Journal of the European Mathematical Society (JEMS) (J. Eur. Math. Soc. (JEMS)) (20230101), 25, no.~2, 633-679. ISSN: 1435-9855 (print).eISSN: 1435-9863.
Subject
14 Algebraic geometry -- 14E Birational geometry 14E30 Minimal model program
Much of algebraic geometry concerns families of varieties. In such families, singularities arise frequently and play an important role; while the general fibre may be smooth (or have mild singularities), special fibres may be more singular. The paper under review considers families of (log) Calabi-Yau varieties, where the general fibre is log Calabi-Yau. The authors allow mild (klt) singularities on the general fibre, so that by the singular version of the Calabi-Yau theorem, the general fibre admits a weak Kähler-Einstein metric [P. Eyssidieux, V. Guedj and A. Zériahi, J. Amer. Math. Soc. {\bf 22} (2009), no.~3, 607--639; MR2505296]. Essentially, here the authors consider the ``transverse'' behaviour of these weak Kähler-Einstein metrics, and their behaviour across singularities. \par The main result is a result that gives a strong analytic criterion for the family to be isotrivial: assuming that a natural Hermitian metric on the pushforward of (powers of) the relative log canonical class is actually flat, the family is locally trivial. Further, the fibrewise Kähler-Einstein metrics can be glued to a global positive current. Some applications to questions around the construction of big vector bundles on the base of families are given. \par The techniques are a combination of those involved in the solution of the Calabi conjecture and its variants, along with techniques arising in the study of singular metrics in complex geometry.