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We prove that for any field of characteristic , any separated scheme of finite type over , and any overconvergent -isocrystal over , the rigid cohomology and rigid cohomology with compact supports are finite-dimensional vector spaces over an appropriate -adic field. We also establish Poincaré duality and the Künneth formula with coefficients. The arguments use a pushforward construction in relative dimension , based on a relative version of Crew's [Cr] conjecture on the quasi-unipotence of certain -adic differential equations
Eisenbud, Popescu, and Walter  have constructed certain special sextic hypersurfaces in as Lagrangian degeneracy loci. We prove that the natural double cover of a generic Eisenbud-Popescu-Walter (EPW) sextic is a deformation of the Hilbert square of a -surface and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type ; thus we get an example similar to that (discovered by Beauville and Donagi ) of the Fano variety of lines on a cubic -fold. Conversely, suppose that is a numerical , suppose that is an ample divisor on of square for Beauville's quadratic form, and suppose that the map is the composition of the quotient for an antisymplectic involution on followed by an immersion ; then is an EPW sextic, and is the natural double cover
Fix an algebraic space , and let and be separated Artin stacks of finite presentation over with finite diagonals (over ). We define a stack classifying morphisms between and . Assume that is proper and flat over , and assume fppf locally on that there exists a finite finitely presented flat cover with an algebraic space. Then we show that is an Artin stack with quasi-compact and separated diagonal
A metric space has Markov-type if for any reversible finite-state Markov chain (with chosen according to the stationary distribution) and any map from the state space to , the distance from to satisfies for some . This notion is due to K.Ball , who showed its importance for the Lipschitz extension problem. Until now, however, only Hilbert space (and metric spaces that embed bi-Lipschitzly into it) was known to have Markov-type 2. We show that every Banach space with modulus of smoothness of power-type (in particular, for ) has Markov-type ; this proves a conjecture of Ball (see [2, Section 6]). We also show that trees, hyperbolic groups, and simply connected Riemannian manifolds of pinched negative curvature have Markov-type . Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in [28, Section 2] by showing that for , any Lipschitz mapping from a subset of to has a Lipschitz extension defined on all of
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