Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds
[Classes pseudo-effectives sur les variétés irréductibles holomorphes symplectiques]
Denisi, Francesco Antonio1
1Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris (France)
Annales de l'Institut Fourier, Online first, 33 p.

We show that Kovács’ result on the cone of curves of a K3 surface generalizes to any projective irreducible holomorphic symplectic manifold $X$. In particular, we show that if $\rho (X)\ge 3$, the pseudo-effective cone $\bar{\mathrm{Eff} (X)}$ is either circular or equal to $\overline{\sum _{E}\mathbf{R}^{\ge 0} [E]}$, where the sum runs over the prime exceptional divisors of $X$. The proof goes through hyperbolic geometry and the fact that (the image of) the Hodge monodromy group $\mathrm{Mon}^2_{\mathrm{Hdg}}(X)$ in $\mathrm{O}^+(N^1(X))$ is of finite index. If $X$ belongs to one of the known deformation classes, carries a prime exceptional divisor $E$, and $\rho (X)\ge 3$, we explicitly construct an additional integral effective divisor, not numerically equivalent to $E$, with the same monodromy orbit as that of $E$. To conclude, we provide some consequences of the main result of the paper, for instance, we obtain the existence of uniruled divisors on certain primitive symplectic varieties.

Nous montrons que le résultat de Kovács sur le cône des courbes d’une surface K3 se généralise à toute variété irréductible holomorphe symplectique projective $X$. En particulier, nous démontrons que si $\rho (X) \ge 3$, le cône pseudo-effectif $\bar{\mathrm{Eff}(X)}$ est soit circulaire, soit égal à $\overline{\sum _{E}\mathbf{R}^{\ge 0} [E]}$, où la somme est prise sur les diviseurs premiers exceptionnels de $X$. La preuve s’appuie sur la géométrie hyperbolique et sur le fait que (l’image de) le groupe de monodromie de Hodge $\mathrm{Mon}^2_{\mathrm{Hdg}}(X)$ dans $\mathrm{O}^+(N^1(X))$ est d’indice fini. Si $X$ appartient à l’une des classes de déformation connues, possède un diviseur premier exceptionnel $E$ et si $\rho (X) \ge 3$, nous construisons explicitement un autre diviseur effectif entier, non numériquement équivalent à $E$, qui appartient à la même orbite de monodromie que $E$. Pour conclure, nous donnons quelques conséquences du résultat principal de l’article. Par exemple, nous établissons l’existence de diviseurs uniréglés sur certaines variétés primitives symplectiques.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3769
Classification : 14J42, 14Exx
Keywords: Irreducible holomorphic symplectic manifolds, cones of divisors, pseudo-effective cone, Mori dream spaces, uniruled divisors.
Mots-clés : Variétés irréductibles holomorphes symplectiques, cônes de diviseurs, cône pseudo-effectif, espaces de rêve Mori, diviseurs uniréglés.

Denisi, Francesco Antonio  1

1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris (France) 
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Denisi, Francesco Antonio. Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds. Annales de l'Institut Fourier, Online first, 33 p.

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