@MISC{Pokorny11thebergman, author = {Florian T. Pokorny}, title = {The Bergman Kernel on Toric Kähler Manifolds}, year = {2011} }

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Abstract

Let (L; h) ! (X;!) be a compact toric polarized Kahler manifold of complex dimension n. For each k 2 N, the bre-wise Hermitian metric hk on Lk induces a natural inner product on the vector space C1(X;Lk) of smooth global sections of Lk by integration with respect to the volume form! n n!. The orthogonal projection Pk: C1(X;Lk) ! H0(X;Lk) onto the space H0(X;Lk) of global holomorphic sections of Lk is represented by an integral kernel Bk which is called the Bergman kernel (with parameter k 2 N). The restriction k: X! R of the norm of Bk to the diagonal in X X is called the density function of Bk. On a dense subset of X, we describe a method for computing the coecients of the asymp-totic expansion of k as k!1 in this toric setting. We also provide a direct proof of a result which illuminates the o-diagonal decay behaviour of toric Bergman kernels. We x a parameter l 2 N and consider the projection Pl;k from C1(X;Lk) onto those global holomorphic sections of Lk that vanish to order at least lk along some toric submanifold of X. There exists an associated toric partial Bergman kernel Bl;k giving rise to a toric partial density function l;k: X! R. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of X as k! 1. Euler-Maclaurin sums and Laplace's method are utilized as important tools for this. We discuss the case of a polarization of CPn in detail and also investigate the non-compact Bargmann-Fock model with imposed vanishing at the origin. We then discuss the relationship between the slope inequality and the asymptotics of Bergman kernels with vanishing and study how a version of Song and Zelditch's toric local-ization of sums result generalizes to arbitrary polarized Kahler manifolds. Finally, we construct families of induced metrics on blow-ups of polarized Kahler manifolds. We relate those metrics to partial density functions and study their properties for a specic blow-up of Cn and CPn in more detail.