Seminar in Real and Complex Geometry

Thursday, January 6, 2022, 16:00-17:30, online (via zoom)

Michael Polyak

Refined tropical counting, ribbon structures and the quantum torus


Tropical geometry is a powerful instrument in algebraic geometry, allowing for a simple combinatorial treatment of various enumerative problems. Tropical curves are planar metric graphs with certain requirements of balancing, rationality of slopes and integrality. An addition of a ribbon structure (and a removal of rationality/integrality requirements) lead to a particularly simple combinatorial construction of moduli of ribbon (pseudo)tropical curves. Refined Block-Goettsche counting of rational tropical curves turns into a construction of some simple top-dimensional cycles on these moduli and maps of spheres. These cycles turn out to be closely related to associative algebras; curves with "flat" vertices necessitate a passage from associative to Lie algebras. In particular, counting of (both complex and real) curves in toric varieties is related to the quantum torus algebra. More complicated counting invariants (the so-called Gromov-Witten descendants, or relative Welschinger invariants) are treated similarly and are related to the super-Lie structure on the quantum torus. As a by-product we obtain a new one-parameter family of weights for a refined counting of the descendants.