Bochner Laplacian and Bergman kernel expansion of semipositive line bundles on a Riemann surface (Q6583568)

\textit{R. Montgomery} [Commun. Math. Phys. 168, No. 3, 651--675 (1995; Zbl 0827.58076)] conducted a study on the spectrum, specifically focusing on the smallest eigenvalue, of the Bochner Laplacian on tensor powers of a Hermitian line bundle \(L\) over a Riemann surface. His analysis was predicated on the assumption that the curvature of the line bundle vanishes transversally along a curve.\N\NThe first theorem presented in the reviewed paper establishes the most general asymptotic behavior for the smallest eigenvalue of the Bochner Laplacian on tensor powers \(L^k\). More precisely, if the curvature vanishes up to order \(r\) at any given point (refer to (1.3) for the precise definition), then the first eigenvalue of the Bochner Laplacian grows asymptotically as \(C k^{2/r}\) for some constant \(C>0\) as \(k \to \infty\) with the first eigenfunction concentrating on the vanishing locus.\N\NUnder additional assumptions, the paper provides results on the asymptotics of the Weyl counting function and a complete asymptotic expansion of the smallest eigenvalue of the Bochner Laplacian.\N\NIn the holomorphic context, the analogous study involves the Bergman kernel of a holomorphic line bundle on a complex manifold. The Bergman kernel is the Schwartz kernel of the projection from smooth sections of the holomorphic line bundle onto holomorphic sections. The analysis of the Bergman kernel and holomorphic sections associated with tensor powers has been extensively investigated, as exemplified by the Hirzebruch-Riemann-Roch theorem and holomorphic Morse inequalities. For positive line bundles, \textit{G. Tian} [J. Differ. Geom. 32, No. 1, 99--130 (1990; Zbl 0706.53036)] established the leading asymptotic behavior of the Bergman kernel along the diagonal, which has proven crucial in addressing the Yau-Tian-Donaldson conjecture.\N\NSince, as demonstrated by \textit{H. Donnelly} [Math. Z. 245, No. 1, 31--35 (2003; Zbl 1036.58027)], there is no spectral gap for the Kodaira Laplacian on tensor powers on complex manifolds of dimension at least 2, the third main result of the paper examines the pointwise asymptotic expansion along the diagonal of the Bergman kernel for a semipositive line bundle over a compact Riemann surface, where the Chern curvature vanishes to finite order at any point.