  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
hyperk\"{a}hler varieties of dimension $\leq 6$; (ii) all known hyperk\"{a}hler
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 hyperk\"{a}hler
varieties and Calabi-Yau varieties. We prove that the fourth Todd classes are
"fakely effective" for all hyperk\"{a}hler varieties and general complete
intersection Calabi-Yau varieties in products of projective spaces.
