East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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RELATIVE TWISTED KÄHLER-RICCI FLOWS ON FAMILIES OF COMPACT KÄHLER MANIFOLDS

  • Received : 2021.07.30
  • Accepted : 2021.09.23
  • Published : 2021.09.30
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Abstract

Let p : X → D be a proper surjective holomorphic submersion where X is a Kähler manifold and D is the unit disc in ℂ. Let Ω be a d-closed semi-positive real (1, 1)-form on X. If each Xs := p-1(s) for s ∈ D satisfies $-c_1(X_s)+{\Omega}{\mid}_{X_s}$ is Kähler, then the Kähler-Ricci flow twisted by ${\Omega}{\mid}_{X_s}$ has a long time solution by Cao's theorem. This family of twisted Kähler-Ricci flows induces a relative Kähler form ω(t) on the total space X. In this paper, we prove that the positivity of ω(t) is preserved along the twisted Kähler-Ricci flow.

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Acknowledgement

This work was supported by a 2-Year Research Grant of Pusan National University.

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