A generalized defect relation for holomorphic maps into algebraic varieties

We establish a generalized defect relation for holomorphic maps into algebraic varieties intersecting arbitrary families of linearly equivalent divisors. Based on this, we also give an improvement of many previous second main theorems for holomorphic maps intersecting divisors in subgeneral position. Moreover, by the relationship between Diophantine approximation and Nevanlinna theory, the corresponding Schmidt subspace theorems are also given.