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From arXiv

Counting real pseudo-holomorphic discs and spheres in dimension four and six

First, we provide another proof that the signed count of the real JJ-holomorphic spheres (or JJ-holomorphic discs) passing through a generic real configuration of kk points is independent of the choice of the real configuration and the choice of JJ, if the dimension of the Lagrangian submanifold LL (fixed points set of the involution) is two or three, and also if we assume LL is orientable and relatively spin, and MM is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the degree of evaluation maps from the moduli spaces of JJ-holomorphic discs. Then, we define the invariant count of discs intersecting cycles of a symplectic manifold at fixed interior marked points, and intersecting real points at the boundary under certain assumptions. The last result is new and was not proved by Welshinger's method.
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Updated on April 25, 2006
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