200 research outputs found
Translated tori in the characteristic varieties of complex hyperplane arrangements
We give examples of complex hyperplane arrangements for which the top
characteristic variety contains positive-dimensional irreducible components
that do not pass through the origin of the character torus. These examples
answer several questions of Libgober and Yuzvinsky. As an application, we
exhibit a pair of arrangements for which the resonance varieties of the
Orlik-Solomon algebra are (abstractly) isomorphic, yet whose characteristic
varieties are not isomorphic. The difference comes from translated components,
which are not detected by the tangent cone at the origin.Comment: Revised and expanded; 16 pages, 10 figures; to appear in Topology and
its Application
Higher homotopy groups of complements of complex hyperplane arrangements
We generalize results of Hattori on the topology of complements of hyperplane
arrangements, from the class of generic arrangements, to the much broader class
of hypersolvable arrangements. We show that the higher homotopy groups of the
complement vanish in a certain combinatorially determined range, and we give an
explicit Z\pi_1-module presentation of \pi_p, the first non-vanishing higher
homotopy group. We also give a combinatorial formula for the \pi_1-coinvariants
of \pi_p.
For affine line arrangements whose cones are hypersolvable, we provide a
minimal resolution of \pi_2, and study some of the properties of this module.
For graphic arrangements associated to graphs with no 3-cycles, we obtain
information on \pi_2, directly from the graph. The \pi_1-coinvariants of \pi_2
may distinguish the homotopy 2-types of arrangement complements with the same
\pi_1, and the same Betti numbers in low degrees.Comment: 24 pages, 3 figure
Intersections of translated algebraic subtori
We exploit the classical correspondence between finitely generated abelian
groups and abelian complex algebraic reductive groups to study the intersection
theory of translated subgroups in an abelian complex algebraic reductive group,
with special emphasis on intersections of (torsion) translated subtori in an
algebraic torus.Comment: 20 pages; accepted for publication by the Journal of Pure and Applied
Algebr
Boundary manifolds of projective hypersurfaces
We study the topology of the boundary manifold of a regular neighborhood of a
complex projective hypersurface. We show that, under certain Hodge theoretic
conditions, the cohomology ring of the complement of the hypersurface
functorially determines that of the boundary. When the hypersurface defines a
hyperplane arrangement, the cohomology of the boundary is completely determined
by the combinatorics of the underlying arrangement and the ambient dimension.
We also study the LS category and topological complexity of the boundary
manifold, as well as the resonance varieties of its cohomology ring.Comment: 31 pages; accepted for publication in Advances in Mathematic
Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures
We study the basic properties of Higgs sheaves over compact K\"ahler
manifolds and we establish some results concerning the notion of semistability;
in particular, we show that any extension of semistable Higgs sheaves with
equal slopes is semistable. Then, we use the flattening theorem to construct a
regularization of any torsion-free Higgs sheaf and we show that it is in fact a
Higgs bundle. Using this, we prove that any Hermitian metric on a
regularization of a torsion-free Higgs sheaf induces an admissible structure on
the Higgs sheaf. Finally, using admissible structures we proved some properties
of semistable Higgs sheaves.Comment: 18 pages; some typos correcte
Bergman kernel and complex singularity exponent
We give a precise estimate of the Bergman kernel for the model domain defined
by where
is a holomorphic map from to ,
in terms of the complex singularity exponent of .Comment: to appear in Science in China, a special issue dedicated to Professor
Zhong Tongde's 80th birthda
Characteristic varieties and Betti numbers of free abelian covers
The regular \Z^r-covers of a finite cell complex X are parameterized by the
Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording
when the Betti numbers b_1,..., b_i of the corresponding covers are finite
carves out certain subsets \Omega^i_r(X) of the Grassmannian.
We present here a method, essentially going back to Dwyer and Fried, for
computing these sets in terms of the jump loci for homology with coefficients
in rank 1 local systems on X. Using the exponential tangent cones to these jump
loci, we show that each \Omega-invariant is contained in the complement of a
union of Schubert varieties associated to an arrangement of linear subspaces in
H^1(X,\Q).
The theory can be made very explicit in the case when the characteristic
varieties of X are unions of translated tori. But even in this setting, the
\Omega-invariants are not necessarily open, not even when X is a smooth complex
projective variety. As an application, we discuss the geometric finiteness
properties of some classes of groups.Comment: 40 pages, 2 figures; accepted for publication in International
Mathematics Research Notice
Analytic curves in algebraic varieties over number fields
We establish algebraicity criteria for formal germs of curves in algebraic
varieties over number fields and apply them to derive a rationality criterion
for formal germs of functions, which extends the classical rationality theorems
of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to
arbitrary algebraic curves over a number field.
The formulation and the proof of these criteria involve some basic notions in
Arakelov geometry, combined with complex and rigid analytic geometry (notably,
potential theory over complex and -adic curves). We also discuss geometric
analogues, pertaining to the algebraic geometry of projective surfaces, of
these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor
of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200
The Quantum McKay Correspondence for polyhedral singularities
Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's
G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral
singularity C^3/G. The classical McKay correspondence describes the classical
geometry of Y in terms of the representation theory of G. In this paper we
describe the quantum geometry of Y in terms of R, an ADE root system associated
to G. Namely, we give an explicit formula for the Gromov-Witten partition
function of Y as a product over the positive roots of R. In terms of counts of
BPS states (Gopakumar-Vafa invariants), our result can be stated as a
correspondence: each positive root of R corresponds to one half of a genus zero
BPS state. As an application, we use the crepant resolution conjecture to
provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].Comment: Introduction rewritten. Issue regarding non-uniqueness of conifold
resolution clarified. Version to appear in Inventione
Anti-commutative Groebner-Shirshov basis of a free Lie algebra
One of the natural ways to prove that the Hall words (Philip Hall, 1933)
consist of a basis of a free Lie algebra is a direct construction: to start
with a linear space spanned by Hall words, to define the Lie product of Hall
words, and then to check that the product yields the Lie identities (Marshall
Hall, 1950). Here we suggest another way using the Composition-Diamond lemma
for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962)
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