Sasakian Geometry on Sphere Bundles

TL;DR

This paper explores the construction of Sasakian structures on sphere bundles over algebraic varieties, identifying conditions for extremal and constant scalar curvature metrics, and introduces new inequivalent structures.

Contribution

It applies fiber join constructions to produce new Sasaki structures, analyzing their extremal properties and scalar curvature, expanding the understanding of Sasakian geometry on sphere bundles.

Findings

Existence of extremal Sasaki metrics in certain cones

Identification of new inequivalent cone indecomposable structures

Computation of cohomology groups for sphere bundles over Riemann surfaces

Abstract

The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety $N$ with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki \cite{Yam99} for K-contact manifolds to the Sasaki case. This construction depends on the choice of $d+1$ integral K\"ahler classes $[\omega_j]$ on $N$ that are not necessarily colinear in the K\"ahler cone. We show that the colinear case is equivalent to a subclass of a different join construction orginally described in \cite{BG00a,BGO06}, applied to the spherical case by the authors in \cite{BoTo13,BoTo14a} when $d=1$, and known as cone decomposable \cite{BHLT16}. The non-colinear case gives rise to infinite families of new inequivalent cone indecomposable Sasaki contact CR on certain sphere bundles. We prove that the Sasaki cone for some of these structures contains an open set of extremal Sasaki metrics and, for certain specialized cases, the regular ray within this cone is shown to have constant scalar curvature. We also compute the cohomology groups of all such sphere bundles over a product of Riemann surfaces.