Computation of Refined Enumerative Invariants in Real and Tropical Geometry
Thomas Blomme · HAL (Le Centre pour la Communication Scientifique Directe) · 2020
Tropical geometry enabled the computation of numerous invariants in complex geometry (Gromov-Witten invariants), as well as in real geometry (Welschinger invariants) using correspondence theorems. These theorems reveal a deep connection between tropical geometry and classical geometry. The richness of tropical objects coupled with their simplicity of use also enabled the definition of tropical refined invariants, whose interpretation on the classical geometry side remains quite mysterious, although several conjectures, the Göttsche-Shende conjecture, suggest an even deeper connection to other classical geometric quantities. One such interpretation is proposed by Mikhalkin in 2015, through the counting of real rational curves in toric surfaces, according to the value of a so-called "quantum index". The refined count of curves, which have to pass through some real and complex conjugated points chosen on the toric boundary of the surface, happens to depend only on the number of complex points on each divisor. In the case where all the chosen points are real, Mikhalkin related the obtained invariant to tropical refined invariants. After giving a way of computing the quantum index of rational curves, we extend this relation between classical and tropical invariants in the case where some of the points of the configuration are purely imaginary, and we give a recursive formula that allows one to compute the involved tropical refined invariants. Finally, we propose a generalization of these refined tropical invariants in toric varieties of higher dimension.