arXiv subjects

Léonard Pille-Schneider

Publications and source records attributed to Léonard Pille-Schneider.

5 recordsLinked to original sources

arXiv2026

Monge-Ampère measures on balanced polyhedral spaces

We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct Monge--Ampère measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge--Ampère equations on balanced polyhedral spaces via a variational approach, providing sufficient conditions for the existence of solutions as well as explicit counterexamples. Finally, we relate our framework to non-archimedean pluripotential theory and explore its connection with the non-archimedean Monge--Ampère equation.

math.AG↗

arXiv2025

Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces

Using a construction by Yamamoto of tropical contractions, we construct a non-archimedean SYZ fibration on the Berkovich analytification of a class of maximally degenerate hypersurfaces in projective space. We furthermore prove that under a discrete symmetry assumption, the potential for the non-archimedean Calabi-Yau metric is constant along the fibers of the retraction. The proof uses the work of Li on the Fermat degeneration of hypersurfaces, and an explicit description of toric plurisubharmonic metrics on the hybrid space associated to a complex toric variety.

math.AG↗

arXiv2023

Global pluripotential theory on hybrid spaces

Let A be an integral Banach ring, and X/A be a projective scheme of finite type, endowed with a semi-ample line bundle L. We define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof. We focus in particular on the case where A is a hybrid ring of complex power series and X/A is a smooth variety, so that X^an is the hybrid space associated to a degeneration X of complex varieties over the punctured disk. We then prove that when L is ample, any plurisubharmonic metric on L with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space X^hyb . We also discuss the continuity of the family of Monge-Ampère measures associated to a continuous plurisubharmonic hybrid metric. In the case where X is a degeneration of canonically polarized manifolds, we prove that the canonical psh extension is continuous on Xhyb and describe it explicitly in terms of the canonical model (in the sense of MMP) of the degeneration.

math.AG↗

arXiv2023

Toric geometry and integral affine structures in non-archimedean mirror symmetry

We study integral dlt models of a proper C((t))-variety X along a toric stratum of the special fiber. We prove that the associated Berkovich retraction - from the non-archimedean analytification of X onto the dual complex of the model - is an affinoid torus fibration around the simplex corresponding to the toric stratum, which extends results from Nicaise-Xu-Yu. This allows us to construct new types of non-archimedean retractions for maximally degenerate families of quartic K3 surfaces and quintic 3-folds, by gluing several non-archimedean SYZ fibrations, each one toric along a codimension one stratum. We then show that the new retractions induce the same singular integral affine structures that arise on the dual complex of toric degenerations in the Gross-Siebert program, as well as on the Gromov-Hausdorff limit of the family.

math.AG↗

arXiv2020

Hybrid convergence of Kähler-Einstein measures

We compute the hybrid limit (in the sense of Boucksom-Jonsson) of the family of Kähler-Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make some remarks on the non-archimedean Monge-Ampère operator and hybrid continuity of Kähler-Einstein potentials in this context.

math.DG↗