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Mésures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. (French) Zbl 0579.32012
Let X be an irreducible n-dimensional Stein space and \(\phi\) : \(X\to [- \infty,R[\) a continuous psh (plurisubharmonic) exhaustion function. Each level set \(S(r)=\{x\in X;\quad \phi (x)=r\},\) \(r<R\), is shown to carry an intrinsic positive measure \(\mu_ r\); the measure \(\mu_ r\) is given by the (2n-1)-form \((dd^ c\phi)^{n-1}\wedge d^ c\phi\) when \(\phi\) is smooth, and otherwise \(\mu_ r\) is constructed by means of the Monge- Ampère operators introduced by Bedford and Taylor. In this context, a general Lelong-Jensen formula \[ \mu_ r(V)=\int^{r}_{- \infty}dt\int_{\phi <t}dd^ cV\wedge (dd^ c\phi)^{n- 1}+\int_{\phi <r}V(dd^ c\phi)^ n, \] is proved and used to study the growth and convexity properties of plurisubharmonic or holomorphic functions. If \((dd^ c\phi)^ n=0\) on \(\{\phi >r_ 0\}\), the function \(r\to \mu_ r(V)\) is shown to be convex and increasing in the interval \(]r_ 0,R[\). Furthermore, if the volume \(\tau (r)=\int_{\phi <r}(dd^ c\phi)^ n\) has moderate growth, i.e. if \(\tau (r)=o(r)\), then bounded holomorphic functions on X are constant; using Siegel’s method, we prove also in that case that the ring of holomorphic functions with \(\phi\)- polynomial growth has a transcendance degree \(\leq n\). This last result is then applied in order to obtain a necessary and sufficient geometric criterion characterizing affine algebraic manifolds: X is algebraic iff it has finite Monge-Ampère volume and if the Ricci-curvature of the metric \(dd^ c(\exp (\phi))\) is bounded below by \(-dd^ c\psi,\) where \(\psi \leq A\phi +B.\)

MSC:
32C30 Integration on analytic sets and spaces, currents
32U05 Plurisubharmonic functions and generalizations
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
32J10 Algebraic dependence theorems
32J99 Compact analytic spaces
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