Formality theorem for gerbes

The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an _L∞$L_{\infty }$ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe.     

Formality theorem for g-manifolds

With any $\mathfrak{g}$-manifold M are associated two dglas $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))$ and $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$, whose cohomologies $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ and $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_{\infty }$ quasi-isomorphism $\mathrm{\Phi }:\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))\to \mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$ whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold M is an isomorphism of Gerstenhaber algebras from $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ to $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$.     

Formality in an equivariant setting

We define and discuss -formality for certain spaces endowed with an action by a compact Lie group. This concept is essentially formality of the Borel construction of the space in a category of commutative differential graded algebras over . These results may be applied in computing the equivariant cohomology of their loop spaces.

     

Ric-semisymmetric submanifolds

This article surveys results obtained on various classes of submanifolds with the interior geometry of Riccisemisymmetric spaces.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 23, pp. 29–66, 1991.     

Semisymmetric submanifolds

This article surveys new work on semisymmetric and k-parallel submanifolds M^m in Eⁿ and Mⁿ(c).Translated from Itogi Nauki i Tekhniki Seriya Problemy Geometrii, Vol. 23, pp. 3–28, 1991.     

Formality of equivariant intersection cohomology of algebraic varieties

We present a proof that the equivariant intersection cohomology of any complete algebraic variety acted by a connected algebraic group is a free module over .

     

Discriminantal Arrangements, Fiber Polytopes and Formality

Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3–formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.     

Volume of submanifolds

We propose a geometric definition of the volume of submanifolds using tubular neighborhoods.     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Asymptotic curves of submanifolds

Translated from Matematicheskie Zametki, Vol. 52, No. 5, pp. 3–12, November, 1992.     

Isoparametric families of submanifolds

An extension of Cartan's theory of isometric families of hypersurfaces is presented. The leaves of the new families are submanifolds of higher codimension. It is proved that the leaves have mean curvature vector with constant norm. If the distribution defined by the normal spaces of the leaves is integrable, the theory can be developed along the same lines as in the usual case (i.e. families of hypersurfaces). These families are thus obtained constructing “parallel” submanifolds. A generalization of Cartan's fundamental formula is obtained and a simple geometrical application of it is given.     

Reflections in submanifolds

This is a study of geodesic symmetries of a Riemannian manifold with respect to a submanifold. We analyze the significance of the property to be volume-preserving, isometric or preserving the mean curvature of the tubular neighborhoods.Work supported in part by the National Science Foundation and the Max-Planck-Institut für Mathematik, Bonn.     

Generalized chen submanifolds

As a generalization of Chen submanifolds,k-th Chen submanifolds are defined. A characterization for them is proved. Spherical 2nd Chen submanifolds are discussed. For a compact submanifoldM with parallel second fundamental form it is proved thatM is ak-th Chen submanifold if and only ifM is ofk-type.Dedicated to Prof. A. Barlotti for his 70- th birthdayThe first author was partially supported by the Provincial Scientific Research Fund from Guangdong Province, China.     

Coordinate finite-type submanifolds

A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of . We prove that the compact coordinate finite-type submanifolds are minimal submanifolds of quadratic hypersurfaces of Euclidean spaces. Moreover, we classify the compact coordinate finite-type submanifolds of codimension 2.     

Transcendental submanifolds of Rn

In this paper we show how the restriction of the complex algebraic cycles to real part of a complex algebraic set is related to the real algebraic cycles of the real part. As a corollary we give examples of smooth submanifolds of a Euclidean space which can not be isotoped to real parts of complex nonsingular subvarieties in the corresponding projective space. Dedicated to memory of Mario Raimondo