Finsler geometry of projectivized vector bundles (Q1890283)

This is a long paper. As in many theorems the words ample (nef) are used we shall give here the definition: A holomorphic line bundle \(L\) over \(M\) is said to be ample (resp. nef) if there exists a Hermitian metric \(h\) along the fibers such that the first Chern form \(c_{1}(L,h)\) is positive definite (resp. positive semi-definite). A holomorphic vector bundle \(E\) of rank \(r>2\) is ample (resp. nef) if the line bundle \({\mathcal L}_{P(E^{*})}\) over \(P(E^{*})\) is ample (resp. nef). The dual bundle \(E^{*}\) is said to be ample (resp. nef) if the line bundle \({\mathcal L}_{P(E)}\) over \(P(E) = E_{*}/C^{*}\) is ample (resp. nef). The main theorem is the following: Let \(E\) be a rank \(r\geq 2\) holomorphic vector bundle over a compact complex manifold \(M\). For any positive integer \(k\), denote by \(\odot ^{k}E\) the \(k\)-fold symmetric product and by \({\mathcal L}_{P(\odot ^{k}E)}\) the dual of the tautological line bundle over the projectivized bundle \(P(\odot ^{k}E)\). Then the following statements are equivalent: (1) \(E^{*}\) is ample; (2) \({\mathcal L}_{P(E)}\) is ample; (3) \(\odot ^{k}E^{*}\) is ample for some positive integer \(k\); (4) \({\mathcal L}_{P(\odot ^{k}E)}\) is ample for some positive integer \(k\); (5) \(\odot ^{k}E^{*}\) is ample for all positive integer \(k\); (6) \({\mathcal L}_{P(\odot ^{k}E)}\) is ample for all positive integer \(k\); (7) there exists a Finsler metric along the fibers of \(E\) with negative mixed holomorphic bisectional curvature; (8) for some positive integer \(k\) there exists a Finsler metric along the fibers of \(\odot ^{k}E\) with negative mixed holomorphic bisectional curvature; (9) for all positive integers \(k\) there exists a Finsler metric along the fibers of \(\odot ^{k}E\) with negative mixed holomorphic bisectional curvature; (10) there exists a positive integer \(m\) and a Hermitian metric along the fibers of \(\odot ^{m}E\) with negative mixed holomorphic bisectional curvature. The theorem is also formulated in the dual space. To obtain this theorem the Riemannian metric on \(TTM\), the tangent bundle of a holomorphic vector bundle, the tangent bundle of a projectivized vector bundle, curvature of tensor products of vector bundles and Finsler metrics are studied.