We define the geometric complex associated to a Morse– Bott–Smale vector field, cf. [AB95], and its associated spectral sequence. We prove an extension of the Bismut–Zhang theorem to Morse–Bott–Smale functions, see Theorem 2 below. The proof is based on the Bismut–Zhang theorem for Morse–Smale functions, see [BZ92].

It is proved that the properties of being Dolbeault formal and geometrically-Bott-Chern-formal are not closed under holomorphic deformations complex structure. Further, we construct a compact manifold which satisfies ∂∂‾-lemma but admits non vanishing Aeppli-Bott-Chern-Massey product.

The Bott periodicity theorem is of fundamental importance in many areas of mathematics, from algebraic topology to functional analysis. It appears unexpectedly in different guises and I would like to explain some of these as well as the influence it has had on the development of different fields. I will concentrate on two roles that periodicity plays. First, periodicity allows one to deloop cla...

“Quantum holonomy” as defined by Berry and Simon, and based on the parallel transport of Bott and Chern, can be considerably extended. There is a natural “parallelity” W*dW = (dW)* W within the Hilbert--Schmidt operators W This defines parallelity and holonomy along curves of density operators e = WW*. There is an intrinsic nonlinearity in the parallel transport which dissolves for curves of pr...

We construct Bott-type Floer homology groups for the sym-plectic manifold (T S 1 ; can) and Bott-type Morse homology groups for the energy functional on the loop space of S 1. Both objects turn out to be isomorpic to the singular homology of the loop space of S 1. So far our objects depend on all choices involved, but the above isomorphism suggests further investigation to show independence of ...

We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent ...

We show that the special case of Serre duality involved in the Borel-Bott-Weil theorem can be formulated and proved in the context of equi-variant Kasparov theory by combining the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of a flag variety that figure in the ...

We formulate and prove the special case of Serre duality involved in the Borel-Bott-Weil theorem in the language of equivariant Kasparov theory. The method is to combine the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of the flag variety figuring in the Borel-B...

We present a novel proof of Thomason’s theorem relating Bott inverted algebraic K-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic K-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varie...