Families of conic Kähler-Einstein metrics

Abstract

Let $p:X\to Y$ be an holomorphic surjective map between compact Kähler manifolds and let $D$ be an effective divisor on $X$ with generically simple normal crossings support and coefficients in $(0,1)$. Provided that the adjoint canonical bundle $K_{X_y}+D_y$ of the generic fiber is ample, we show that the current obtained by glueing the fiberwise conic Kähler-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 $X$.

arXiv comment24 pages; v2: there was a gap in section 5 which led us to weaken the statement on the regularity of the current outside D (bounded instead of smooth)