Let X be a nonsingular projective n-fold (n>1) of general type with ample canonical bundle or Fano projective manifold over an algebraic closed field of any characteristic. We will introduce a new method to get a bunch of inequalities in terms of Chern classes by Schubert calculus in the Chow group of Grassmannian under Gauss map. As an application, we give an affirmative answer to a generalized open question posted by Hunt on geography problem on nonsingular projective manifolds to all dimensions. This is a joint work with Hao Sun.