Autour de la cohomologie de Bott-Chern

Higher-Page Bott-Chern and Aeppli Cohomologies and Applications

For every positive integer $r$, we introduce two new cohomologies, that we call $E_r$-Bott-Chern and $E_r$-Aeppli, on compact complex manifolds. When $r=1$, they coincide with the usual Bott-Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when $r\geq 2$. They provide analogues in the Bott-Chern-Aeppli context of the $E_r$-cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-$(r-1)$-$\partial\bar\partial$-manifolds that we introduced very recently. We also…

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.

Cohomologie des fibrés en droites sur les variétés magnifiques de rang minimal

The Borel-Weil-Bott theorem describes the cohomology of line bundles over flag varieties. Here, one generalizes this theorem to a wider class of projective varieties : the wonderful varieties of minimal rank.

Blow-up formulae for twisted cohomologies with supports

We study twisted cohomologies with paracompactifying families of supports. The Kunneth theorems, Leray-Hirsch theorems and self-intersection formulae are established. Based on these results, we eventually give explicit expressions of complex blow-up formulae for twisted Dolbeault cohomology on arbitrary complex manifolds and the ones of generalized blow-ups formulae for twisted de Rham cohomology on arbitrary oriented smooth manifolds. These expressions are induced by the morphisms of (simple or double) complexes of spaces of forms and currents rather than just the maps between cohomologies, w…

广义Calabi--Eckmann三维流形的整Bott--Chern上同调

Integral Bott--Chern cohomology of Generalized Calabi--Eckmann threefolds

本文计算了复维3的广义Calabi--Eckmann流形的整Bott--Chern上同调，证明非双全纯流形间该上同调环同构，并用特征映射区分不同上同调。