Two results on branched coverings of Grassmannians (Q1290845)

The article contains two main results, both about branched coverings of Grassmannians. The first asserts that if \(P: G(r,n) \to \mathbb{P}^{\binom{n}{r}-1}\) is the Plücker embedding of \(r\)-dimensional subspaces of \(\mathbb{C}^n\) and \(n\geq r+2\geq 1\), then there exists a projective manifold \(Y\) and branched covering \(f: Y\to G(r,n)\) of degree \(\dim G(r,n)\) such that there is no branched covering \(f': Y' \to \mathbb{P}^{\binom{n}{r}-1}\) of a projective manifold \(Y'\) with \(f\) the pullback of \(f'\) under \(P\). The second result shows that an obvious and natural way to construct branched coverings only leads to trivial examples. More precisely, if \(G(r,n')\subset G(r,n)\) is the immersion defined by a field embedding \(\mathbb{C}^{n'} \subset \mathbb{C}^n\), and \(X\) is a subvariety of \(G(r,n)\) whose homology class is a positive multiple of the homology class of \(G(r,n')\). Then \(X\) is a translation of \(G(r,n')\) by an automorphism of \(G(r,n)\).