Perturbations for bare curve counts

Date: 2020-11-12

Time: 14:00 - 15:00

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Tobias Ekholm

We describe how to construct perturbations for counts of bare holomorphic curves in a 3-dimensional Calabi-Yau manifold X with boundary on a Maslov zero Lagrangian submanifold L. We use the polyfolds of Hofer-Wyzocki-Zehnder  as configuration spaces. A brief description with more details is as follows. Consider the space Z of stable maps from Riemann surfaces to (X,L) and the bundle W of complex anti-linear formal differentials there. The Cauchy-Riemann operator gives a section in this bundle. Using a class of perturbation sections of W that we call local target-smooth we produce a section s of W which vanishes over any component of any stable map of symplectic area zero. Curves without such components are called bare and we show that for generic sections s, with this property, bare solutions are transversely cut out and we control the topology of the maps in generic one-parameter families. A key technical result is that for solutions corresponding to the perturbing sections we use, bubbling off a non-rational constant component is a codimension two phenomenon. A key end-result is an identification of nodal curve solutions with hyperbolic and elliptic boundary nodes with associated solutions on the normalizatioin of the nodal curves.