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29
Differentiable Stacks and Gerbes
, 2008
"... We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹ ..."
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Cited by 45 (4 self)
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We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S¹gerbes over manifolds. We develop a ChernWeil theory of characteristic classes in this general setting by presenting a construction of Chern classes and DixmierDouady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S¹bundles and S¹gerbes extending the wellknown result of Weil and Kostant. In particular, we give an explicit construction of S¹central extensions with prescribed curvaturelike 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 Zariskidense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X → Y with Y either a DMcurve or a Shimura modular stack. 1. ..."
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Cited by 12 (3 self)
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Abstract. Suppose X is a smooth quasiprojective variety over C and ρ: π1(X, x) → SL(2, C) is a Zariskidense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X → Y with Y either a DMcurve 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, DettweilerReiter, Harao ..."
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Cited by 6 (0 self)
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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, DettweilerReiter, HaraokaYokoyama. 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 CrawleyBoevey. 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 pRANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES
"... ABSTRACT. We prove results about the intersection of the prank 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 prank f ..."
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Cited by 6 (4 self)
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ABSTRACT. We prove results about the intersection of the prank 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 prank 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 prank. The first application is that a generic hyperelliptic curve of genus g ≥ 3 and prank 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 prank over finite fields. 1.
On the geometry of DeligneMumford stacks
, 2006
"... Abstract. General structure results about Deligne–Mumford stacks are summarized, 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 identified, defined to be those that embed as a (locally) closed s ..."
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Abstract. General structure results about Deligne–Mumford stacks are summarized, 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 identified, 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 wellstudied hypotheses. 1.