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Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below. (English) Zbl 1462.32007

This interesting paper gives a pointwise estimate for the Bergman kernel of weighted spaces \(A^2(M, h, \mu )\), \(\mu= e^{-2 \psi} {\text{Vol}}\), of holomorphic, square integrable functions on a complete Hermitian manifold with Ricci curvature bounded from below. The estimates are of the form \(K_\mu (z,w) e^{-\psi (z) - \psi (w)} \le C \frac{e^{-\gamma d(z,w)}}{\sqrt{{\text{Vol}}(z,1){\text{Vol}}(w,1)}}\). In addition, the authors establish an interesting exponential decay of canonical solutions of the \(\overline \partial\)-equation: \(|f(w)|^2 \, e^{-2\psi (w)} \le \frac{C}{{\text{Vol}}(w,R)} \, e^{-\gamma d(z,w)} \, \int_{B(z,R)} |u|_h^2 \, d\mu.\) In order to study coercivity of the corresponding Kohn Laplacian \(\Box_{h,\mu}\) they prove an interesting version of the basic identity of the Kohn Laplacian, where the torsion tensor of the Chern connection is involved. They mention that their version of the basic identity follows from [P. A. Griffiths, Am. J. Math. 88, 366–446 (1966; Zbl 0147.07502)]. It is also indicated that the exponential decay of Bergman kernels has already been established in the literature for various special cases, see for instance [M. Christ, J. Geom. Anal. 1, No. 3, 193–230 (1991; Zbl 0737.35011)] for the one-dimensional case, or [H. Delin, Ann. Inst. Fourier 48, No. 4, 967–997 (1998; Zbl 0918.32007); the second author, Adv. Math. 285, 1706–1740 (2015; Zbl 1329.32022); A. P. Schuster and D. Varolin, J. Reine Angew. Math. 691, 173–201 (2014; Zbl 1309.32002)], where the point of view is mainly complex analytic; in [X. Ma and G. Marinescu, Math. Ann. 362, No. 3–4, 1327–1347 (2015; Zbl 1337.32011)] a similar estimate is proved in the more general setting of Hermitian line bundles over symplectic manifolds.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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