Abstract
Abstract We define a dual of the Chow transformation of currents on the complex projective space. This transformation factorizes a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear diferential operator. In such a way we complete the general scheme of integral geometry for the Chow transformation. On another hand we prove the existence of a well defined closed positive conormal current associated to every closed positive current on the projective space. This is a consequence of the existence of a dual current, defined on the dual projective space. This allows us to extend to the case of a closed positive current the known inversion formula for the conormal of the Chow divisor of an effective algebraic cycle.
Highlights
For T a (q, q)-current on PN, we denote by C(T) the Chow transform of T, which is a (, )-current on the Grassmannian Gq−,N of (q − )-dimensional projective subspaces of PN
We de ne a dual of the Chow transformation of currents on the complex projective space. This transformation factorizes a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear di erential operator
In such a way we complete the general scheme of integral geometry for the Chow transformation
Summary
For T a (q, q)-current on PN, we denote by C(T) the Chow transform of T, which is a ( , )-current on the Grassmannian Gq− ,N of (q − )-dimensional projective subspaces of PN. We prove the existence of a dual integral transform C∗, de ned for ( , )-currents on Gq− ,N, with values in. C is injective and a left inverse for C is PC∗ In this way we complete for the Chow transformation the general scheme of integral geometry (see [18]). We know by [24] that the Chow transformation of currents de ned on a complex projective manifold is injective and we recall here the proof by slicing of this property (see Proposition 2 and Lemma 8). When we take for T the integration current [Z] associated to an algebraic cycle Z in PN of codimension q, the injectivity of C∗C is a consequence of the following result (see Proposition 10). Con(T) is a well de ned closed positive current on T∗PN (see Proposition 9 and Theorem 4). We retrieve a distribution on the linear group GL(N + , C), which satis es an equivalent system of PDE (see Theorem 3)
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