We show that for very general hypersurfaces in odd-dimensional simplicial projective toric varieties satisfying an effective combinatorial property the Hodge conjecture holds. This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.

中文翻译：

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关于复曲面变种中超曲面的霍奇猜想

我们表明，对于满足有效组合性质的奇数维简单射影复曲面品种中非常普遍的超曲面，霍奇猜想成立。这在Oda猜想和Hodge猜想之间建立了联系。我们还给出了一个明确的标准，该标准取决于要验证的组合条件的非常普遍的超曲面的程度。