Regularized determinants of Laplacians for Hermitian line bundles over projective spaces
Regularized determinants of Laplacians for Hermitian vector bundles play a very important role in modern mathematics. While a lot of progress has been made in the qualitative study of such a concept, comparably few results in the quantitative study are known. In this paper, we give the precise values for the regularized determinant for all line bundles \({\mathcal O}(m)\) over projective spaces \(\mathbb{P}^n\) with respect to the Fubini-Study metric, in terms of \(m \) and \(n\). The value of the regularized determinant, by definition, depends on the eigenvalues and their multiplicities of the associated Laplacians. In practice, there are two ways to compute it. One is done by a certain general formalism. The other is done by using the precise eigenvalues and their multiplicities, which are, in general, hard to find. In this paper, we adopt the first method. More precisely, we will consider the projective space as an arithmetic variety. Then we use the arithmetic Riemann-Roch theorem to evaluate the regularized determinant in question. For doing so, we need to know a certain arithmetic intersection and the \(L^2\) contribution to the Quillen metric on the associated determinant line bundle. During this process, one then needs to use the classical Bott-Chern secondary characteristic class. When \(m=0\), the result was first obtained by Gillet, Soulé and Zagier, by using the second method. For the corresponding result on the sphere in Riemannian geometry, we refer to the paper: ``Analytic torsion of spheres'', by the author and \textit{Y. You} in Int. J. Math. 7, No. 1, 109-125 (1996; Zbl 0854.58043).
Bott-Chern secondary characteristic class
scientific article; zbMATH DE number 880133