
TL;DR
This paper proves that fiberwise conic K"ahler-Einstein metrics on a fibration of compact K"ahler manifolds induce a positive current on the total space, extending across divisors under certain ampleness conditions.
Contribution
It establishes the positivity and extendability of a current constructed from fiberwise conic K"ahler-Einstein metrics in a family of K"ahler manifolds with divisors.
Findings
The current from fiberwise metrics is positive.
The current extends across divisors.
The current is bounded outside the divisor.
Abstract
Let be an holomorphic surjective map between compact K\"ahler manifolds and let be an effective divisor on with generically simple normal crossings support and coefficients in . Provided that the adjoint canonical bundle of the generic fiber is ample, we show that the current obtained by glueing the fiberwise conic K\"ahler-Einstein metrics on the regular locus of the fibration is positive. Moreover, we prove that this current is bounded outside the divisor and that it extends to a positive current on .
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