Higher-Page Bott-Chern and Aeppli Cohomologies and Applications

非紧复流形上 Bott-Chern 和 Aeppli 上同调的对偶性

Duality between Bott-Chern and Aeppli Cohomology on Non-Compact Complex Manifolds

本文研究了非紧复流形上 Bott-Chern 和 Aeppli 上同调的对偶性，证明了在伪凸性假设下其对偶性成立，并通过反例展示了无此假设时结果失效。

The jumping phenomenon of the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups

Let $X$ be a compact complex manifold, and let $π: \mathcal{X} \rightarrow B$ be a small deformation of $X$, the dimensions of the Bott-Chern cohomology groups $H_{\rm BC}^{p,q}(X(t))$ and Aeppli cohomology groups $H_{\rm A}^{p,q}(X(t))$ may vary under this deformation. In this paper, we will study the deformation obstructions of a $(p,q)$ class in the central fiber $X$. In particular, we obtain an explicit formula for the obstructions and apply this formula to the study of small deformations of the Iwasawa manifold.

Aeppli-Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey-Lawson's spark complex

By comparing Deligne complex and Aeppli-Bott-Chern complex, we construct a differential cohomology $\widehat{H}^*(X, *, *)$ that plays the role of Harvey-Lawson spark group $\widehat{H}^*(X, *)$, and a cohomology $H^*_{ABC}(X; \Z(*, *))$ that plays the role of Deligne cohomology $H^*_{\mathcal{D}}(X; \Z(*))$ for every complex manifold $X$. They fit in the short exact sequence $$ 0\rightarrow H^{k+1}_{ABC}(X; \Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{δ_1}{\rightarrow} Z^{k+1}_I(X, p, q) \rightarrow 0$$ and $\widehat{H}^{\bullet}(X, \bullet, \bullet)$ possess ring structure and refin…

Autour de la cohomologie de Bott-Chern

The goal of the memoir is to develop a new cohomology theory which encompasses De Rham and Dolbeault cohomology as well as Deligne Beilinson cohomology, in the context of general complex analytic manifolds. The special case of the Iwasawa manifold is investigated as a typical example of what occurs in the non Kähler case. Elementary applications to the Kodaira-Spencer deformation theory and to the calculation of Chern classes are given.

Cohomologie et $K$-theorie equivariantes des tours de Bott et des varietes de drapeaux. Application au calcul de Schubert

We determine the structure of the equivariant cohomology and $K$-theory of Bott towers. By restriction, we obtain similar results for Bott-Samelson varieties. This results allow us to describe more precisely the equivariant cohomology and $K$-theory of the flag variety $X$. We give an expression for the cup product in $H_T^*(X)$.

Tensor product decompositions for cohomologies of Bott-Samelson varieties

Let $T$ be a maximal torus of a semisimple complex algebraic group, $\mathrm{BS}(s)$ be the Bott-Samelson variety for a sequence of simple reflections $s$ and $\mathrm{BS}(s)^T$ be the set of $T$-fixed points of $\mathrm{BS}(s)$. We prove the tensor product decompositions for the image of the restriction $H^\bullet_T(\mathrm{BS}(s),k)\to H_T^\bullet(X,k)$, where $X\subset\mathrm{BS}(s)^T$ is defined by some special not overlapping equations $γ_iγ_{i+1}\cdotsγ_j=w_{i,j}$ with right-hand sides belonging to the Weyl group.