Volume 2005, Issue 587

A conjecture of Thurston asserts that a geometrically finite hyperbolic 3-manifold is completely determined by the bending data of its convex core. We prove this conjecture for small deformations of fuchsian manifolds.

A resolution Z  →  X of a Poisson variety X   is called Poisson if every Poisson structure on X   lifts to a Poisson structure on Z . For symplectic varieties, we prove that Poisson resolutions coincide with symplectic resolutions. It is shown that for a smooth Poisson surface S , the natural resolution S   [ n   ]  →  S   ( n   ) is a Poisson resolution. Furthermore, if ∣− K S  ∣ is base-point-free, we prove that this is the unique projective Poisson resolution for S  ( n   ) .

We prove that Friedlander’s generalized isomorphism conjecture on the cohomology of algebraic groups, and hence the Isomorphism Conjecture for the cohomology of the complex algebraic Lie group G(ℂ) made discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G( F  ), where F   is the algebraic closure of a finite field.

Let be the open unit ball in ℂ d , d  ≧ 1, and H d 2 be the space of analytic functions on determined by the reproducing kernel (1 − 〈  z ,  λ  〉) −1 . This reproducing kernel Hilbert space serves a universal role in the model theory for d  -contractions, i.e. tuples T  = ( T 1 ,…, T d  ) of commuting operators on a Hilbert space such that || T 1 x 1  + ⋯ +  T d  x d  || 2  ≦ || x 1 || 2 + ⋯ + || x d  || 2 for all x 1 , … , x d  ∈ . If  is a separable Hilbert space then we write H d 2 ( ) ≅  H d 2  ⊗  for the space of -valued H d 2 functions and we use M z  = (,…, ) to denote the tuple of multiplication by the coordinate functions. We consider M z -invariant subspaces ℳ ⊆ H d 2 ( ). The fiber dimension of ℳ  is defined to be . We show that if ℳ  has finite positive fiber dimension m , then the essential Taylor spectrum of M z  |ℳ ,  σ e ( M z  |ℳ ), equals ∂ plus possibly a subset of the zero set of a nonzero bounded analytic function on and ind( M z  −  λ ) |ℳ = (−1) d m   for every λ  ∈  \ σ e ( M z  |ℳ ). As a corollary we prove that if T  = ( T 1 ,…, T d  ) is a pure d -contraction of finite rank, then σ e ( T  ) ∩  is contained in the zero set of a nonzero bounded analytic function and (−1) d  ind( T  −  λ ) =  κ  ( T  ) for all λ  ∈  \ σ e ( T  ). Here κ( T  ) denotes Arveson’s curvature invariant. We will also show that for d  > 1 there are such d -contractions with σ e ( T  ) ∩  ≠ ∅. These results answer a question of Arveson, [ William Arveson , The Dirac operator of a commuting d -tuple, J. Funct. Anal. 189(1) (2002), 53–79]. We also prove related results for the Hardy and Bergman spaces of the unit ball and unit polydisc of ℂ d .

Let k be a field of characteristic zero, V   a smooth, positive-dimensional, quasiprojective variety over k , and Q   a nonempty divisor on V . Let K   be the function field of V , and A  ⊂  K   the semilocal ring of Q . We prove the Diophantine undecidability of: (1)  A , in all cases; (2)  K , when k   is real and V    has a real point; (3)  K , when k   is a subfield of a p  -adic field, for some odd prime p . To achieve this, we use Denef’s method: from an elliptic curve E   over ℚ, without complex multiplication, one constructs a quadratic twist ℰ of E over ℚ( t  ), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose ƒ  in K , vanishing at Q , such that the group ℰ( K  ) deduced from the field extension ℚ( t  ) →̃ ℚ( ƒ   ) ↪  K   is equal to ℰ(ℚ( t  )). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p  -adic case).

We study how to minimize the number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of corank one finitely determined holomorphic germ ƒ   : (ℂ n , 0) → (ℂ n , 0). According to a result of Gaffney, these are the 0-stable invariants and all polar multiplicities which appear in the stable types of a stable deformation of the germ. First we describe all stable types, then we show how the invariants in the source and the target are related and reduce the number using these relations. We also investigate the relationship between the local Euler obstruction and the polar multiplicities of the stable types. We show an algebraic formula for the local Euler obstruction in terms of the polar multiplicities and show that the Euler obstruction is an invariant for the Whitney equisingularity.

Let Γ be a discrete finitely generated group. Let  →  T be a Γ-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary Z . We assume that Γ →  →   |  Γ is a Galois covering of a compact manifold with boundary. Let ( D   + (θ)) θ ∈  T   be a Γ-equivariant family of Dirac-type operators. Under the assumption that the boundary family is L 2 -invertible, we define an index class in K 0 ( C   0 ( T  ) ⋊ r   Γ ). If, in addition, Γ is of polynomial growth, we define higher indices by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indices: the structure of the formula is as in the seminal work of Atiyah, Patodi and Singer, with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family. Under similar assumptions we extend our theorem to any G  -proper manifold, with G an étale groupoid. We employ this generalization in order to establish a higher Atiyah-Patodi-Singer index formula on certain foliations with boundary. Fundamental to our work is a suitable generalization of Melrose b  -pseudodifferential calculus as well as the superconnection proof of the index theorem on G  -proper manifolds recently given by Gorokhovsky and Lott in [ A. Gorokhovsky and J. Lott , Local index theory over étale groupoids, J. reine angew. Math. 560 (2003), 151–198].