Nonproper intersection products and generalized cycles

We develop intersection theory in terms of the $${{\mathscr {B}}}$$ B -group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $${{\mathscr {B}}}$$ B -classes have well-defined multiplicities at each point. We focus on a $${{\mathscr {B}}}$$ B -analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global $${{\mathscr {B}}}$$ B -classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce $${{\mathscr {B}}}$$ B -analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $${{\mathscr {B}}}$$ B -variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.