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

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