Étude d'une équation de Monge-Ampère sur les variétés kählériennes compactes

On a compact Kähler manifold of complex dimension m ? 2, let us consider the change of Kähler metric $\text{g′}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gm</mtext>}}\text{= g}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gm</mtext>}}\text{+ ?}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gm</mtext>}}\text{φ}$. Let F?C^∞(V × R) be a function everywhere > 0 and v a real number ≠ 0. When $\text{0 < C}{}^{\mathrm{?1}}\text{? F(x, t) ? C(¦t¦}{}^{\mathrm{a}}\text{+ 1)}$ for all (x, t) ?V × ] ?∞, t₀], where C and t₀ are constants and $\text{1 ? a <}\text{m}\text{(m ? 1)}$, one exhibits a function φ?C^∞ (V) such that $\text{¦g′∥g¦}{}^{\mathrm{?1}}\text{= e}{}^{\mathrm{<mtext>\nu </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gf</mtext>}}\text{F(x, φ ?}\text{}\text{?}\text{gf) (¦g¦}\text{and}\text{¦g′¦}$ the determinants of the metrics g and $\text{g′,}\text{}\text{?}\text{gf = (}\text{mes}\text{V)}{}^{\mathrm{?1}}\text{∝ φ dV)}$.