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 (Xy B/Xy) is generically klt, KX=Y + B is relatively trivial and p* (m(KX=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 (Xy; By) such that k(KXy + By) = 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