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- Pokorny, Florian T., 1980-
(författare)
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The Bergman Kernel on Toric Kähler Manifolds
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
- Pokorny, Florian T., et al.
(författare)
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Toric partial density functions and stability of toric varieties
- 2014
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 358:3-4, s. 879-923
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Tidskriftsartikel (refereegranskat)abstract
- Let (L, h) -> (X, omega) denote a polarized toric Kahler manifold. Fix a toric submanifold Y and denote by (rho) over cap (tk) : X -> R the partial density function corresponding to the partial Bergman kernel projecting smooth sections of L-k onto holomorphic sections of L-k that vanish to order at least tk along Y, for fixed t > 0 such that tk is an element of N. We prove the existence of a distributional expansion of (rho) over cap (tk) as k -> infinity, including the identification of the coefficient of k(n-1) as a distribution on X. This expansion is used to give a direct proof that if omega has constant scalar curvature, then (X, L) must be slope semi-stable with respect to Y (cf. Ross and Thomas in J Differ Geom 72(3): 429-466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.
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