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Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle. (English) Zbl 1078.32017

A complex manifold \(X\) is said to have the Oka-Grauert property if for all Stein manifolds \(S\) the inclusion of the space of holomorphic maps from \(S\) to \(X\) into the space of continuous maps, both with the compact-open topology, is a weak homotopy equivalence. This intriguing paper reinterprets the Oka-Grauert property in purely holomorphic terms, it is equivalent to the simpicial sheaf \(sO(\cdot,X)\) being a finite homotopy sheaf on the Stein site.
The paper is very well presented, nevertheless some familiarity with homotopy theory would certainly aid its comprehension. Ultimately the results point the way to a completely new interaction between complex geometry and homotopical algebra.

MSC:

32Q28 Stein manifolds
18F10 Grothendieck topologies and Grothendieck topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18G55 Nonabelian homotopical algebra (MSC2010)
32E10 Stein spaces
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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References:

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