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New stubs Oka principle, Oka manifold (with redirect Oka map) and Franc Forstnerič. Jardine has shown that one can use the Toen-Vezzosi like engineering with his intermediate model structure on the category of simplicial presheaves on a simplicial version of the Stein site. The $(\infty,1)$-stacks/fibrants will be Oka maps; those cofibrants which are represented by complex manifolds are in fact Stein manifolds.
Added several complex varaibles. I wanted to quote it from some other entries but it seems problems with nlab again.
Have you seen http://arxiv.org/abs/math/0303355, Zoran?
Model Structures and the Oka Principle
Authors: Finnur Larusson
Abstract: We embed the category of complex manifolds into the simplicial category of prestacks on the simplicial site of Stein manifolds, a prestack being a contravariant simplicial functor from the site to the category of simplicial sets. The category of prestacks carries model structures, one of them defined for the first time here, which allow us to develop “holomorphic homotopy theory”. More specifically, we use homotopical algebra to study lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. We prove that holomorphic maps satisfy certain versions of the Oka Principle if and only if they are fibrations in suitable model structures. We are naturally led to a simplicial, rather than a topological, approach, which is a novelty in analysis.
Larusson is in the same department with Jardine, and he has far developed initial observations of Jardine mentioned above. It is a very interesting development. I remember few years ago being pretty excited when I saw the abstract you quote above, but did not have the time to go reading this. I was reminded of it few days ago when planing to go to the seminar by Forstnerič, where the analytic aspects where exhibited, rather than the model structure. If I will have the time to go to Ljubljana in that period, Larusson visits Ljubljana in early December for a colloqium.
Stein manifolds should be viewed as an analogues of affine schemes; thus the role of Stein site by analogy with approaches to derived algebraic geometry and ${A}^1$-homotopy theory. It would be interesting to also have an analogue of birational theory (bimeromorphic homotopy theory - like birational motives of Bruno Kahn). I have attended once a talk on ${B}^1$-homotopy theory in rigid analytic geomtry which has similar structure like Voevodsky’s theory. The situation there should be even closer to the phenomena in several complex variables.
No. Larusson is at Adelaide (my university), but he was previously in Canada with Jardine.
I did not know that he moved and now is that far. I do not know why my LaTeX for A1 homotopy and B1 homotopy does not display in 4.
I tried your code and it did not work. I then tried ${A}^1$ with ’braces’ around the A and that seems to work.
Thanks, I have now updated with that strange correction.
Intersting. So in Larusson’s model structure fibrant replacement of representables is weak Oka-fication of complex manifolds. What would this be used for in practice? Where does one care about complex manifolds only if they are weakly Oka?
Well, read the easy What is… Notices survey of Oka principle listed in the entry. The Oka property is about a homotopy solution to a section-like problem, deforming it and thus obtaining a holomorphic solution. Existence of various kinds of functions, maps and so on in several complex variables very much depends on the subtype of a complex domain one deals with. This is very complex and intricate. For example, the Stein property (which one should look at as an analogue of affine in sheaf picture) is in fact just one of many variants of weak convexity properties; one of the characterizations is in terms of plurisubharmonic functions. I do not know exactly when the weak Oka variant is interesting, we should read more.
New entry Finnur Lárusson.
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