## Abstract

We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if p : W (X; B) → Y is a Kähler fiber space such that (X_{y} B/_{Xy}) is generically klt, K_{X=Y} + B is relatively trivial and p_{* }(m(K_{X=Y} + B)) is Hermitian flat for some suitable integer m, then p 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 p : (X; B) → Y of klt pairs (X_{y}; B_{y}) such that k(K_{Xy} + B_{y}) = 0. Finally, we disprove a folkore conjecture by exhibiting a one-dimensional family of elliptic curves whose relative (Ricci-)flat metric is not semipositive.

Original language | English (US) |
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Pages (from-to) | 633-679 |

Number of pages | 47 |

Journal | Journal of the European Mathematical Society |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - 2023 |

## Keywords

- Kähler fiber space
- conic Kähler metrics
- direct image of log pluricanonical bundles
- log Calabi-Yau manifolds

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics