Volume 2007, Issue 610

A deformation U , of a graded K -algebra A is said to be of PBW type if gr U is A . It has been shown for Koszul and N -Koszul algebras that the deformation is PBW if and only if the relations of U satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a finite and explicitly determined linear algebra problem. We extend these results to an arbitrary graded K -algebra, using the notion of central extensions of algebras and a homological constant attached to A which we call the complexity of A .

We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of vector bundles together with N sections on a smooth complex projective curve C . We show that when N is large, these intersection numbers can equally be computed on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C . The result has applications to the calculation of the intersection theory of the moduli space of semistable bundles on C .

Quantum Lefschetz theorem by Coates and Givental [ T. Coates, A. B. Givental , Quantum Riemann-Roch, Lefschetz and Serre, math.AG/0110142.] gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle ℒ on X . We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semisimplicity and the Virasoro conjecture for the Gromov-Witten theories of (not necessarily Fano) projective toric manifolds.

We show how methods from K -theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M ∞ -stable, homotopy-invariant, excisive K -theory of algebras over a fixed unital ground ring H, ( A , B ) ↦ kk * ( A , B ), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K -theory, KH . We prove that, if H is commutative and A is central as an H-bimodule, then We show further that some calculations from operator algebra KK -theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk .

We extend the construction of Dedekind sums to the case of an arbitrary totally real number field of class number one. Our method is based on the choice of some convenient analogue of the logarithm of Dedekind's η function in this context. We deduce its modular transformation from a Kronecker limit formula established by Asai. It allows us to introduce a generalization of Rademacher's Φ function. We use this function to define the corresponding Dedekind sums and derive their main properties. These sums are not rational numbers but real-analytic functions.

By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups.

We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL p , when p is an odd prime. In particular, we determine their additive structures completely, and we reduce the problem of determining their multiplicative structures to a problem in invariant theory.

We show that if p is an odd prime, the class ρ in the Chow ring of the classifying space of PGL p is not a polynomial in Chern classes of representations.