Twisted cotangent bundle of Hyperkähler manifolds (with an appendix by Simone Diverio)

Anella, Fabrizio; Höring, Andreas

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1429-1457.

Soit $X$ une variété hyperkählérienne, et $H$ un diviseur ample sur $X$. Nous donnons une borne inférieure en fonction de la forme de Beauville-Bogomolov-Fujiki $q\left(H\right)$ pour la pseudo-effectivité du faisceau cotangent tordu ${\Omega }_X\otimes H$. Si $X$ est équivalente par déformation au schéma de Hilbert ponctuel d’une surface K3, cette borne inférieure peut être calculée explicitement et nous étudions son optimalité.

Let $X$ be a Hyperkähler manifold, and let $H$ be an ample divisor on $X$. We give a lower bound in terms of the Beauville–Bogomolov–Fujiki form $q\left(H\right)$ for the pseudoeffectivity of the twisted cotangent bundle ${\Omega }_X\otimes H$. If $X$ is deformation equivalent to the punctual Hilbert scheme of a K3 surface, the lower bound can be written down explicitly and we study its optimality.

Classification : 14J42, 14J60, 14J28
Keywords: Hyperkähler manifold, cotangent bundle, positivity of vector bundles
Mot clés : Variété hyperkählérienne, fibré cotangent, positivité des fibrés vectoriels

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