Transverse Kähler holonomy in Sasaki Geometry and S-Stability

We study the transverse Kähler holonomy groups on Sasaki manifolds (M, 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(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n 1) × U(n 2)) as well as the fiber join operation preserve S-stability.

HODGE NUMBER DISTRIBUTION FOR CICY MANIFOLDS

The method of exact and spectral sequences is applied to compute the Hodge numbers of all known complete intersection Calabi–Yau manifolds. When plotted on a two-dimensional chart, the distribution of their values reveals an interesting periodic pattern. Next, we introduce a seven-dimensional crystal-like structure which seems to be relevant to study the systematics of the Hodge number distribution.