Some Remarks on the Semipositivity Theorems

  • Osamu Fujino

    Osaka University, Japan
  • Taro Fujisawa

    Tokyo Denki University, Japan
  • Morihiko Saito

    Kyoto University, Japan


We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semipositivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semipositivity theorem for admissible variations of mixed Hodge structure in [FF] by using the theories of Cattani, Kaplan, Schmid, Steenbrink, and Zucker. This generalizes Kawamata's classical result in the pure case.

Cite this article

Osamu Fujino, Taro Fujisawa, Morihiko Saito, Some Remarks on the Semipositivity Theorems. Publ. Res. Inst. Math. Sci. 50 (2014), no. 1, pp. 85–112

DOI 10.4171/PRIMS/125