Positivity for individual moduli spaces [04AX]
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Positivity for individual moduli spaces
In general the bordism current receives contributions from many -dimensional moduli spaces of holomorphic curves. We now focus on one moduli space by fixing the choice of the Lagrangian intersections and the homotopy type of , and consider a connected open subset of the moduli space which contains only automatically transverse curves. We observe:
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For a fixed automatically transverse holomorphic curve , along between any two successive corners, by the nowhere vanishing of , the Jacobian of the boundary evaluation map cannot change sign. That is, either the orientation of the universal family agrees with the orientation of (resp. ) along at every point along the boundary portion of , or the two orientations disagree at every point.
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The Lagrangians are graded by assumption, and the orientations are canonically determined by . The corner behaviour (cf. Remark 3.7) implies that at the degree one self intersections, does not change sign. On the other hand, at the and the ends along , the 1-form changes orientation sign. Thus at a fixed automatically transverse curve, the orienation of the universal family and either completely agree along every point of , or completely disagree.
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As we deform among automatically transverse curves, the orientation signs cannot change. Thus either all these holomorphic curves contribute positively to , or they all contribute negatively.
The above discussion also suggests the limitation of the positivity condition: if we encounter a holomorphic curve in the moduli space, which is not automatically transverse, then it is possible to switch orientation signs. For arbitrary exact immersed Lagrangians, it seems unreasonable to expect the positivity condition, and it is conceivable that counterexamples may arise from -principle constructions. Whether counterexamples occur for more restrictive Lagrangians seems less clear, and we leave the following sample questions as food for thought:
Question 6.
How does the positivity condition behave under exact isotopy with surgery?
Question 7.
Are there examples of exact Calabi-Yau manifolds such that the positivity condition is satisfied for bordism currents between all exact, almost calibrated, unobstructed immersed Lagrangians equipped with suitable brane structures? What if the Lagrangians are quantitatively almost calibrated (cf. (3))?
Remark 3.9.
If the Fukaya category is defined over , we can require all holonomy factors to be integer valued. The positivity condition requires all the contributions to to have the same orientation sign. This has the amusing consequence that all holonomy factors associated with -dimensional moduli spaces contributing to , must in fact all be . Intuitively, this means there is a unique such holomorphic curve through any generic point of , and the boundary evaluation of universal family to is transverse.