On degeneracy loci (Q2785292)

Let \(X_r\) be the degeneracy locus of rank \(r\) of a morphism \(\varphi\) of vector bundles over a smooth irreducible variety \(X\), and \({\mathcal I}_{X_r}\) the ideal sheaf of \(X_r\).NEWLINENEWLINENEWLINEWe quote from the author's introduction: Our aim is to study the cohomology of \({\mathcal I}_{X_r}\). In particular we want to know if \(H^1 ({\mathcal I}_{X_r})=0\), since this implies the connectedness of \(X_r\). For this, we use resolutions of the ideal sheaves. Indeed, when one has a resolution NEWLINE\[NEWLINE0\to K^l \to K^{l-1} \dots \to K^1\to {\mathcal I}_{X_r} \to 0NEWLINE\]NEWLINE then \(H^{i+k} (K^i)=0\) for all \(i\) and fixed \(k\geq 0\) implies \(H^{k+1} ({\mathcal I}_{X_r}) =0\), as can be seen by cutting the resolutions into short exact sequences. The Dolbeault cohomology of \({\mathcal I}_{X_r}\) can be studied in the same way. In this way, one can partially settle a conjecture of \textit{W. Fulton} and \textit{R. Lazarsfeld} [Acta Math. 146, 271-283 (1981; Zbl 0469.14018)] concerning the connectedness of \(X_r\) for symmetric maps \(\varphi: E^*\to E\) with ample vector bundle \(S^2E\), at least for certain types of varieties \(X\) including the abelian and toric ones.