1997/11/04 by H. Gillet, Henri Gillet, Christophe Soulé +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Numerical methods in inverse problems #dg-ga #math.DG
paper · pdf · doi:10.1007/s002200050496
22 pages, plain TeX
arxiv created 1997/11/04 · openalex publication_date 1998/12/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E be a holomorphic vector bundle on a compact Kähler manifold X. If we fix a metric h on E, we get a Laplace operator Δ acting upon smooth sections of E over X. Using the zeta function of Δ, one defines its regularized determinant det'(Δ). We conjectured elsewhere that, when h varies, this determinant det'(Δ) remains bounded from above. In this paper we prove this in two special cases. The first case is when X is a Riemann surface, E is a line bundle and dim(H0 (X,E)) + dim(H1 (X,E)) ≤ 2, and the second case is when X is the projective line, E is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis.