A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties

Druel, Stéphane; Guenancia, Henri; Păun, Mihai

Comptes Rendus. Mathématique, Complex algebraic geometry, in memory of Jean-Pierre Demailly, Tome 362 (2024), pp. 93-118

Let $X$ be a $\mathbb{Q}$-Fano variety admitting a Kähler–Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler–Einstein $\mathbb{Q}$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_X$ by ${𝒪}_X$ is polystable with respect to the anticanonical polarization.

Soit X une variété $\mathbb{Q}$-Fano admettant une métrique de Kähler–Einstein. Nous montrons, qu’à un revêtement fini quasi-étale près, X est un produit de variétés $\mathbb{Q}$-Fano admettant une métrique de Kähler–Einstein dont le fibré tangent est stable relativement au diviseur anticanonique. La démonstration repose notamment sur un théorème de décompostion pour les feuilletages algébriquement intégrables. Nous montrons également que l’extension canonique de $T_X$ par ${𝒪}_X$ est polystable à nouveau relativement au diviseur anticanonique.

Classification : 14B05, 14J45, 32Q20, 37F75
Keywords: $\mathbb{Q}$-Fano varieties, singular Kähler–Einstein metrics, stable reflexive sheaves, algebraically integrable foliations
Mots-clés : Variétés $\mathbb{Q}$-Fano, métriques de Kähler–Einstein singulières, faisceaux réflexifs stables, feuilletages algébriquement intégrables

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