Bastien Philippe | Searcharxiv

arXiv2026

A note on the transcendental basepoint-free conjecture for Calabi-Yau manifolds

In this note, we prove that the transcendental basepoint-free conjecture for Calabi-Yau manifolds holds if it holds for its hyperk{\"a}hler factors in its Beauville-Bogomolov decomposition. Based on a contraction theorem due to Bakker and Lehn, we show that the conjecture holds for a big and nef class $\alpha$ on a hyperk{\"a}hler manifold under a mild condition on the dimension of the space generated by classes of rational curves on which $\alpha$ vanishes.

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arXiv2025

Hyperbolicity properties of moduli spaces of marked hyperk{\"a}hler manifolds

We study the hyperbolicity properties of moduli spaces of marked hyperk{\"a}hler manifolds along directions corresponding to families having positivity properties for their Hodge bundle. In particular, we show that the Kobayashi pseudo-distance computed using disks tangent to these directions vanishes. As an intermediate step, we establish the existence of families of marked hyperk{\"a}hler manifolds over arbitrary curves having a prescribed period map to the corresponding period domain. This generalizes a recent theorem of Greb and Schwald [GS24] to the case of hyperk{\"a}hler manifolds of arbitrary dimensions and nonnecessarily compact curves. Finally, using Nevanlinna theory, we establish restrictions on families of hyperk{\"a}hler manifolds over C having positive Hodge bundle.

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