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# Jihun Yum, Characterization of Diederich-Fornaess and Steinness Indices in Complex Manifolds

## April 7 @ 4:00 pm - 6:00 pm KST

Let Ω be a relatively compact pseudoconvex domain in a complex manifold *X* with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by

*DF*(Ω) := sup* _{ρ}* { 0 <

*η*< 1 : -(-

*ρ*)

*is strictly plurisubharmonic on Ω ∩*

^{η}*U*for some neighborhood

*U*of ∂Ω },

*S*(Ω) := inf* _{ρ}* {

*η*> 1 :

*ρ*

*is strictly plurisubharmonic on Ω*

^{η}^{c}∩

*U*for some neighborhood

*U*of ∂Ω },

where *ρ* is a defining function for Ω.

In the previous talk, we have seen that two indices are completely characterized by D’Angelo 1-form when the ambient space is *X *=* C^{n}*. In this talk, we generalize the formulas for a relatively compact pseudoconvex domains in a (general) complex manifold

*X*. Since the formulas do not hold anymore in general, unfortunately, we introduce 4 kinds of each of the Diederich-Fornaess and Steinness indices. Then we give some non-degeneracy conditions for these indices agree. Also, we exam the geometric meaning of the D’Angelo 1-form when the boundary ∂Ω is Levi-flat.