The Riemann-Roch theorem on algebraic curves

Abstract

Séminaire de Théorie spectrale et géométrie (Chambéry-Grenoble), 1988-1989, tous droits réservés. L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Séminaire de théorie spectrale et géométrie CHAMBÉRY-GRENOBLE 1988-1989(115-121) THE RIEMÀNN-ROCH THEOREM ON ALGEBRAIC CURVES

L^2-Riemann-Roch for singular complex curves

“…In [BPS90], Brüning, Peyerimhoff and Schröder proved that h 0,0 s (X * ) − h 0,1 s (X * ) = m − g and h 0,0 w (X * ) − h 0,1 w (X * ) = m − g + deg Z − |Z| by computing the indices of the differential operators ∂ s and ∂ w . Schröder generalized this result for vector bundles in [Sch89].…”

“…The first part of Corollary 4.8 was discovered by Haskell [Has89], and from that one can deduce the second statement of Theorem 1.2 by use of L 2 -Serre duality. Moreover, Theorem 1.2 was proved in essence by Brüning, Peyerimhoff and Schröder in [BPS90] and [Sch89] by computing the indices of the ∂ w -and the ∂ s -operator.…”

“…From that, we draw also a new understanding of weakly holomorphic functions (Theorem 1.4) and of the divisor Z − |Z|. Moreover, all the previous work has been done only for forms with values in the trivial bundle (except of [Sch89]), whereas we incorporate line bundles. This is essential for applications as we illustrate by the examples mentioned above.…”