Koszul cohomology and the geometry of projective varieties. Appendix: The nonvanishing of certain Koszul cohomology groups (by Mark Green and Robert Lazarsfeld).(English)Zbl 0559.14008

In the paper under review the author proves various results on the Koszul cohomology (such as duality theorem, vanishing theorems, Lefschetz theorems etc.) thus developing algebraic techniques for describing the equations defining the image of a complex manifold under an embedding in a projective space. As an application, the author describes the structure of a minimal free resolution of the ideal of the embedding of a smooth curve C by a complete linear system of divisors of a sufficiently large degree. Another application concerns the so-called Arbarello-Sernesi module AS(X,L): if X is a compact complex manifold and L is an analytic line bundle over X, then $$AS(X,L)=\oplus_{q\in {\mathbb{Z}}}H^ 0(X,K_ X\otimes L^ q)$$ viewed as a module over $$S(H^ 0(X,L))$$ (where S denotes the symmetric algebra). The author shows that if $$| L|$$ does not have fundamental points and maps X onto an n-dimensional variety $$(n=\dim X)$$, then, with a few exceptions, AS(X,L) is generated in degree $$\leq n-1$$ and its relations are generated in degrees $$\leq n$$ (this generalizes Petri’s result for curves). Other applications include various local Torelli theorems.
Reviewer: F.L.Zak

MSC:

 14F99 (Co)homology theory in algebraic geometry 32C35 Analytic sheaves and cohomology groups 32Q99 Complex manifolds 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32L20 Vanishing theorems
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