4,980 research outputs found
Moment-angle manifolds and complexes. Lecture notes KAIST'2010
These are notes of the lectures given during the Toric Topology Workshop at
the Korea Advanced Institute of Science and Technology in February 2010. We
describe several approaches to moment-angle manifolds and complexes, including
the intersections of quadrics, complements of subspace arrangements and level
sets of moment maps. We overview the known results on the topology of
moment-angle complexes, including the description of their cohomology rings, as
well as the homotopy and diffeomorphism types in some particular cases. We also
discuss complex-analytic structures on moment-angle manifolds and methods for
calculating invariants of these structures.Comment: 26 pages, minor change
Complex surfaces with CAT(0) metrics
We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and
construct such metrics on CP^2 with various orbifold structures. In particular,
in relation to questions of Gromov and Davis-Moussong we construct such metrics
on a compact quotient of the two-dimensional unite complex ball. In the course
of the proof of these results we give criteria for Sasakian 3-manifolds to be
globally CAT(1). We show further that for certain Kummer coverings of CP^2 of
sufficiently high degree their desingularizations are of type K(pi,1).Comment: Revised version accepted in GAF
Geometric structures on moment-angle manifolds
The moment-angle complex Z_K is cell complex with a torus action constructed
from a finite simplicial complex K. When this construction is applied to a
triangulated sphere K or, in particular, to the boundary of a simplicial
polytope, the result is a manifold. Moment-angle manifolds and complexes are
central objects in toric topology, and currently are gaining much interest in
homotopy theory, complex and symplectic geometry.
The geometric aspects of the theory of moment-angle complexes are the main
theme of this survey. We review constructions of non-Kahler complex-analytic
structures on moment-angle manifolds corresponding to polytopes and complete
simplicial fans, and describe invariants of these structures, such as the Hodge
numbers and Dolbeault cohomology rings. Symplectic and Lagrangian aspects of
the theory are also of considerable interest. Moment-angle manifolds appear as
level sets for quadratic Hamiltonians of torus actions, and can be used to
construct new families of Hamiltonian-minimal Lagrangian submanifolds in a
complex space, complex projective space or toric varieties.Comment: 60 page
Real line arrangements with Hirzebruch property
A line arrangement of lines in satisfies Hirzebruch
property if each line intersect others in points. Hirzebruch asked if all
such arrangements are related to finite complex reflection groups. We give a
positive answer to this question in the case when the line arrangement in
is real, confirming that there exist exactly four such
arrangements.Comment: Minor changes and the misattributed names of complex reflection
arrangements are correcte
Cohomology of face rings, and torus actions
In this survey article we present several new developments of `toric
topology' concerning the cohomology of face rings (also known as
Stanley-Reisner algebras). We prove that the integral cohomology algebra of the
moment-angle complex Z_K (equivalently, of the complement U(K) of the
coordinate subspace arrangement) determined by a simplicial complex K is
isomorphic to the Tor-algebra of the face ring of K. Then we analyse Massey
products and formality of this algebra by using a generalisation of Hochster's
theorem. We also review several related combinatorial results and problems.Comment: 28 pages, more minor changes, to be published in the LMS Lecture
Note
Manin triples of real simple Lie algebras. Part 2
We classify Manin triples of the type up to weak
and gauge equivalence.Comment: Latex, 12page
On the cohomology of quotients of moment-angle complexes
We describe the cohomology of the quotient Z_K/H of a moment-angle complex
Z_K by a freely acting subtorus H in T^m by establishing a ring isomorphism of
H*(Z_K/H,R) with an appropriate Tor-algebra of the face ring R[K], with
coefficients in an arbitrary commutative ring R with unit. This result was
stated in [BP02, 7.37] for a field R, but the argument was not sufficiently
detailed in the case of nontrivial H and finite characteristic. We prove the
collapse of the corresponding Eilenberg-Moore spectral sequence using the
extended functoriality of Tor with respect to `strongly homotopy
multiplicative' maps in the category DASH, following Munkholm [Mu74]. Our
collapse result does not follow from the general results of Gugenheim-May and
Munkholm.Comment: 3 page
Foliations with unbounded deviation on the two-dimensional torus
There exists a smooth foliation with 3 singular points on the two-dimensional
torus such that any lifting of a leaf of this foliation on the universal
covering of the torus is a dense subset of the covering.Comment: 6 pages, 3 figure
On variants of -measures and compensated compactness
We introduce new variant of -measures defined on spectra of general
algebra of test symbols and derive the localization properties of such
-measures. Applications for the compensated compactness theory are given. In
particular, we present new compensated compactness results for quadratic
functionals in the case of general pseudo-differential constraints. The case of
inhomogeneous second order differential constraints is also studied
Impact of interparticle dipole-dipole interactions on optical nonlinearity of nanocomposites
In this paper, effect of dipole-dipole interactions on nonlinear optical
properties of the system of randomly located semiconductor nanoparticles
embedded in bulk dielectric matrix is investigated. This effect results from
the nonzero variance of the net dipole field in an ensemble. The analytical
expressions describing the contribution of the dipole-dipole coupling to
nonlinear dielectric susceptibility are obtained. The derived relationships are
applicable over the full range of nanoparticle volume fractions. The factors
entering into the contribution and depending on configuration of the dipoles
are calculated for several cases. It is shown that for the different
arrangements of dipole alignments the relative change of this contribution does
not exceed 1/3.Comment: 5 pages, 1 figur
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