Homotopy groups of complements to ample divisors

We study the homotopy groups of complements to reducible divisors on non-singular projective varieties with ample components and isolated non normal crossings. We prove a vanishing theorem generalizing conditions for commutativity of the fundamental groups. The calculation of supports of non vanishing homotopy groups as modules over the fundamental group in terms of the geometry of the locus of non-normal crossings is discussed. We review previous work on the local study of isolated non-normal crossings and relate the motivic zeta function to the local polytopes of quasiadjunction. As an application, we obtain information about the support loci of homotopy groups of arrangements of hyperplane

Hodge decomposition of Alexander invariants

Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property of polytopes of quasiadjunction is discussed

On Mordell-Weil group of isotrivial abelian varieties over function fields

We show that the Mordell Weil rank of an isotrivial abelian variety with a cyclic holonomy depends only on the fundamental group of the complement to the discriminant provided the discriminant has singularities in the introduced here CM class. This class of singularities includes all unibranched plane curves singularities. As a corollary we give a family of simple Jacobians over field of rational functions in two variable for which the Mordell Weil rank is arbitrary large.Comment: Minor corrections. Final version. To appear in Mathematische Annale

Automorphisms of crepant resolutions for quotient spaces

A formula for calculating the Lefschetz number of an automorphism acting on a crepant resolution for a quotient of a Kahler manifold derived from an equivariant version of McKay correspondence. The latter is proven in some cases. As an application the Lefschetz numbers of of involutions acting on Calabi-Yau threefolds and their mirrors are compared.Comment: 11 pages, PlainTe

Albanese varieties of abelian covers

We show that Albanese varieties of abelian covers of projective plane are isogenous to product of isogeny components of abelian varieties associated with singularities of the ramification locus. In particular Albanese varieties of abelian covers of projective plane ramified over arrangements of lines and uniformized by unit ball are isogenous to a product of Jacobians of Fermat curves. Periodicity of the sequence of (semi-abelian) Albanese varieties of unramified cyclic covers of complements to a plane singular curve is shown.Comment: Substantial revision. Main changes in theorem 4.1 and sections 4 and 5. To appear in Journal of Singularitie

Eigenvalues for the monodromy of the Milnor fibers of arrangements

We decribe upper bounds for the orders of the eigenvalues of the monodromy of Milnor fibers of arrangements given in terms of combinatorics

On combinatorial invariance of the cohomology of Milnor fiber of arrangements and Catalan equation over function field

We discuss combinatorial invariance of the betti numbers of the Milnor fiber for arrangements of lines with points of multiplicity at most three and describe a link between this problem and enumeration of solutions of the Catalan equation over function field in the case when its coefficients are products of linear forms and the equation defines an elliptic curve.Comment: Corrections in the statements and proofs of theorems 1.1,1.2 and 3.1. Remarks 4.1,4.2 adde

Non vanishing loci of Hodge numbers of local systems

We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of the theorem. This yields the "Hodge decomposition" of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomials. Several explicit calculations for complements to arrangements are made.Comment: Final version. To appear in Manuscripta Mathematic

Elliptic genus of phases of N=2 theories

We discuss an algebro-geometric description of Witten's phases of N=2 theories and propose a definition of their elliptic genus provided some conditions on singularities of the phases are met. For Landau-Ginzburg phase one recovers elliptic genus of LG models proposed in physics literature in early 90s. For certain transitions between phases we derive invariance of elliptic genus from an equivariant form of McKay correspondence for elliptic genus. As special cases one obtains Landau-Giznburg/Calabi-Yau correspondence for elliptic genus of weighted homogeneous potentials as well as certain hybrid/CY correspondences.Comment: 19p. Comments are welcom

Homotopy groups of the complements to singular hypersurfaces,II

The homotopy group $\pi_{n-k} ({\bf C}^{n+1}-V)$ where $V$ is a hypersurface with a singular locus of dimension $k$ and good behavior at infinity is described using generic pencils. This is analogous to the van Kampen procedure for finding a fundamental group of a plane curve. In addition we use a certain representation generalizing the Burau representation of the braid group. A divisibility theorem is proven that shows the dependence of this homotopy group on the local type of singularities and behavior at infinity. Examples are given showing that this group depends on certain global data in addition to local data on singularities.Comment: 34 pages, Plain TEX, Version 3.