Volume 1999, Issue 508

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.

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 .

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 ).

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.

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.

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 .

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.

We investigate algebraic and analytic subvarieties of ℂ n with automorphisms which cannot be extended to the ambient space.