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The Bergman Kernel on Toric Kähler Manifolds
School of Mathematics, The University of Edinburgh.ORCID iD: 0000-0003-1114-6040
2011 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

Let $(L,h)\to (X, \omega)$ be a compact toric polarized Kähler manifold of complex dimension $n$. For each $k\in N$, the fibre-wise Hermitian metric $h^k$ on $L^k$ induces a natural inner product on the vector space $C^{\infty}(X, L^k)$ of smooth global sections of $L^k$ by integration with respect to the volume form $\frac{\omega^n}{n!}$. The orthogonal projection $P_k:C^{\infty}(X, L^k)\to H^0(X, L^k)$ onto the space $H^0(X, L^k)$ of global holomorphic sections of $L^k$ is represented by an integral kernel $B_k$ which is called the Bergman kernel (with parameter $k\in N$). The restriction $\rho_k:X\to R$ of the norm of $B_k$ to the diagonal in $X\times X$ is called the density function of $B_k$.

On a dense subset of $X$, we describe a method for computing the coefficients of the asymptotic expansion of $\rho_k$ as $k\to \infty$ 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 $l\in N$ and consider the projection $P_{l,k}$ from $C^{\infty}(X, L^k)$ onto those global holomorphic sections of $L^k$ that vanish to order at least $lk$ along some toric submanifold of $X$. There exists an associated toric partial Bergman kernel $B_{l, k}$ giving rise to a toric partial density function $\rho_{l, k}:X\to R$. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of $X$ as $k\to \infty$. Euler-Maclaurin sums and Laplace's method are utilized as important tools for this. We discuss the case of a polarization of $CP^n$ 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 $C^n$ and $CP^n$ in more detail.

Place, publisher, year, edition, pages
The University of Edinburgh , 2011. , 130 p.
National Category
URN: urn:nbn:se:kth:diva-53788OAI: diva2:479144
Sponsorship: Engineering and Physical Sciences Research Council (EPSRC), United KingdomAvailable from: 2012-01-19 Created: 2011-12-30 Last updated: 2012-02-23Bibliographically approved

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Pokorny, Florian T.

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