Logarithmic Trace of Toeplitz Projectors

TLDR: In this paper, it was shown that the Toeplitz logarithmic trace vanishes identically for all contact forms on the three-sphere and that the invariance of the Szego kernel only depends on the contact structure defined by the boundary pseudo-convex CR structure.

Abstract: In [6] we defined Toeplitz projectors on a compact contact manifold, which are analogues of the Szego projector on a strictly pseudo-convex boundary. The kernel of a Toeplitz projector, as the Szego kernel, has a holonomic singularity including a logarithmic term. The coefficient of the logarithmic term is well defined, so as its trace (the integral over the diagonal). Here we show that this trace only depends on the contact structure and not on the choice of the Toeplitz operator (for a given contact structure there are many possible choices). This generalizes a result of K. Hirachi [16] for the Szego kernel, and also shows that his invariant (the trace of the logarithmic coefficient of the Szego kernel) only depends on the contact structure defined by the boundary pseudo-convex CR structure. Finally we show that the Toeplitz logarithmic trace vanishes identically for all contact forms on the three-sphere.