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 $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ 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 $\mathbb CP^2$ 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 $(g(R),W,g(R)\oplus g(R))$ 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 HH-measures and compensated compactness

We introduce new variant of $H$-measures defined on spectra of general algebra of test symbols and derive the localization properties of such $H$-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