Abstract
We define a refined Gromov–Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4‑manifold that are self-transverse as an element in a capped version of the Chekanov–Eliashberg dg‑algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point.
We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5‑sphere.
Funding Statement
GDR is supported by the grant KAW 2016.0198 from the Knut and Alice Wallenberg Foundation. TE is supported by the Knut and Alice Wallenberg Foundation as a Wallenberg Scholar KAW2020.0307 and by the Swedish Research Council VR2020-04535. DT is partially supported by the Simons Foundation grant #385573, Simons Collaboration on Homological Mirror Symmetry.
Citation
Georgios Dimitroglou Rizell. Tobias Ekholm. Dmitry Tonkonog. "Refined disk potentials for immersed Lagrangian surfaces." J. Differential Geom. 121 (3) 459 - 539, July 2022. https://doi.org/10.4310/jdg/1664378618
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