Results 1 - 10
of
29
Differentiable Stacks and Gerbes
, 2008
"... We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹-bundles and S¹-gerbes over differentiable stacks. In particular, we establish the relationship between S¹-gerbes and groupoid S¹-central extensions. We define connections and curvings for groupoid S¹ ..."
Abstract
-
Cited by 45 (4 self)
- Add to MetaCart
We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹-bundles and S¹-gerbes over differentiable stacks. In particular, we establish the relationship between S¹-gerbes and groupoid S¹-central extensions. We define connections and curvings for groupoid S¹-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S¹-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S¹-bundles and S¹-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S¹-central extensions with prescribed curvature-like data.
On the classification of rank two representations of quasiprojective fundamental groups
"... Abstract. Suppose X is a smooth quasiprojective variety over C and ρ: π1(X, x) → SL(2, C) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X → Y with Y either a DM-curve or a Shimura modular stack. 1. ..."
Abstract
-
Cited by 12 (3 self)
- Add to MetaCart
(Show Context)
Abstract. Suppose X is a smooth quasiprojective variety over C and ρ: π1(X, x) → SL(2, C) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X → Y with Y either a DM-curve or a Shimura modular stack. 1.
KATZ’S MIDDLE CONVOLUTION ALGORITHM
, 2006
"... Abstract. This is an expository account of Katz’s middle convolution operation on local systems over P 1 − {q1,...,qn}. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Harao ..."
Abstract
-
Cited by 6 (0 self)
- Add to MetaCart
Abstract. This is an expository account of Katz’s middle convolution operation on local systems over P 1 − {q1,...,qn}. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Haraoka-Yokoyama. Kostov’s program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and Crawley-Boevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle. 1.
THE p-RANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES
"... ABSTRACT. We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p ≥ 3. Using this, we prove that the Z/ℓmonodromy of every irreducible component of the stratum H f g of hyperelliptic curves of genus g and p-rank f ..."
Abstract
-
Cited by 6 (4 self)
- Add to MetaCart
(Show Context)
ABSTRACT. We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p ≥ 3. Using this, we prove that the Z/ℓmonodromy of every irreducible component of the stratum H f g of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3, f ≥ 1 and ℓ ̸ = p is an odd prime. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g ≥ 3 and p-rank 0 is not supersingular. Other applications are about absolutely simple Jacobians and the generic behavior of class groups and zeta functions of hyperelliptic curves of given genus and p-rank over finite fields. 1.
On the geometry of Deligne-Mumford stacks
, 2006
"... Abstract. General structure results about Deligne–Mumford stacks are sum-marized, applicable to stacks of finite type over a field. When the base field has characteristic 0, a class of “(quasi-)projective ” Deligne–Mumford stacks is iden-tified, defined to be those that embed as a (locally) closed s ..."
Abstract
-
Cited by 6 (0 self)
- Add to MetaCart
(Show Context)
Abstract. General structure results about Deligne–Mumford stacks are sum-marized, applicable to stacks of finite type over a field. When the base field has characteristic 0, a class of “(quasi-)projective ” Deligne–Mumford stacks is iden-tified, defined to be those that embed as a (locally) closed substack of a smooth proper Deligne–Mumford stack having projective coarse moduli space. These conditions are shown to be equivalent to some well-studied hypotheses. 1.
