Variation of singular Kähler–Einstein metrics: Kodaira dimension zero (with an appendix by Valentino Tosatti)
Junyan Cao
Université Paris 6, FranceHenri Guenancia
Université de Toulouse; CNRS, UPS, Toulouse, FranceMihai Păun
Universität Bayreuth, Germany
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Abstract
We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if is a Kähler fiber space such that is generically klt, is relatively trivial and is Hermitian flat for some suitable integer , then is locally trivial. Motivated by questions in birational geometry, we investigate the regularity of the relative singular Ricci-flat Kähler metric corresponding to a family of klt pairs such that . Finally, we disprove a folkore conjecture by exhibiting a one-dimensional family of elliptic curves whose relative (Ricci-)flat metric is not semipositive.
Cite this article
Junyan Cao, Henri Guenancia, Mihai Păun, Variation of singular Kähler–Einstein metrics: Kodaira dimension zero (with an appendix by Valentino Tosatti). J. Eur. Math. Soc. 25 (2023), no. 2, pp. 633–679
DOI 10.4171/JEMS/1184