Let X be a smooth complex variety of dimension at most two, and let F be its function field. We prove that the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.

We define and study a candidate for ‘arithmetic’ cohomology theory with values in crystalline p-adic smooth sheaves. We show that it injects into arithmetic étale cohomology and prove a duality result.

Vogan has conjectured that the cohomologically induced modules A$_ q$(λ) in the weakly fair range exhaust all unitary representations of U(p, q) with certain kinds of real integral infinitesimal character. To prove a statement like this, it is essential to identify these modules among the set of all irreducible Harish-Chandra modules. Barbasch and Vogan have parametrized this latter set in terms of their annihilators and asymptotic supports (or, equivalently, associated varieties). In this paper, we identify the weakly fair A$_ q$(λ) in this parametrization by combining known results about their asymptotic supports together with an explicit computation of their annihilators. In particular, this determines all vanishing and coincidences among the A$_ q$(λ) in the weakly fair range, and gives the Langlands parameters of these modules.

The paper extends the rigidity of the mixing expanding repellers theorem of D. Sullivan announced at the 1986 IMC. We show that, for a regular conformal, satisfying the ‘Open Set Condition’, iterated function system of countably many holomorphic contractions of an open connected subset of a complex plane, the Radon–Nikodym derivative dμ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and μ is the unique probability invariant measure equivalent with m. Next, we introduce the concept of nonlinearity for iterated function systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the following rigidity result: If h, the topological conjugacy between two nonlinear systems F and G, transports the conformal measure mF to the equivalence class of the conformal measure mG, then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, we prove that the hyperbolic system associated to a given parabolic system of countably many holomorphic contractions is nonlinear, which allows us to extend our rigidity result to the case of parabolic systems.

We prove a Reider type theorem for separating any cluster by an adjoint system to a pseudoeffective divisor on a normal surface. As a corollary we get a Reider type theorem for adjoint linear systems (to a nef $\Bbb Q$-divisor) on normal log surfaces. This theorem is new even for smooth surfaces.

The space of elliptic virtual representations of a p-adic group is endowed with a natural inner product EP(,), defined analytically by Kazhdan and homologically by Schneider–Stuhler. Arthur has computed EP in terms of analytic R-groups. For Iwahori spherical representations, we show that EP can also be expressed in terms of a corresponding inner product on space of elliptic virtual representations of Weyl groups. This leads to an explicit description of both elliptic representation theories, in terms of the Kazhdan–Lusztig and Springer correspondences

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form χA is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of χA. Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.

Let $\Bbb V$ be n-dimensional complex vector space. The aim of this paper is to give an elementary proof of the isomorphism ($\cal O$t$\Bbb _V$∧[n] ≃ $\cal O$t$\Bbb _V$*, which quantizes the Fourier–Sato transform of the conic sheaf $\cal O$t$\Bbb _V$ of temperate holomorphic functions

I give the necessary and sufficient conditions for the existence of Unitary local systems with prescribed local monodromies on $\mathbb P$1 − S where S is a finite set. This is used to give an algorithm to decide if a rigid local system on $\mathbb P$1 − S has finite global monodromy, thereby answering a question of N. Katz. The methods of this article (use of Harder–Narasimhan filtrations) are used to strengthen Klyachko's theorem on sums of Hermitian matrices. In the Appendix, I give a reformulation of Mehta–Seshadri theorem in the SU(n) setting.

Let x: Ln → $\mathbb S$2n+1 ⊂ $\mathbb R$2n+2 be a minimal submanifold in $\mathbb S$2n+1. In this note, we show that L is Legendrian if and only if for any A ∈ su(n + 1) the restriction to L of 〈Ax, √(−1)x〉 satisfies Δf = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least ½(n(n + 3)). Moreover if the multiplicity equals to ½(n(n + 3)), then Ln is totally geodesic.

We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F4 root system is included in it.

Let L/K be a finite Galois extension of number fields. We use complexes arising from the étale cohomology of $\Bbb Z$ on open subschemes of Spec $\cal O$L to define a canonical element of the relative algebraic K-group K0($\Bbb Z$[Gal(L/K)], $\Bbb R$. We establish some basic properties of this element, and then use it to reinterpret and refine conjectures of Stark, of Chinburg and of Gruenberg, Ritter and Weiss. Our results precisely explain the connection between these conjectures and the seminal work of Bloch and Kato concerning Tamagawa numbers. This provides significant new insight into these important conjectures and also allows one to use powerful techniques from arithmetic algebraic geometry to obtain new evidence in their favour.

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s [les ] 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.