Taras Panov

On the cohomology of torus manifolds

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orie ntation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geo metry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.

Toric cohomological rigidity of simple convex polytopes

A simple convex polytope P is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over P. Not every P has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this paper we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes, including products of simplices. The cohomological rigidity of P is related to the bigraded Betti numbers of its Stanley�Reisner ring, another important invariant coming from combinatorial commutative algebra

Geometric structures on moment-angle manifolds

A moment-angle complex ZK is a cell complex with a torus action constructed from a finite simplicial complex K. When this con- struction 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 and complex and symplec- tic geometry. The geometric aspects of the theory of moment-angle com- plexes are the main theme of this survey. Constructions of non-Kahler complex-analytic structures on moment-angle manifolds corresponding to polytopes and complete simplicial fans are reviewed, and invariants of these structures such as the Hodge numbers and Dolbeault cohomology rings are described. 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. Bibliography: 59 titles.

Colimits, Stanley-Reisner algebras, and loop spaces

We study diagrams associated with a finite simplicial complex Kin various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups);Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz’s spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops Ω DJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for Ω DJ(K) for an arbitrary complex K,and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.

Moment-angle complexes from simplicial posets

We extend the construction of moment-angle complexes to simplicial posets by associating a certain Tm-space ZS to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space ZS has many important topological properties of the original moment-angle complex ZK associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of ZS is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of ZS from below by proving the toral rank conjecture for the moment-angle complexes ZS.

Полиэдральные произведения и коммутанты прямоугольных групп Артина и Коксетера@@@Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

Конструкция полиэдрального произведения использована для построения моделей классифицирующих пространств прямоугольных групп Артина и Коксетера, общих граф-произведений групп и их коммутантов. В качестве приложения получен критерий свободности коммутанта графпроизведения групп и явно описан минимальный набор образующих для коммутанта прямоугольной группы Коксетера. Библиография: 21 название.

Hamiltonian-minimal Lagrangian submanifolds in toric varieties

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebrated Arnold conjecture on the number of fixed points of a Hamiltonian symplectomorphism. In the previous works the authors defined and studied a family of H-minimal Lagrangian submanifolds in complex space arising from intersections of Hermitian quadrics. Here we extend this construction to define H-minimal submanifolds in toric varieties.

Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the Stanley―Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg―Moore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.

Геометрические структуры на момент-угол-многообразиях@@@Geometric structures on moment-angle manifolds

Момент-угол-комплекс ZK представляет собой клеточный комплекс с действием тора, сопоставляемый конечному симплициальному комплексу K . Если K является триангуляцией сферы или, в частности, границей симплициального многогранника, то соответствующий момент-угол-комплекс ZK является многообразием. Момент-угол-многообразия и комплексы являются одними из основных объектов изучения в торической топологии и в настоящее время привлекают большое внимание в теории гомотопий, комплексной и симплектической геометрии. Данный обзор посвящен геометрическим аспектам теории момент-уголкомплексов. Мы рассматриваем конструкции некэлеровых комплексных структур на момент-угол-многообразиях, соответствующих многогранникам и полным симплициальным веерам, и описываем инварианты этих структур, такие как числа Ходжа и кольца когомологий Дольбо. Также большой интерес представляют симплектические и лагранжевы аспекты теории момент-угол-многообразий. Эти многообразия возникают как множества уровней квадратичных гамильтонианов для действий тора и могут быть использованы для построения новых семейств гамильтоново-минимальных лагранжевых подмногообразий в комплексном пространстве, проективном пространстве и торических многообразиях. Библиография: 59 названий.

Toric Kempf-Ness Sets

In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology,” we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox “geometric invariant theory” approach to toric varieties. We proceed by studying the cohomology of these “toric” Kempf-Ness sets. In the case of projective nonsingular toric varieties the Kempf-Ness sets can be described as complete intersections of real quadrics in a complex space.