Andrés Jaramillo Puentes (Universität Tübngen): A Wall-Crossing Formula for Motivic Gromov-Witten Invariants

In enumerative geometry, Gromov-Witten invariants play a central role in counting curves on algebraic varieties, and their variations under different conditions provide a rich framework for understanding moduli spaces. In recent years, there has been significant progress in developing enriched versions of these invariants within the framework of motivic homotopy theory, leading to what we now call motivic Gromov-Witten invariants. Motivic invariants encode additional algebraic structure over the Grothendieck-Witt ring of a base field, allowing for finer distinctions in curve counts, particularly over fields with nontrivial real structure.

In this talk, we discuss a wall-crossing formula for motivic Gromov-Witten invariants. Specifically, we explore how variations in point conditions and configurations influence the values of these invariants, and demonstrate how these changes can be systematically tracked using a motivic analogue of classical wall-crossing phenomena. We will illustrate how this formula provides a mechanism to relate invariants associated with distinct configurations by tracking contributions along certain “walls” in the parameter space, which play an analogous role to wall-crossing in real enumerative geometry.

Additionally, we will present applications of this formula to specific enumerative problems, showcasing how the motivic perspective not only recovers known real and complex cases but also opens new pathways for counting problems over arbitrary fields. This development lays the groundwork for future research, providing a powerful tool to bridge combinatorial and motivic techniques in tropical and algebraic geometry.

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