Bergman kernel on toric Kahler manifolds

Abstract

Let (L,h) → (X,ω) be a compact toric polarized Kahler manifold of complex dimension n. For each k ε N, the fibre-wise Hermitian metric hk on Lk induces a natural inner product on the vector space C∞(X,Lk) of smooth global sections of Lk by integration with respect to the volume form ωn /n! . The orthogonal projection Pk : C∞(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 ε N). The restriction ρk : 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 coefficients of the asymptotic expansion of ρk as k → ∞ in this toric setting. We also provide a direct proof of a result which illuminates the off-diagonal decay behaviour of toric Bergman kernels. We fix a parameter l ε N and consider the projection Pl,k from C∞(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 ρl,k : X → R. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of X as k → ∞. 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 localization 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 specific blow-up of Cn and CPn in more detail.