Affine hypersurfaces admitting a pointwise symmetry

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T _p M) for all p ∈ M, which preserves (pointwise) the affine metric h, the difference tensor K (resp. the cubic form) and the affine shape operator S. In this paper, we deal with locally strongly convex affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K, S) under the action of SO(3) on a Euclidean vector space (V, h) and find a representative (canonical form of K and S) of each SO(3)/G-orbit. Then, we classify hypersurfaces admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart of Z ₂). Besides well-known hypersurfaces (for Z ₂ × Z ₂ we get the locally homogeneous hypersurface (x ₁ ?) 1/2x ₃2 (x ₂ ?) 1/2x ₄2) = 1) we obtain e.g. warped products of two-dimensional affine spheres (resp. quadrics) and curves.     

A version of Turán’s problem for positive definite functions of several variables

We present a systematic approach to solving the problem of affine homogeneity of real hypersurfaces in the three-dimensional complex space. This question is an important part of the general problem of holomorphic classification of homogeneous real hypersurfaces in three-dimensional complex spaces. In contrast to the two-dimensional case, the whole problem (just as its affine part) has not yet been fully studied, although there exist a large number of examples of homogeneous manifolds. We study only the class of tubular type surfaces, which is defined by conditions imposed on the 2-jet of their canonical equations and generalizes the class of tube manifolds. We discuss the procedure of describing all matrix Lie algebras corresponding to the homogeneous manifolds under consideration. In the class that we study, we distinguish four cases depending on the third-order Taylor coefficients of the canonical equations; in three of these cases, the Lie algebras and the corresponding affine homogeneous surfaces are completely described. The key point of the proposed approach is the solution of a large system of quadratic equations that corresponds to each of the homogeneous surfaces.     

Geometry of compact complex manifolds with maximal torus action

We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S ¹) ^k . We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ? T _? = (?*) ^k . On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kähler form ω _F . As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.     

Geometry of Holomorphic Distributions of Real Hypersurfaces in a Complex Projective Space

We characterize homogeneous real hypersurfaces M's of type (A ₁), (A ₂) and (B) of a complex projective space in the class of real hypersurfaces by studying the holomorphic distribution T ⁰ M of M.     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?H ⁿ (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

On complexity of three-dimensional hyperbolic manifolds with geodesic boundary

The nonintersecting classes ? _p,q are defined, with p, q ?? ? and p ?? q ?? 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ?? ? _p,q , then the complexity c(M) and the Euler characteristic ??(M) of M are related by the formula c(M) = p???(M). The classes ? _q,q , q ?? 1, and ?_2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ?_3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ?-invariants of manifolds.     

Kähler geometry of the universal Teichmüller space and coadjoint orbits of the Virasoro group

The Kähler geometry of the universal Teichmüller space and related infinite-dimensional Kähler manifolds is studied. The universal Teichmüller space T may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The classical Teichmüller spaces T(G), where G is a Fuchsian group, are contained in T as complex Kähler submanifolds. The homogeneous spaces Diff₊(S ¹)/Möb(S ¹) and Diff₊(S ¹)/S ¹ of the diffeomorphism group Diff₊(S ¹) of the unit circle are closely related to T. They are Kähler Frechet manifolds that can be realized as coadjoint orbits of the Virasoro group (and exhaust all coadjoint orbits of this group that have the Kähler structure).     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

On the geometry of flat pseudo-Riemannian homogeneous spaces

Let M = ? _s ⁿ /Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ? Iso(? _s ⁿ ) its fundamental group and G the Zariski closure of Γ in Iso(? _s ⁿ ). We show that the G-orbits in ? _s ⁿ are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ? _s ⁿ to the G-orbits is nondegenerate, then M has abelian linear holonomy. If additionally G is not abelian, then G contains a certain subgroup of dimension 6. In particular, for non-abelian G, orbits with non-degenerate metric can appear only if dim G ≥ 6. Moreover, we show that ? _s ⁿ is a trivial algebraic principal bundle G → M → ? ^n?k . As a consquence, M is a trivial smooth bundle G/Γ → M → ? ^n?k with compact fiber G/Γ.     

Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] ^G ]?=?[k[X]] ^G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ k[X] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?     

Real Hypersurfaces in Complex Two-Plane Grassmannians

 The complex two-plane Grassmannian G ₂(C ^m+2 in equipped with both a K?hler and a quaternionic K?hler structure. By applying these two structures to the normal bundle of a real hypersurface M in G ₂(C ^m+2 one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G ₂ C ^m+2 , m≥3, for which these two distributions are invariant under the shape operator of M. Received 13 November 1996; in revised form 3 March 1997     

Period integrals of CY and general type complete intersections

We develop a global Poincaré residue formula to study period integrals of families of complex manifolds. For any compact complex manifold X equipped with a linear system V ^? of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on X. Two important ingredients of this construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize the construction to CY and general type complete intersections. When X is an algebraic manifold having a sufficiently large automorphism group G and V ^? is a linear representation of G, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper Lian, Song and Yau (arXiv:1105.2984v1, 2011). The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.     

The Conformal Willmore Functional: A Perturbative Approach

The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds (M,g)) is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds (?³,g _? )—where g _? is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds (M,g) of dimension three.     

Sequences of contractions on L1-spaces

Let T_n, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L¹ for which T_ng → g weakly in L¹ for g?M. If T_n1 → 1 strongly, then T_nf → f strongly for all f in the closed vector sublattice in L¹ generated by M.This result can be applied to the determination of Korovkin sets and shadows in L¹. Given a set G ? L¹, its shadow S(G) is the set of all f?L¹ with the property that T_nf → f strongly for any sequence of contractions T_n, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L¹. For instance, if 1 ?G, then, where M is the linear hull of G and $\text{B}$_M is the sub-σ-algebra of $\text{B}$ generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements.     

Locally symmetric complex affine hypersurfaces

In this paper we study non-degenerate locally symmetric complex affine hypersurfaces Mⁿ of the complex affine space, i.e. hypersurfaces satisfying R=0, where is the affine connection induced on Mⁿ by the complex affine structure on the complex affine space, and R is the curvature tensor of . We classify the non-degenerate locally symmetric hypersurfaces Mⁿ, n > 2, and the minimal non-degenerate locally symmetric hypersurfaces Mⁿ, n > 1.Aspirant N.F.W.O. (Belgium)     

Nonpositive curvature: A geometrical approach to Hilbert-Schmidt operators

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that GM, the Banach-Lie group generated by M, acts isometrically and transitively on M. Moreover, GM admits a polar decomposition relative to M, namely GM?M×K as Hilbert manifolds (here K is the isotropy of p=1 for the action ), and also GM/K?M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM?Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed ea, we obtain ea=exevex with ex∈M and v orthogonal to M at p=1. As a corollary we obtain decompositions for the full group of invertible elements G?M×exp(T₁M^⊥)×K.     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

Gaiotto’s Lagrangian Subvarieties via Derived Symplectic Geometry

Let Bun_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T^?Bun_G. We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.