The Chern invariants for parabolic bundles at multiple points (Q2844292)

Let \(X'\) be a smooth complex surface and let \(D' = D'_1+\cdots +D'_k\subset X\) be an effective reduced divisor with smooth components. Let \(E'\) be a vector bundle on \(X'\) provided with a filtration and a parabolic weight on each \(D'_i\). If \(D'\) has normal crossing, then \(E'\) is a locally abelian parabolic bundle on \((X',D')\) for which the parabolic Chern classes were computed (see. e.g., [\textit{N. Borne}, Indiana Univ. Math. J. 58, No. 1, 137--180 (2009; Zbl 1186.14016); \textit{J. N. N. Iyer} and \textit{C. T. Simpson}, Math. Ann. 338, No. 2, 347--383 (2007; Zbl 1132.14006); \textit{C. H. Taher}, Manuscr. Math. 132, No. 1-2, 169--198 (2010; Zbl 1210.14046)]). Here the author considers the general case (with each \(D'\) smooth) taking an embedded resolution \((X,D)\) of \((X',D')\) obtained with finitely many blowing ups. He get an extension of \(E'\) from \(X\setminus D = X'\setminus D'\) to \(X\) better than the other ones and get parabolic Chern classes for this extension satisfying a local Bogomolov-Gieseker inequality.