The Bergman Kernel on Toric Kähler Manifolds

Let (L, h) → (X,ω) be a compact toric polarized Kähler manifold of complex dimension n. For each k ∈ N, the fibre-wise Hermitian metric h on L induces a natural inner product on the vector space C∞(X,Lk) of smooth global sections of L by integration with respect to the volume form ω n n! . The orthogonal projection Pk : C∞(X,Lk) → H(X,L) onto the space H(X,L) of global holomorphic sections of L is represented by an integral kernel Bk which is called the Bergman kernel (with parameter k ∈ 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 ∈ N and consider the projection Pl,k from C∞(X,Lk) onto those global holomorphic sections of L 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 CP 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 Kähler manifolds. Finally, we construct families of induced metrics on blow-ups of polarized Kähler manifolds. We relate those metrics to partial density functions and study their properties for a specific blow-up of C and CP in more detail.