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Let M be a simple hyperkähler manifold. Kuga-Satake construction gives an embedding of H 2 (M , C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H • (M , C) → H • + l (T , C) for some l , which is compatible with the Hodge structures and the Poincaré pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl (2) -triples on M. [ABSTRACT FROM AUTHOR]

For arbitrary field coefficients K , we show that K -perverse sheaves on a complex affine torus satisfy the so-called propagation package , i.e., the generic vanishing property and the signed Euler characteristic property hold, and the corresponding cohomology jump loci satisfy the propagation property and codimension lower bound. The main ingredient used in the proof is Gabber–Loeser's Mellin transformation functor for K -constructible complexes on a complex affine torus, and the fact that it behaves well with respect to perverse sheaves. As a concrete topological application of our sheaf-theoretic results, we study homological duality properties of complex algebraic varieties, via abelian duality spaces . We provide new obstructions on abelian duality spaces by showing that their cohomology jump loci satisfy a propagation package. This is then used to prove that complex abelian varieties are the only complex projective manifolds which are abelian duality spaces. We also construct new examples of abelian duality spaces. For example, we show that if a smooth quasi-projective variety X satisfies a certain Hodge-theoretic condition and it admits a proper semi-small map (e.g., a closed embedding or a finite map) to a complex affine torus, then X is an abelian duality space. [ABSTRACT FROM AUTHOR]

We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern–Gauss–Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd , we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section. [ABSTRACT FROM AUTHOR]

Associated to any manifold equipped with a closed form of degree >1 is an ‘ L ∞ -algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L ∞ -morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L ∞ -algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’ arises this way. [ABSTRACT FROM AUTHOR]

By Hantzsche–Wendt manifold (for short HW - manifold ) we understand any oriented closed Riemannian manifold of dimension n with a holonomy group ( Z 2 ) n − 1 . Two HW - manifolds M 1 and M 2 are cohomological rigid if and only if a homeomorphism between M 1 and M 2 is equivalent to an isomorphism of graded rings H ⁎ ( M 1 , F 2 ) and H ⁎ ( M 2 , F 2 ) . We prove that HW - manifolds are cohomological rigid. [ABSTRACT FROM AUTHOR]

We construct a 1-convex surface X such that its universal covering X ˜ has the property that H 1 ( X ˜ , O X ˜ ) is not separated. [ABSTRACT FROM AUTHOR]

Let M 2 n be a Poisson manifold with Poisson bivector field Π . We say that M is b -Poisson if the map Π n : M → Λ 2 n ( T M ) intersects the zero section transversally on a codimension one submanifold Z ⊂ M . This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of ( M , Π ) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology. [ABSTRACT FROM AUTHOR]

Abstract: Let be a subanalytic compact pseudomanifold. We show a de Rham theorem for forms on the nonsingular part of . We prove that their cohomology is isomorphic to the intersection cohomology of in the maximal perversity. [Copyright &y& Elsevier]

Let M n be a closed manifold of almost nonnegative sectional curvature and nonzero first de Rham cohomology group. Using a topological argument, we show that the Morse-Novikov cohomology group H p (M n , θ) vanishes for any p and θ ∈ H d R 1 (M n) , θ ≠ 0. Based on a new integral formula, we also show that a similar result holds for a closed manifold of almost nonnegative Ricci curvature under the additional assumption that its curvature operator is uniformly bounded from below. [ABSTRACT FROM AUTHOR]