Intersection Cohomology on Nonrational Polytopes⋆

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image R_* ( is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf , called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if is rational). Using we define the intersection cohomology space IH(). It is conjectured that a strictly convex piecewise linear function on acts as a Lefschetz operator on IH(). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.     

The de Rham Bundle on a Compactification of Moduli Space of Abelian Varieties

In this paper, we show that the Chern classes c _k of the de Rham bundle defined on any good toroidal compactification of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of , for g=4, 5 and k>0.     

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials P _x , _w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where (the symmetric group on n letters) and the permutation w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1+q) ^l(w) if and only if w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety X _w to have a small resolution. We conclude with a simple method for completely determining the singular locus of X _w when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (B C _n , F ₄, G ₂).     

The -theory of toric manifolds

Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an -dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the -theory of all toric manifolds and certain singular toric varieties.

     

On the universal Gröbner bases of toric ideals of graphs

The universal Gröbner basis of an ideal is a Gröbner basis with respect to all term orders simultaneously. We characterize in graph theoretical terms the elements of the universal Gröbner basis of the toric ideal of a graph. We also provide a new degree bound. Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.     

On <Emphasis Type="Italic">n</Emphasis> × <Emphasis Type="Italic">m</Emphasis>-valued Łukasiewicz-Moisil algebras

n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM _n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM _n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM _n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM _n×m -algebra is computed.      

Flavones from Callus Tissue of <Emphasis Type="Italic">Iris ensata</Emphasis>

Flavonoids uncharacteristic of intact plants were isolated from callus tissue of Iris ensata and were identified as 5-hydroxy-4′-methoxyflavone, 5-hydroxy-3′-methoxyflavone, and 5-hydroxy-2′-methoxyflavone using PMR and mass spectrometry. It was proposed that the lack of growth of callus tissue after changing cultivation conditions was related to the inhibiting effect of these flavones on cell proliferation. __________ Translated from Khimiya Prirodnykh Soedinenii, No. 5, pp. 440–442, September–October, 2005.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

On a theorem of Tate

We study applications of divisibility properties of recurrence sequences to Tate’s theory of abelian varieties over finite fields.