Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Guenancia, Henri

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1291-1330.

Soit $X$ une variété compacte kählerienne et $\Delta$ un $ℝ$-diviseur dont le support est à croisements normaux simples et à coefficients entre $1/2$ et $1$. En supposant $K_X+\Delta$ ample, on prouve l’existence et l’unicité d’une métrique de Kähler-Einstein à courbure négative sur $X\setminus \mathrm{Supp}\left(\Delta \right)$ ayant des singularités mixtes Poincaré et coniques suivant les coefficients de $\Delta$. Nous appliquons ensuite ce résultat pour prouver un théorème d’annulation concernant certains champs de tenseurs holomorphes naturellement attachés à la paire $(X,\Delta )$.

Let $X$ be a compact Kähler manifold and $\Delta$ be a $ℝ$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \mathrm{Supp}\left(\Delta \right)$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta$. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.

Classification : 32Q05, 32Q10, 32Q15, 32Q20, 32U05, 32U15
Keywords: Kähler-Einstein metrics, cone singularities, Poincaré singularities, cusps, orbifold tensors, complex Monge-Ampère equation
Mot clés : métriques de Kähler-Einstein, singularités coniques, singularités Poincaré, cusps, tenseurs orbifoldes, équation de Monge-Ampère complexe

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