Talk:Stable map

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[edit] Semipositivity

Perhaps I missed it, but it seems to me that the semi-positive condition has not been emphasized here. The pseudocycle construction requires the compactified moduli space to have "boundary" of co-dimension 2. In general, this is not true. One must put conditions on the Chern numbers of spheres that can bubble off, hence the semi-positivity condition used in e.g. McDuff-Salamon. SammyBoy 05:07, 10 April 2006 (UTC)

You are right. The simplest fix is to assume semipositivity. Perhaps we should say "under suitable/reasonable assumptions" the boundary can be shown to have codimension at least 2. Taken literally, this is a vacuous statement, so perhaps we should at least define semipositivity and explain how it's used? I'm never sure how much precision is desirable in an encyclopedia article as difficult as this. Your improvements are welcome. Joshua Davis 19:30, 10 April 2006 (UTC)

Sorry I disappeared for so long. How about under "some simplifying assumptions on the geometry of the symplectic manifold" ? I think you are right that too much precision is undesirable. On the other hand, the huge amount of work that has been done in trying to explain and formalize the virtual moduli cycle would seem like a waste of time without some precision. (This basically is the construction of something like a fundamental class of the moduli space of pseudoholomorphic spheres, even though the boundary strata may be of too high a dimension. I don't understand it, alas.) SammyBoy 07:49, 16 September 2006 (UTC)