Global Koppelman Formulas on (Singular) Projective Varieties

Journal article, 2019

Let i: X→ PN be a projective manifold of dimension n embedded in projective space PN, and let L be the pullback to X of the line bundle OPN(1). We construct global explicit Koppelman formulas on X for smooth (0 , ∗) -forms with values in Ls for any s. The same construction works for singular, even non-reduced, X of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves AX∗ of (0 , ∗) -currents with mild singularities at Xsing. In particular, if s≥regX-1, where regX is the Castelnuovo–Mumford regularity, we get an explicit representation of the well-known vanishing of H,q(X, Ls-q) , q≥ 1. Also some other applications are indicated.