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In this article, we get properties for singular (dual) Nakano semi-positivityand obtain singular type vanishing theorem involving $L^2$-subsheaves on weaklypseudoconvex manifolds by $L^2$-estimates and $L^2$-type Dolbeaultisomorphisms. As applications, Fujita's conjecture type theorem with singularHermitian metrics is presented.

Nakano-Nadel type, Bogomolov-Sommese type vanishing and singular dual Nakano semi-positivity