Bergman kernels and the pseudoeffectivity of relative canonical bundles

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A criterion for the pseudoeffectivity of (twisted ) relative canonical bundles of surjective projective maps.

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A decomposition theorem for projective manifolds with nef anticanonical bundle (2017, methodology): “In fact, by using [Ber09, BP08], over some Zariski open set Y 0 of Y , we can construct a continuous metric h on f ⋆ ( O X ( mK X/Y + L )) ⊗ O Y 0 such that it is positively curved.”
Kodaira dimension of algebraic fiber spaces over abelian varieties (2015, unknown): “In this context we recall the construction of the Bergman metric e−φX/Y on the bundle KX/Y + L; we refer to [2] for further details.”
A solution of an L 2 extension problem with an optimal estimate and applications (2015, background): “The above result is due to [5] and [3] in the case of example (1) and due to [7] in the case of example (2).”
A solution of an $L^{2}$ extension problem with optimal estimate and applications (2013, background): “The above result is due to [5] and [3] in the case of example 1), and due to [8] in the case of example 2).”
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, unknown): “By [4], II(t) is subharmonic on B1(0) ∗ = B1(0)\{0} and √ −1∆IIdz ∧ dz̄ extends to a positive current on B1(0).”
Positivity of twisted relative pluricanonical bundles and their direct images (2014, methodology): “We stress on the fact that the statement above is implicit in [3], it is not mentioned explicitly.”