Sasaki几何中的横向Kähler完整性与${\oldmathcal S}$ -稳定性
Transverse Kähler holonomy in Sasaki Geometry and ${\oldmathcal S}$-Stability
摘要：
研究了Sasaki流形$(M,{\oldmathcal S})$上的横向Kähler和乐群及其在Reeb向量场对特征叶理的横向全纯变形下的稳定性.特别地，我们证明了当第一贝蒂数$b_1(M)$和基本Hodge数$h^{0,2}_B({\oldmathcal S})$为零时，${\oldmathcal S}$在横向Kähler流的变形下是稳定的.此外，我们还证明了不可约横截面超Kähler Sasaki结构是${\oldmathcal S}$ -不稳定的，而当$\dim M\geq 7$满足时，不可约横截面Calabi-Yau Sasaki结构是${\oldmathcal S}$ -稳定的.最后，我们证明了标准的Sasaki连接操作（横向holonomy $U(n_1)\times U(n_2)$）以及纤维连接操作保持${\oldmathcal S}$稳定性。
英文摘要：
We study the transverse Kähler holonomy groups on Sasaki manifolds $(M,{\oldmathcal S})$ and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number $b_1(M)$ and the basic Hodge number $h^{0,2}_B({\oldmathcal S})$ vanish, then ${\oldmathcal S}$ is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is ${\oldmathcal S}$-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is ${\oldmathcal S}$-stable when $\dim M\geq 7$. Finally, we prove that the standard Sasaki join operation (transverse holonomy $U(n_1)\times U(n_2)$) as well as the fiber join operation preserve ${\oldmathcal S}$-stability.
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