40 research outputs found
Complements of hypersurfaces, variation maps and minimal models of arrangements
We prove the minimality of the CW-complex structure for complements of
hyperplane arrangements in by using the theory of Lefschetz
pencils and results on the variation maps within a pencil of hyperplanes. This
also provides a method to compute the Betti numbers of complements of
arrangements via global polar invariants
The -genus and a regularization of an -equivariant Euler class
We show that a new multiplicative genus, in the sense of Hirzebruch, can be
obtained by generalizing a calculation due to Atiyah and Witten. We introduce
this as the -genus, compute its value for some examples and
highlight some of its interesting properties. We also indicate a connection
with the study of multiple zeta values, which gives an algebraic interpretation
for our proposed regularization procedure.Comment: 14 pages; version to appear in J. Phys.
Cohomology of bundles on homological Hopf manifold
We discuss the properties of complex manifolds having rational homology of
including those constructed by Hopf, Kodaira and
Brieskorn-van de Ven. We extend certain previously known vanishing properties
of cohomology of bundles on such manifolds.As an application we consider
degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex
variables and Complex Geometry. Xiamen. Chin
Geometric and homological finiteness in free abelian covers
We describe some of the connections between the Bieri-Neumann-Strebel-Renz
invariants, the Dwyer-Fried invariants, and the cohomology support loci of a
space X. Under suitable hypotheses, the geometric and homological finiteness
properties of regular, free abelian covers of X can be expressed in terms of
the resonance varieties, extracted from the cohomology ring of X. In general,
though, translated components in the characteristic varieties affect the
answer. We illustrate this theory in the setting of toric complexes, as well as
smooth, complex projective and quasi-projective varieties, with special
emphasis on configuration spaces of Riemann surfaces and complements of
hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics
and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201
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
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
Exotic smooth structures on 4-manifolds with zero signature
For every integer , we construct infinite families of mutually
nondiffeomorphic irreducible smooth structures on the topological -manifolds
and (2k-1)(\CP#\CPb), the connected sums of
copies of and \CP#\CPb.Comment: 6 page