We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein–Lazarsfeld–Mustaţa–Nakamaye–Popa. As an application, we show that finite time non-collapsing singularities of the Kahler–Ricci flow on compact Kahler manifolds always form along analytic subvarieties, thus answering a question of Feldman–Ilmanen–Knopf and Campana. We also extend the second author’s results about noncollapsing degenerations of Ricci-flat Kahler metrics on Calabi–Yau manifolds to the nonalgebraic case.

Kähler currents and null loci

We give a new proof of Grauert’s theorem on Oka’s principle [Gra1, Gra2, Gra3] (see also [C]) in the case of (smooth) Stein manifolds, which does not use induction over the base dimension Instead we use induction over the levels of a strictly plurisubharmonic exhausting function (Grauert’s bump method) The present paper is an edited version of our preprint [H-L2] from 1986 (which is difficult to find and of bad printing quality) We did not publish the paper in a journal at that time, because we planned to write a book containing it But the book has not been written until now On the other hand, in the meantime, some interest to this proof appeared (see, eg, the work of Gromov [Gro]) Therefore we think a publication of our proof could be useful even with a delay of 10 years This proof gives also some Oka principle for arbitrary pseudoconvex manifolds (see Theorem 13 below) For vector bundles (vector bundles without further mention will refer to topological vector bundles with fibre Cr ) this Oka principle contains the following

The oka-grauert principle without induction over the base dimension

Certain Generating Subspaces for Vertex Operator Algebras

Let X be a complex space and A c= X a closed subset. A is called complete locally pluripolar [6] if for any x0 e A there is an open neighbourhood U of x0 and a plurisubharmonic function φ : U —> [— oo, oo) such that A n U = {φ = — oo}. Clearly any closed analytic subset of X is complete locally pluripolar (see [6] for some non-trivial examples). There is also [8] the notion of locally pluripolar set: the set A is called locally pluripolar if locally A is contained in {φ = —oo} for some plurisubharmonic function φ. It has been proved in [8] that in a Stein space X any locally pluripolar set A c: X is globally pluripolar, i. e. there exists a plurisubharmonic function ψ on X with Αα{ψ = — oo}.

Complete locally pluripolar sets.

Suppose that X is an analytic subvariety of a Stein manifold M and that φ is a plurisubharmonic (psh) function on X which is dominated by a continuous psh exhaustion function u of M . Given any number c > 1, we show that φ admits a psh extension to M which is dominated by cu on M . We use this result to prove that any ω-psh function on a subvariety of the complex projective space is the restriction of a global ω-psh function, where ω is the Fubini-Study Kahler form. Introduction Let X ⊂ Cn be a (closed) analytic subvariety. In the case when X is smooth it is well known that a plurisubharmonic (psh) function on X extends to a psh function on Cn [Sa] (see also [BL, Theorem 3.2]). Using different methods, Coltoiu generalized this result to the case when X is singular [Co, Proposition 2]. In this article we follow Coltoiu’s approach and show that it is possible to obtain extensions with global growth control: Theorem A. Let X be an analytic subvariety of a Stein manifold M and let φ be a psh function on X. Assume that u is a continuous psh exhaustion function on M so that φ(z) 1 there exists a psh function ψ = ψc on M so that ψ |X = φ and ψ(z) R− ρ(z) for all z ∈ CN \D with ‖z‖ = R. (ii) χ(R) > u(r(z)) − ρ(z) for all z ∈ ∂D with ‖z‖ = R. Then ũ(z) = { max{u(r(z)), χ(‖z‖) + ρ(z)}, if z ∈ D, χ(‖z‖) + ρ(z), if z ∈ CN \D, is a continuous psh exhaustion function on CN and ũ = u on V . Employing the methods of Coltoiu [Co] we now construct psh extensions with growth control over bounded sets in Cn. Proposition 1.2. Let χ be a psh function on a subvariety X ⊂ Cn and let v be a continuous psh function on Cn with χ 0, there exists a psh function χ = χR on C n so that χ | X = χ and χ(z) 1 is an increasing sequence defined in terms of the mj’s. Theorem A will follow by showing that it is possible to choose {mj} rapidly increasing so that lim γj is arbitrarily close to 1. We fix next an increasing sequence {mj}j≥−1 so that m−1 = m0 = 0 γj−1 > 1 for all j > 1. Proposition 1.3. Let X, φ, u be as in Theorem A with M = Cn, and let {mj}, {γj} be as above. There exists a psh function ψ on Cn so that ψ |X = φ and for all z ∈ Cn we have ψ(z) φj(z) for z ∈ X , ∫ X∩Kj−1 (ψj − φj) < 2 . (4) ψj(z) ≥ ρj(z) for z ∈ Dj , ψj(z) = ρj(z) for z ∈ C \Dj. (5) ψj(z) < ψj−1(z) for z ∈ Kj−1, where ψ0 = ρ0 = max{u, 0}. (6) Here the integral in (4) is with respect to the area measure on each irreducible component, i.e. ∫

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Extension of plurisubharmonic functions with growth control

The conclusion "obtained from a Stein space by blowing up finitely many points" means precisely that there exist: a compact analytic set S C X with d imxS>0 for any x ~ S; a Stein space 14, a finite set A C Y and a proper holomorphic map p : X ~ Y inducing a biholomorphism X\S ~ Y\A and which satisfies p, C!~x ~ Cr~r. Following a customary terminology X is called an 1-convex space, S its exceptional set and Y the Remmert reduction of X. Theorem 1.1. was conjectured by Fornaess-Narashimhan in [5]. In this paper we prove the converse:

Strongly Plurisubharmonic Exhaustion Functions on 1-Convex Spaces

Traces of Runge domains on analytic subsets

Let D C Cn , n > 3, be an open set such that for any linear hyperplane H C Cn the intersection H n D is Stein. According to a theorem of Lelong [13] it then follows, that D is itself Stein. Using the solution of the Levi problem by Docquier-Grauert [7], this result can easily be generalized to open subsets in Stein manifolds. Therefore, it is natural to raise the following question (see also [4]):

Mihnea Coltoiu

Papers

Strongly Plurisubharmonic Exhaustion Functions on 1-Convex Spaces

Traces of Runge domains on analytic subsets

Open sets with Stein hypersurface sections in Stein spaces

Coverings of 1-convex manifolds with 1-dimensional exceptional set

On Levi's problem on complex spaces and envelopes of holomorphy