We give a partial answer to the effective non-vanishing problem for algebraic surfaces in positive characteristic, and also give counterexamples for the Kawamata-Viehweg vanishing and the logarithmic semipositivity on ruled surfaces in positive characteristic.

On a compact manifold $M$, we consider the affine space $A$ of non self-adjoint perturbations some invertible elliptic operator acting on sections Hermitian bundle, by differential lower order. We construct and classify all complex analytic functions Fr\'echet vanishing exactly over elements, having minimal order which are obtained local renormalizations, concept coming from quantum field theor...