An extension of Kodaira vanishing in arbitrary codimension

One of the most celebrated theorems in complex algebraic geometry is Kodaira vanishing, together with its extension due to Kawamata
and Viehweg. In this talk I will discuss about a joint work with Greg G.Smith, wwhere we generalize Kodaira vanishing to arbitrary codimension. I will present a few applications to this work; by giving answers to questions of projective normality and bounding the multigraded regularity.
I will also dicuss some future projects that derive nicely from this work.