Non-Archimedean methods in complex geometry

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 very recent resolution of the Yau–Tian–Donaldson conjecture in its general form 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, including also the transcendental setting of Kähler manifolds. 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 ≈ 0, at least for “maximally degenerate” families. Some of these conjectures involve non-Archimedean geometry, notably the Monge–Ampère equation, and have seen great recent progress through the work of Yang Li and his resolution of the weak SYZ conjecture for all maximally degenerate Calabi–Yau families. We plan to study this situation further, in part by considering special cases, and also by looking at “intermediate degeneration” families.

• 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 varieties 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.