Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds

We show that Kovács’ result on the cone of curves of a K3 surface generalizes to any projective irreducible holomorphic symplectic manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . In particular, we show that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , the pseudo-effective cone <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="normal">Eff</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> is either circular or equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:msub> <mml:mo>∑</mml:mo> <mml:mi>E</mml:mi> </mml:msub> <mml:msup> <mml:mi mathvariant="bold">R</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>E</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , where the sum runs over the prime exceptional divisors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . The proof goes through hyperbolic geometry and the fact that (the image of) the Hodge monodromy group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal">Mon</mml:mi> <mml:mi mathvariant="normal">Hdg</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal">O</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is of finite index. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> belongs to one of the known deformation classes, carries a prime exceptional divisor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , we explicitly construct an additional integral effective divisor, not numerically equivalent to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> , with the same monodromy orbit as that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> . 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.