On the fundamental groups of compact Sasakian manifolds

Abstract

Abstract. We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that the fundamental group of any compact Hodge manifold is Sasaki. In particular, all finite groups are Sasaki. On the other hand, we show that there exist many restrictions on Sasaki groups. We also study the Abel-Jacobi map of a compact Sasakian manifold and its applications to Sasaki groups.

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

“…There are some well-known topological obstructions to the existence of a Sasakian structure on a compact K-contact manifold (refer to, e.g. [6,2,4,9,1]). …”

Kähler hyperbolic manifolds and Chern number inequalities

“…Our second main purpose in this article is to generalize the concepts of Kähler hyperbolicity and non-ellipticity by terming a condition by "Kähler exactness", which has been used in [CY17], and show that a Kähler exact manifold of general type has ample canonical bundle. Recall that on a compact Kähler manifold any Kähler form is closed but can never be exact, which motivates us to introduce the following notion.…”

“…Recently B.-L. Chen and X. Yang made some important progress towards this question and the Hopf Conjecture 1.1 in two articles [CY18] and [CY17]. In the first one [CY18], They showed that a compact Kähler manifold homotopy equivalent to a negatively curved compact Riemannian manifold admits a Kähler-Einstein metric of negative Ricci curvature ([CY18, Thm 1.1]).…”

“…with equality holds if and only if M is covered by the unit ball in C n . In their second article [CY17], they presented some sufficient conditions related to Kähler forms and fundamental groups for compact Kähler manifold to be Kähler hyperbolic or non-elliptic ([CY17, Thms 1.5, 1.6, 1.7]). Consequently this settles the Hopf Conjecture 1.1 in these situations.…”

“…The main purpose of this article is to take a step further towards this question by showing that Kähler hyperbolic manifolds as well as Kähler non-elliptic manifolds indeed satisfy a family of sharp Chern number inequalities (Theorems 2.1 and 2.4). In addition to these, we shall term a condition "Kähler exactness" used in [CY17], which include Kähler hyperbolic and non-elliptic manifolds, and show that a general type Kähler exact manifold has ample canonical bundle (Theorem 2.8).…”