for this workshop
Non-Archimedean methods in complex geometry
at the
American Institute of Mathematics, Pasadena, California
organized by
Mattias Jonsson, Valentino Tosatti, and Yueqiao Wu
This workshop, sponsored by AIM and the NSF, will be devoted to non-Archimedean methods in complex geometry. The main topics for the workshop are:
- K-stability and canonical metrics. The recent resolution of the Yau–Tian–Donaldson conjecture for Fano varieties, and partial results in the general polarized case reveals deep connections between the existence of canonical metrics and various flavors of K-stability. The latter is typically understood as an algebro-geometric condition, but can also be understood through valuations and non-Archimedean geometry over trivially valued fields. One topic here is to understand the connection between the different K-stability notions. Another is to use this circle of ideas to further study the existence of canonical metrics on non-compact Kähler manifolds.
- Degenerations of Calabi–Yau manifolds and the SYZ Conjecture. For a polarized family $X_t$ of Calabi–Yau manifolds parametrized by the punctured unit disc, meromorphic at the puncture $t=0$, powerful conjectures by Strominger–Yau–Zaslow (SYZ), Kontsevich–Soibelman and Gross–Wilson give strong predictions on the geometry of $X_t$, equipped with the Ricci flat metric, for $t\approx0$, at least for "maximally degenerate" families. Some of these conjectures involve non-Archimedean geometry, notably the Monge–Ampère equation, and has seen great recent progress through the work of Yang Li. We plan to study this situation further, in part by considering special cases.
- Algebro-geometric and tropical aspects of degenerations. An important property of Berkovich spaces over non-Archimedean fields is that they can often be contracted onto some subspace with piecewise affine structure, a so called skeleton. Inspired by SYZ mirror symmetry, Kontsevich–Soibelman defined an essential skeleton of a degeneration of varities with a nontrivial pluricanonical form, which is conjectured to be topologically equivalent to the base of an SYZ fibration. Furthermore, a wide-ranging program of Gross–Siebert, showed how a degeneration can be reconstructed from the skeleton equipped with its affine structure, using tropical and logarithmic geometry. Many open questions remain, including for example determining the homeomorphism type of the skeleton for degenerations of Calabi–Yau and hyperkähler manifolds.
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
Space and funding is available for a few more participants. If you would like to participate, please apply by filling out the on-line form no later than July 1, 2025. Applications are open to all, and we especially encourage women, underrepresented minorities, junior mathematicians, and researchers from primarily undergraduate institutions to apply.
Before submitting an application, please read the description of the AIM style of workshop.
For more information email workshops@aimath.org
