The vanishing of the log term of the Szego kernel and Tian–Yau–Zelditch expansion

This thesis consists in two results. In [Z. Lu, G. Tian, The log term of Szego kernel, Duke Math. J. 125, N 2 (2004), 351-387], the authors conjectured that given a Kähler form ω on CPn in the same cohomology class of the Fubini–Study form ωFS and considering the hyperplane bundle (L; h) with Ric(h) = ω, if the log–term of the Szego kernel of the unit disk bundle Dh L vanishes, then there is an automorphism φ : CPn → CPn such that φω = ωFS. The first result of this thesis consists in showing a particular family of rotation invariant forms on CP2 that confirms this conjecture. In the second part of this thesis we find explicitly the Szego kernel of the Cartan–Hartogs domain and we show that this non–compact manifold has vanishing log–term. This result confirms the conjecture of Z. Lu for which if the coefficients aj of the TYZ expansion of the Kempf distortion function of a n– dimensional non–compact manifold M vanish for j > n, then the log–term of the disk bundle associated to M is zero.