Global Koppelman formulas on (singular) projective varieties

Let $i\colon X\to \Pk^N$ be a projective manifold of dimension $n$ embedded in projective space $\Pk^N$, and let $L$ be the pull-back to $X$ of the line bundle $\Ok_{\Pk^N}(1)$. We construct global explicit Koppelman formulas on $X$ for smooth $(0,*)$-forms with values in $L^s$ for any $s$. %The formulas are intrinsic on $X$. The same construction works for singular, even non-reduced, $X$ of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves $\A_X^*$ of $(0,*)$-currents with mild singularities at $X_{sing}$. In particular, if $s\ge \reg X -1$, where $\reg X$ is the Castelnuovo-Mumford regularity, we get an explicit %%% representation of the well-known vanishing of $H^{0,q}(X, L^{s-q})$, $q\ge 1$. Also some other applications are indicated.