## Bergman kernel on toric Kahler manifolds

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##### Date

28/06/2011##### Author

Pokorny, Florian Till

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##### 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.