We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the \(\partial {\bar{\partial }}\)-lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

我们表明，在Sasakian结构的任意变形下，Sasakian流形的Hodge数不变。我们还提出了一个上半连续性定理，该类是具有同向可定向的横向黎曼叶型的流形上光滑的横向椭圆算子族的核的尺寸。我们用它来证明\（\ partial {\ bar {\ partial}} \）-引理和作为横向Kähler的是刚性的属性，在横向全同质结构的小变形下保持了叶状结构。我们研究了一个例子，该例子表明横向全同叶的任意变形不是这种情况。最后，我们指出了上半连续性定理在K接触流形上的应用。