Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).     

Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics, when we perturb such systems inside the class of all polynomial systems of degreen     

Fréchet-Urysohn spaces in free topological groups

Let and be respectively the free topological group and the free Abelian topological group on a Tychonoff space . For every natural number we denote by () the subset of () consisting of all words of reduced length . It is well known that if a space is not discrete, then neither nor is Fréchet-Urysohn, and hence first countable. On the other hand, it is seen that both and are Fréchet-Urysohn for a paracompact Fréchet-Urysohn space . In this paper, we prove first that for a metrizable space , () is Fréchet-Urysohn if and only if the set of all non-isolated points of is compact and is Fréchet-Urysohn if and only if is compact or discrete. As applications, we characterize the metrizable space such that is Fréchet-Urysohn for each and is Fréchet-Urysohn for each except for . In addition, however, there is a first countable, and hence Fréchet-Urysohn subspace of () which is not contained in any (). We shall show that if such a space is first countable, then it has a special form in (). On the other hand, we give an example showing that if the space is Fréchet-Urysohn, then it need not have the form.

     

Set-theoretic complete intersections on binomials

Let be an affine toric variety of codimension over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, is a set-theoretic complete intersection on binomials if and only if is a complete intersection. Moreover, if are binomials such that , then . While in the positive characteristic case, is a set-theoretic complete intersection on binomials if and only if is completely -glued.

These results improve and complete all known results on these topics.

     

Standard monomial bases and geometric consequences for certain rings of invariants

Consider the diagonal action ofSL _n (K) on the affine spaceX = V^⊕m ⊕ (V*)^⊕q whereV = K ⁿ ,K an algebraically closed field of arbitrary characteristic andm,q > n. We construct a ‘standard monomial’’ basis for the ring of invariantsK[X] ^SL _n ^(K). As a consequence, we deduce thatK[X] ^SL _n ^(K) is Cohen-Macaulay. We also present the first and second fundamental theorems forSL _n (K)- actions.     

Legendrian Subvarieties of Projective Space

I prove that every smooth Legendrian variety generated by quadrics is a homogeneous variety and, further, I give a list of all such Legendrian varieties. Another result is that no complete intersection is a Legendrian variety.Mathematics Subject Classification (2000): Primary: 14M99, secondary: 17B10, 22E46 *The author was partially supported by Marie Curie Programme.     

Complete toric varieties with reductive automorphism group

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.     

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

We prove that the moduli spaces of polarized Abelian threefolds with polarizations of types D=(1,1,2),(1,2,2),(1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space which parametrizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^-1. Dedicated to the memory of Fabio BardelliMathematics Subject Classification (2000) Primary: 14K10; Secondary: 14H10, 14H30, 14D07     

Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem

An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex naturally determines a Bratteli diagram and a stable AF-algebra . Consider the following problem:

INPUT: a pair of abstract simplicial complexes and ;

QUESTION: is isomorphic to ?

We show that this problem is Gödel incomplete, i.e., it is recursively enumerable but not decidable. This result is in sharp contrast with the recent decidability result by Bratteli, Jorgensen, Kim and Roush, for the isomorphism problem of stable AF-algebras arising from the iteration of the same positive integer matrix. For the proof we use a combinatorial variant of the De Concini-Procesi theorem for toric varieties, together with the Baker-Beynon duality theory for lattice-ordered abelian groups, Markov's undecidability result, and Elliott's classification theory for AF-algebras.

     

Abelian Theorems,Farey Series and the Riemann Hypothesis

The aim of this paper is to provide unconditional estimates for the error terms associated with Farey series that are comparable to error terms in the prime number theorem, and also to provide error terms for Farey series based on implications of the RH().