Vanishing Theorem for Transverse Dirac Operators on Riemannian Foliations

Let X be a compact manifold of dimension 2n equipped with an almost complex structure J : TX → TX, E a Hermitian vector bundle on X, and gX a Riemannian metric on X. Assume that the almost complex structure J is compatible with gX . Consider a Hermitian line bundle L over X endowed with a Hermitian connection ∇L such that its curvature R = (∇L)2 is nondegenerate. Thus, ω = i 2πR L is a symplectic form on X. One can construct canonically a Spin Dirac operator Dk acting on Ω(X, E ⊗ L) = ⊕q=0Ω(X, E ⊗ L), the direct sum of spaces of (0, q)-forms with values in E ⊗ Lk. Under the assumption that J is compatible with ω, Borthwick and Uribe [2] proved that, for sufficiently large k, KerD k = 0, where D k denotes the restriction of Dk to Ω 0,odd(X, E ⊗ Lk). This result generalizes the famous Kodaira vanishing theorem for the cohomology of the sheaf of sections of a holomorphic vector bundle twisted by a large power of a positive line bundle. It has interesting applications in geometric quantization (see [2] and references therein). In [18], Ma and Marinescu gave a proof of the Borthwick-Uribe result, which uses only the Lichnerowicz formula for the Spin Dirac operator. They also show that, if we put