On contact loci of hyperplane arrangements

We give an explicit expression for the contact loci of hyperplane arrangements and show that their cohomology rings are combinatorial invariants. We also give an expression for the restricted contact loci in terms of Milnor fibers of associated hyperplane arrangements. We prove the degeneracy of a spectral sequence related to the restricted contact loci of a hyperplane arrangement and which conjecturally computes algebraically the Floer cohomology of iterates of the Milnor monodromy. We give formulas for the Betti numbers of contact loci and restricted contact loci in generic cases

Monodromy conjecture for log generic polynomials

A log generic hypersurface in P n with respect to a birational modification of P n is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product f = f1 . . . fp of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple (f1, . . . , fp, g) and for the product fg, if g is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for the tuple (f1, . . . , fp, g) and for the product fg, if g is log very-generic. Even the case f = 1 is intricate, the proof depending on nontrivial properties of Bernstein-Sato ideals, and it singles out the class of log (very-) generic hypersurfaces as an interesting class of singularities on its own

Monodromy conjecture for semi-quasihomogeneous hypersurfaces

We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more generally for semi-quasihomogeneous hypersurfaces. We also give a strange generalization allowing a twist by certain differential forms

Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces

We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.Comment: 15 pages, Introduction is modifie

Around the tangent cone theorem

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.Comment: 39 pages; to appear in the proceedings of the Configurations Spaces Conference (Cortona 2014), Springer INdAM serie

Pentaalkoxytriphenylene monoesters and their dyads; structural factors influencing columnar and nematic mesophase behaviour

Ester linkages are frequently employed in the design of discotic liquid crystals, and indeed the first examples of discotic liquid crystals reported by Chandrasekhar, Sadashiva and Suresh were hexaesters of benzene. Within the wide range of liquid crystalline triphenylene systems so far reported, the symmetrical hexa(aryl) esters are particularly noteworthy because they form the relatively rare and technologically important discotic nematic mesophase. An alternative strategy for inducing nematogenic behaviour is to build linked and twinned structures, and here we report examples of materials that combine the two design features. Pentahexyloxy triphenylenes bearing a single aryl ester retain the columnar mesophase. Linked dyad structures promote nematic phase formation and stability is influenced by the link type and bonding arrangement within isomeric series (phthalates) and related constructs

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions

COHOMOLOGY OF CONTACT LOCI

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent

On the length of perverse sheaves on hyperplane arrangements

Abstract. In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with same monodromy around the planes