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Register a new userGeometrical pluripotential theory on Sasaki manifolds
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We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result is to generalize T. Darvas theory on the geometric structure of the space of Kahler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kahler structure.
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We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng cite{CC1, CC2, CC3} on existence of Kahler metrics with constant scalar curvature (csck) on compact Kahler manifolds. Following their strategy, we prove that given a Sasaki structure (with Reeb vector field and complex structure on its cone fixed ), there exists a Sasaki structure with transverse constant scalar curvature (cscs) if and only if the $mathcal{K}$-energy is reduced proper modulo the identity component of the automorphism group which preserves both the Reeb vector field and transverse complex structure. Technically, the proof mainly consists of two parts. The first part is a priori estimates for scalar curvature type equations which are parallel to Chen-Chengs results in cite{CC2, CC3} in Sasaki setting. The second part is geometric pluripotential theory on a compact Sasaki manifold, building up on profound results in geometric pluripotential theory on Kahler manifolds. There are notable, and indeed subtle differences in Sasaki setting (compared with Kahler setting) for both parts (PDE and pluripotential theory). The PDE part is an adaption of deep work of Chen-Cheng cite{CC1, CC2, CC3} to Sasaki setting with necessary modifications. While the geometric pluripotential theory on a compact Sasaki manifold has new difficulties, compared with geometric pluripotential theory in Kahler setting which is very intricate. We shall present the details of geometric pluripotential on Sasaki manifolds in a separate paper cite{HL} (joint work with Jun Li).
On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Amp`ere operator is defined. It is shown that it satisfies a version of theorems of A. D. Alexandrov and Chern-Levine-Nirenberg. These notions and results were previously known in the special case of hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we prove a new multiplicativity property.
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
Let $L_f$ be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial $f.$ In this article, we give ten examples of $2$-connected seven dimensional Sasaki-Einstein manifolds $L_f$ for which $H_{3}(L_f, mathbb{Z})$ is completely determined. Using the Boyer-Galicki construction of links $L_f$ over particular Kahler-Einstein orbifolds, we apply a valid case of Orliks conjecture to the links $L_f $ so that one is able to explicitly determine $H_{3}(L_f,mathbb{Z}).$ We give ten such new examples, all of which have the third Betti number satisfy $10leq b_{3}(L_{f})leq 20$.
We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.
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