Intrinsic pseudo-volume forms for logarithmic pairs
Thomas Dedieu (Universität Bayreuth)

Claire Voisin defined a variant of the Kobayashi-Eisenman pseudo-volume form
of a complex manifold X, by introducing the notion of holomorphic K-correspondance,
in order to replace holomorphic maps between the unit polydisk D^n and X in the
definition by them. I will study an adaptation of these constructions to the frame of
logarithmic pairs (X, D), where X is a complex manifold, and D is a normal crossing
divisor, the positive part of wich is reduced. I will define an intrinsic pseudo-volume
form Phi_{X,D} for every logarithmic pair (X, D). I will prove on the one hand that
Phi_{X,D} is generically non-degenerate if K_X(D) is ample and the positive part of
D is simply normal crossing (this corresponds to the Griffiths and Kobayshi-Ochiai
theorem in the standard case). On the other hand, I will show the vanishing of Phi_{X,D}
for a large class of pairs with trivial logarithmic canonical bundle, which is an important
step towards an analogous of the Kobayashi conjecture on measure-hyperbolicity.

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