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January 20, 2010
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We prove that the curve Y 0 (𝔭) has no 𝔽 2 ( T )-rational points where 𝔭 ⊲ 𝔽 2 [ T ] is a prime ideal of degree at least 3 and Y 0 (𝔭) is the affine Drinfeld modular curve parameterizing Drinfeld modules of rank two over 𝔽 2 [ T ] of generic characteristic with Hecke-type level 𝔭-structure. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over 𝔽 2 ( T ) implying the uniform boundedness conjecture in this particular case. We reach our results with a variant of the formal immersion method. Moreover we show that the group Aut( X 0 (𝔭)) has order two. As a further application of our methods we also determine the prime-to- p cuspidal torsion packet of X 0 (𝔭) where 𝔭 ⊲ 𝔽 q [ T ] is a prime ideal of degree at least 3 and q is a power of the prime p .
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January 20, 2010
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For α > 1 we consider the initial value problem for the dispersive equation i∂ t u + (–Δ) α /2 u = 0. We prove an endpoint L p inequality for the maximal function with initial values in L p -Sobolev spaces, for p ∈ (2 + 4/(d + 1), ∞). This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp L p space-time estimates (local in time) for the same range of p .
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January 20, 2010
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We study the Kähler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kähler-Einstein metric.
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January 29, 2010
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A vector field V on an n -dimensional round sphere S n ( r ) defines a submanifold V ( S n ) of the tangent bundle TS n . The Gluck and Ziller question is to find the infimum of the n -dimensional volume of V ( S n ) among unit vector fields. This volume is computed with respect to the natural metric on the tangent bundle as defined by Sasaki. Surprisingly, the problem is only solved for dimension three [Gluck and Ziller, Math. Helv. 61: 177–192, 1986]. In this article we tackle the question for the 2-sphere. Since there is no globally defined vector field on S 2 , the infimum is taken on singular unit vector fields without boundary. These are vector fields defined on a dense open set and such that the closure of their image is a surface without boundary. In particular if the vector field is area-minimizing it defines a minimal surface of T 1 S 2 ( r ). We prove that if this minimal surface is homeomorphic to ℝ P 2 then it must be the Pontryagin cycle . It is the closure of unit vector fields with one singularity obtained by parallel translating a given vector along any great circle passing through a given point. We show that Pontryagin fields of the unit 2-sphere are area-minimizing.
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January 20, 2010
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We construct Koppelman formulas on Grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual. As an application we obtain new explicit proofs of some vanishing theorems of the Bott-Borel-Weil type by solving the corresponding -equation. We also relate the projection part of our formulas to the Bergman kernels associated to the line bundles.
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January 29, 2010
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We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X // G for the G -action, we introduce a stratification which is defined with respect to orbit types of closed orbits. Our main result is a description of the quotient X // G in terms of quotients by normalizer subgroups associated to the stratification.
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January 20, 2010
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We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph we construct a directed graph E such that is isomorphic to a full corner of C *( E ). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.
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January 20, 2010
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We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.
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February 26, 2010
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Let k be a global field, p an odd prime number different from char( k ) and S , T disjoint, finite sets of primes of k . Let be the Galois group of the maximal p -extension of k which is unramified outside S and completely split at T . We prove the existence of a finite set of primes S 0 , which can be chosen disjoint from any given set ℳ of Dirichlet density zero, such that the cohomology of coincides with the étale cohomology of the associated marked arithmetic curve. In particular, . Furthermore, we can choose S 0 in such a way that realizes the maximal p -extension k 𝔭 ( p ) of the local field k 𝔭 for all 𝔭 ∈ S ∪ S 0 , the cup-product is surjective and the decomposition groups of the primes in S establish a free product inside . This generalizes previous work of the author where similar results were shown in the case T = ∅ under the restrictive assumption p ∤ #Cl( k ) and ζ p ∉ k .