Variation of singular Kähler-Einstein metrics: Kodaira dimension zero

Junyan Cao, Henri Guenancia, Mihai Paun, Valentino Tosatti

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)633-679
Number of pages47
JournalJournal of the European Mathematical Society
Issue number2
StatePublished - 2023


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

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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