Hyperkahler SYZ conjecture and semipositive line bundles

Abstract

Let $M$ be a compact, holomorphic symplectic Kaehler manifold, and $L$ a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of $L$ is effective. This result is related to the hyperkaehler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if $L$ is not big.

21 pages, v. 2.0, many references added, many minor corrections

References in corpus (6)

• $L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)
• Geometric stability of the cotangent bundle and the universal cover of a projective manifold
• Higher dimensional Zariski decompositions
• Non-algebraic Hyperkaehler manifolds
• The Kaehler cone of a compact hyperkaehler manifold
• Rational curves on holomorphic symplectic fourfolds

Cited by in corpus (4)

• Kobayashi pseudometric on hyperkahler manifolds
• Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations
• Families of Lagrangian fibrations on hyperkaehler manifolds
• Parabolic nef currents on hyperkaehler manifolds