We give a relation between two theories of improper intersections, of Tworzewski and of Stückrad-Vogel, for the case of algebraic curves. Given two arbitrary quasiprojective curves V₁,V₂, the intersection cycle V₁ ∙ V₂ in the sense of Tworzewski turns out to be the rational part of the Vogel cycle v(V₁,V₂). We also give short proofs of two known effective formulae for the intersection cycle V₁ ∙ V₂ in terms of local parametrizations of the curves.
We consider the intersection multiplicity of analytic sets in the general situation. We prove that it is a regular separation exponent for complex analytic sets and so it estimates the Łojasiewicz exponent. We also give some geometric properties of proper projections of analytic sets.
We present a construction of an intersection product of arbitrary complex analytic cycles based on a pointwise defined intersection multiplicity.
It is shown that a holomorphically embedded open disk in C2 and a totally real embedded open disk which have a common smooth boundary have nontrivial intersection.
We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic -correspondences. We define an intrinsic logarithmic pseudo-volume form for every pair consisting of a complex manifold and a normal crossing Weil divisor on , the positive part of which is reduced. We then prove that is generically non-degenerate when is projective and ...
Let a smooth projective family and a pseudo-effective line bundle on (i.e. with a non-negative curvature current ). In its works on invariance of plurigenera, Y.-T. Siu was interested in extending sections of (defined over the central fiber of the family ) to sections of . In this article we consider the following problem: to extend sections of . More precisely, we show the following result: assuming the triviality of the multiplier ideal sheaf , any section of extends to ; in other...