Related papers: A geometric description of smooth cohomology

This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product…

We describe the cohomology ring of the moduli space of a flexible polygon in geometrically meaningful terms. We propose two presentations, both are computation friendly: there are simple rules for cup product.

Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes…

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the…

We present two approaches to constructing an integration map for smooth Deligne cohomology. The first is defined in the simplicial model, where a class in Deligne cohomology is represented by a simplicial form, and the second in a related…

This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.

We describe complex conjugation on the primitive middle-dimensional algebraic de Rham cohomology of a smooth projective hypersurface defined over a number field that admits a real embedding. We use Griffiths' description of the cohomology…

In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We…

In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions…

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular, we provide a formula for the cup product which gives the cohomology module an algebra structure and then we describe this…

We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.

Let $X(P,\lambda)$ be a 4-dimensional toric orbifold associated to a polygon $P$ and a characteristic function $\lambda$. Assuming that $X(P,\lambda)$ is locally smooth over a vertex of $P$, we determine the integral cohomology ring…

This paper explicitly describes Hodge structures of complete intersections of ample hypersurfaces in compact simplicial toric varieties.

We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.

We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.

We introduce the notion of a $p$-Cartier smooth algebra. It generalises that of a smooth algebra and includes valuation rings over a perfectoid base. We give several characterisations of $p$-Cartier smoothness in terms of prismatic…

In this survey, we discuss whether the complex projective space can be characterized by its integral cohomology ring among compact complex manifolds.

In this paper, we give an explicit description of tropical cohomology of smooth algebraic varieties over trivially valued fields. We also construct ``monodromy weight'' spectral sequences for tropical cohomology of geometric strictly…

Call a pure Hodge structure geometric if it is contained in the cohomology of a smooth complex projective variety. The main goal is to show that for any set of Hodge numbers (subject to the obvious constraints), there exists a geometric…

We review the recent progress in the study of cyclic cohomology in the presence of Hopf symmetry.