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 Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δ?) be a Lie bialgebra. Let (U?(g),Δ?) a quantization of (g,δ?) through Etingof-Kazhdan functor. We prove the existence of a L_∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T₊U=T₊(U?(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.     

Formality of Donaldson submanifolds

We introduce the concept of s–formal minimal model as an extension of formality. We prove that any orientable compact manifold M, of dimension 2n or (2n−1), is formal if and only if M is (n−1)–formal. The formality and the hard Lefschetz property are studied for the Donaldson submanifolds of symplectic manifolds constructed in [13]. This study permits us to show an example of a Donaldson symplectic submanifold of dimension eight which is formal simply connected and does not satisfy the hard Lefschetz theorem. An erratum to this article is available at .     

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.

     

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.     

Formality for the nilpotent cone and a derived Springer correspondence

Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the triangulated category generated by the Springer perverse sheaf and the derived category of differential graded modules over a dg-ring related to the Weyl group.     

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 .

     

Formality in generalized Kähler geometry

We prove that no nilpotent Lie algebra admits an invariant generalized Kähler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kähler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.     

Formality Theorem with Coefficients in a Module

In this paper, X will denote a manifold. In a very famous paper, Kontsevich [Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque [C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, and showed that it is formal. Denote by the DGLA of E-polyvector fields. Considering M, a module over E, we define the-module of E-polyvector fields with values in M. Similarly, we define the-module of E-polydifferential operators with values in M,. We show that there is a quasi-isomorphism of L _∞-modules over from to . Our result extends Calaque’s (and Kontsevich’s) result.     

Formality of the complements of subspace arrangements with geometric lattices

We show that for an arrangement of subspaces in a complex vector space with geometric intersection lattice, the complement of the arrangement is formal. We prove that the Morgan rational model for such an arrangement complement is formal as a differential graded algebra. Bibliography: 10 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 235–247.     

Hall's theorem for hypergraphs

We prove a hypergraph version of Hall's theorem. The proof is topological. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 83–88, 2000     

Goldie's theorem for semigroups

In this paper, we try to look for a semigroup analogue of Goldie's theorem which characterizes the orders of the full matrix ringM _n (D) over a division ringD. The main result of this paper, Theorem 4.11, does give a characterization of the semigroups which have a classical left quotient semigroupM _n (G ⁰), whereG ⁰ is a group with zero. The author is grateful to the referee for his careful modification of this paper and helpful comments which resulted in several improvements.     

Dyson's theorem for curves

Let K be a number field and X₁ and X₂ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X₁×X₂. If Xi=P₁ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.     

Luzin's theorem for charges

A charge in the Euclidean space is an additive function defined on the family of all bounded BV sets equipped with a suitable topology. We define derivatives of charges and show that each measurable function defined on is equal almost everywhere to the derivative of a charge.

     

Beck's theorem for pseudo-monads

In this work we establish a 2-categorical analogue of Beck's theorem characterizing monadic functors. We show that a 2-functor (a pseudo-functor) U is monadic iff it is a right pseudo-adjoint, it reflects adjoint equivalences and it creates U-absolute pseudo-coequalizers of codescent objects.