The Bergman Kernel on Toric Kähler Manifolds

Florian T. Pokorny
PhD thesis, The University of Edinburgh, 2011


Let $(L,h)\to (X, \omega)$ be a compact toric polarized Kähler manifold of complex dimension $n$. For each $k\in \mathbb{N}$, the fibre-wise Hermitian metric $h^k$ on $L^k$ induces a natural inner product on the vector space $\mathcal{C}^{\infty}(X, L^k)$ of smooth global sections of $L^k$ by integration with respect to the volume form $\frac{\omega^n}{n!}$. The orthogonal projection $P_k:\mathcal{C}^{\infty}(X, L^k)\to H^0(X, L^k)$ 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 $k\in \mathbb{N}$). The restriction $\rho_k:X\to \mathbb{R}$ 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 $l\in \mathbb{N}$ and consider the projection $P_{l,k}$ from $\mathcal{C}^{\infty}(X, L^k)$ 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 $\rho_{l, k}:X\to \mathbb{R}$. 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 $\mathbb{CP}^n$ 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 $\mathbb{C}^n$ and $\mathbb{CP}^n$ in more detail.


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@phdthesis{pokorny2011a, title = {The Bergman Kernel on Toric K{\"a}hler Manifolds}, author = {Pokorny, Florian T.}, year = {2011}, school = {The University of Edinburgh}, doi = {1842/5301}, url = {}, }

Supplementary Video, talk at The University of Cambridge