Birational positivity in dimension $4$

Taji, Behrouz

Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 203-216.

Dans cet article, nous montrons que pour une variété projective lisse, $X$, de dimension au plus $4$ et de dimension de Kodaira non négative, la dimension de Kodaira des sous-faisceaux cohérents de ${\Omega }^p$ est majorée par la dimension de Kodaira de $X$. Cela implique la finitude du groupe fondamental de $X$ lorsque la dimension de Kodaira de $X$ est nulle et sa caractéristique holomorphe d’Euler est non nulle.

In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of ${\Omega }^p$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

Classification : 14J35, 14E30
Keywords: Kodaira dimension, varieties of Kodaira dimension zero, minimal model theory
Mot clés : dimension de Kodaira, variétés avec la dimension de Kodaira nulle, théorie du modéle minimal

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