1999/07/02 by F. Prosmans, Fabienne Prosmans, Prosmans, F. +3
Computer Science · Mathematics · #32C38 #35A27 #58G05 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Digital Image Processing Techniques #FOS: Mathematics #Mathematical Analysis and Transform Methods #Topological and Geometric Data Analysis #math.AG #math.CV #msc:32C38 #msc:35A27 #msc:58G05
paper · pdf · doi:10.48550/arxiv.math/9907012
36 pages, LaTeX + AMS and Xy-pic macros. To appear in Duke Math. Journal
arxiv created 1999/07/02 · openalex publication_date 1999/07/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we prove that any perfect complex of D∞-modules may be reconstructed from its holomorphic solution complex provided that we keep track of the natural topology of this last complex. This is to be compared with the reconstruction theorem for regular holonomic D-modules which follows from the well-known Riemann-Hilbert correspondence. To obtain our result, we consider sheaves of holomorphic functions as sheaves with values in the category of ind-Banach spaces and study some of their homological properties. In particular, we prove that a Künneth formula holds for them and we compute their Poincaré-Verdier duals. As a corollary, we obtain the form of the kernels of ``continuous'' cohomological correspondences between sheaves of holomorphic forms. This allows us to prove a kind of holomorphic Schwartz' kernel theorem and to show that D∞=RHomtop(O,O). Our reconstruction theorem is a direct consequence of this last isomorphism. Note that the main problem is the vanishing of the topological Ext's and that this vanishing is a consequence of the acyclicity theorems for DFN spaces which are established in the paper.