli stack of quasi-parabolic $G$-bundles and identify the spaces of sections of its members to the conformal blocs of Tsuchiya, Ueno and Yamada. We describe the canonical sheaf on these stacks and show that they admit a unique square root, which we will construct explicitly. Finally we show how the results on the stacks apply to the coarse moduli spaces and recover (and extend) the Drezet-Narasimhan theorem. We show moreover that the coarse moduli spaces of semi-stable $SO_r$-bundles are not locally factorial for $r\geq 7$.
This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and unifo... more This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian coefficients), vanishing theorems (e.g. affine Lefschetz), uniformization for the "prime-to-l alteration topology", rigidity for non-abelian coefficients, a new proof of the absolute purity conjecture, duality, etc.
This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and unifo... more This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian coefficients), vanishing theorems (e.g. affine Lefschetz), uniformization for the "prime-to-l alteration topology", rigidity for non-abelian coefficients, a new proof of the absolute purity conjecture, duality, etc.
A G ] 3 1 A ug 2 00 0 On the Hitchin morphism in positive characteristic
Let X be a smooth projective curve over a field of characteristic p > 0. We show that the Hitc... more Let X be a smooth projective curve over a field of characteristic p > 0. We show that the Hitchin morphism, which associates to a Higgs bundle its characteristic polynomial, has a non-trivial deformation over the affine line. This deformation is constructed by considering the moduli stack of t-connections on vector bundles on X and an analogue of the p-curvature, and by observing that the associated characteristic polynomial is, in a suitable sense, a pthpower.
-Soient A un anneau, f unélément simplifiable de A , A le séparé complété de A pour la topologie ... more -Soient A un anneau, f unélément simplifiable de A , A le séparé complété de A pour la topologie (f)-adique. Nous prouvons que la donnée d'un fibré vectoriel sur Spec(A)équivautà celle d'un fibré sur l'ouvert f = 0 de Spec(A) et sur Spec(A) , et d'un isomorphisme de leurs images réciproques sur l'ouvert f = 0 de Spec(A) .
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