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We assign a positive number V , a new conformal invariant, to a Riemann surface R of finite genus in terms of the extremal lengths of certain weak homology classes on R , and determine the range of V . In particular we find algebraic relations among the extremal lengths of homology classes on compact Riemann surfaces.
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Let X n ⊂ or X n ⊂ ℙ n + a be a patch of a C ∞ submanifold of an affine or projective space such that through each point x ∈ X there exists a line osculating to order n + 1 at x . We show that X is uniruled by lines, generalizing a classical theorem for surfaces. We describe two circumstances that imply linear spaces of dimension k osculating to order two must be contained in X , shedding light on some of Ein's results on dual varieties. We present some partial results on the general problem of finding the integer m 0 = m 0 ( k, n, a ) such that there exist examples of patches X n ⊂ ℙ n + a , having a linear space L of dimension k osculating to order m 0 — 1 at each point such that L is not locally contained in X , but if there are k -planes osculating to order m 0 at each point, they are locally contained in X . The same conclusions hold in the analytic category and complex analytic category if there is a linear space osculating to order m at one general point x ∈ X .
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Let G be a reductive p -adic group, we are interested in finitely generated projective smooth G -modules. Let P be such a module, consider it as a З-module, where З is the Bernstein center of the category of smooth G -modules. Then we can form P ⊗ З.χ ℂ for every complex-valued character of З: it is a finite length smooth representation of G . We describe its image in the Grothendieck group of finite length smooth G -modules. To do this, we define under suitable assumptions a З-valued character on the З-admissible (but not admissible!) representation P . The case of ind G K (1) where K is a special compact open subgroup of G is an interesting example. Some of his properties are discussed and extended to other representations of K using Bushnell and Kutzko's theory of types, when G = GL( n ).
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We obtain estimates on the character of the cohomology of an S 1 -equivariant holomorphic vector bundle over a Kähler manifold M in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of M . In particular, we prove and extend inequalities conjectured by Wu and Zhang. The proof is based on constructing a flat family of complex spaces M t ( t ∈ ℂ) such that M t is isomorphic to M for t ≠ 0, while M 0 is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.
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We compute all finite automorphism groups of three-dimensional complex tori which are maximal in the isogeny class. The maximal order of such an automorphism group is 1296.
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We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type B n and type D n . In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A . To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B .
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Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable.
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We investigate algebraic and analytic subvarieties of ℂ n with automorphisms which cannot be extended to the ambient space.
