On positive Sasakian geometry (Q1419407)

By using the transverse Yau theorem [\textit{A. El Kacimi-Alaoui}, Compos. Math. 73, No.~1, 57--106 (1990; Zbl 0697.57014)], the following Sasakian version of the positive Calabi conjecture is proved: Theorem A. Let \({\mathcal S}=(\xi, \eta,\Phi,g)\) be a positive Sasakian structure on a compact manifold \(M\). Then \(M\) admits a Sasakian structure \({\mathcal S}'\) with positive Ricci curvature \(a\)-homologous to \(\mathcal S\). As an application, an alternative proof of the existence of Sasakian metrics with positive Ricci tensor on \(k\#(S^2\times S^3)\) for every positive integer \(k\) is presented (theorem B. cf. \textit{J.-P. Sha} and \textit{D.-G. Yang}, J. Differ. Geom. 33, No.~1, 127--137 (1991; Zbl 0728.53027)]. The outline of the paper is as follows: In section 1, transverse holomorphic invariants of the Sasakian structure \({\mathcal S}=(\xi,\eta,\Phi,g)\) on a smooth manifold \(M\), such as transverse Euler and holomorphic Euler characteristic \(\chi({\mathcal F}_\xi)\) and \(\chi_{\text{hol}} ({\mathcal F}_\xi)\) and the basic cohomology \(H^r_B({\mathcal F}_\xi)\) are reviewed. Here \({\mathcal F}_\xi\) is the characteristic foliation of the Sasakian structure [cf. \textit{C. P. Boyer}, \textit{K. Galicki} and \textit{M. Nakamaye}, Math. Ann. 325, No.~3, 485--524 (2003; Zbl 1046.53027)] hereafter refered to as [1]). In section 2, the El Kacimi-Alaoui's transverse Yau Theorem and its reformulation in [1] are reviewed (theorems 2.2 and 2.3). A transverse analogue of Dolbeault's theorem is not yet obtained, but some partial results such as uniqueness of basic Hodge number, Euler characteristic and Hirzebruch signature of the quasi-regular positive Sasakian structure on a 5-manifold are proved (prop. 2.4). Applying these results, theorem A is proved. Then the universal cover of a complete positive Sasakian 5-manifold is shown to be diffeomorphic to \[ M^5(k,\alpha)=S^5\#k\#(S^2\times S^3)\# M^5_{\alpha_1} \#\cdots\#M^5_{\alpha_r}, \quad\alpha_1 | \alpha_2 |\cdots |\alpha_r, \] where \(M^5_\alpha\) is a compact simply connected 5-manifold with \(H_2(M^5_\alpha, \mathbb{Z})\cong \mathbb{Z}_\alpha \oplus\mathbb{Z}_\alpha\) (theorem 2.9). Existence of positive Sasakian structures on these manifolds is still open. Theorem B is a partial answer to this problem. It is proved to show \(k\#(S^2\times S^3)\) is diffeomorphic to \(C_f\cap S^7\), where \(C_f\) is the cone in \(\mathbb{C}^4\) cutted out by the zero locus of a weighted homogeneous polynomial \(f\) of degree \(k+1\), weights \((1,1,1,k)\), \(k\geq 2\) and Fano index \(I=2\) (Lemma 3.7). The authors say that theorem A also applies to prove the existence of positive Ricci curvature metrics on any homotopy sphere that bounds a parallelizable manifold, and on an infinite number of rational homology 5-spheres [cf. \textit{C. P. Boyer} and \textit{K. Galicki}, Math. Res. Lett. 9, No.~4, 521--528 (2002; Zbl 1025.53017), \textit{C. P. Boyer}, \textit{K. Galicki} and \textit{M. Nakamaye}, Topology 42, No.~5, 981--1002 (2003; Zbl 1066.53089)].