Remarks on Kawamata's effective non-vanishing conjecture for manifolds with trivial first Chern classes

Abstract

Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\le 6$; (ii) all known hyperkähler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi-Yau varieties. We prove that the fourth Todd classes are "fakely effective" for all hyperkähler varieties and general complete intersection Calabi-Yau varieties in products of projective spaces.

17 pages. Comments are welcome; v2: Theorem 1.4 & Proposition 3.5 improved; v3: final version, to appear in Math. Zeit

References in corpus (1)

• M-theory on Calabi-Yau Five-Folds

Cited by in corpus (3)

• Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
• Positivity of Riemann-Roch polynomials and Todd classes of hyperkähler manifolds
• The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds