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Courants résidus et formule de King

We characterize the existence of global liftings of local data of meromorphic forms, satisfying a family of compatibility conditions on their polar sets, in terms of obstructions belonging to the cohomology H.(X, Ω.), where X is the ambient manifold. These obstructions are constructed canonically, using residue-principal value operators.

On the global lifting of meromorphic forms

Let X be a Riemann surface and let ω be a meromorphic differential form of degree 1 on X; in the neighborhood of a point where z is a local coordinate, we have ω = f(z)dz, where f is a meromorphic function. Let Y = {a l } l∈I be the set of poles of ω, wnd res a j (ω) the Cauchy residue of ω at a j . Then, for J a finite subset of I, if for any j ∈ J, γ j is a positively oriented circle whose center is a j and if which γ j is the boundary does not meet Y in any point different from a j and if (n j ) j∈J is a family of elements of ℤ, ℝ or ℂ, we have the residue formula.

General Theory of Multidimensional Residues

This thesis concerns various aspects of the theory of residue currents Particularly, we study residue currents on singular varieties and duality theorems
for such currents On a singular variety, there are various notions of holomorphic functions
In Paper I, we study how to extend the definition of Coleff-Herrera products
and Bochner-Martinelli type residue currents from the case of strongly holomorphic functions to weakly holomorphic functions, and investigate how various properties known in the strongly holomorphic case transform into the
weakly holomorphic case The duality theorem for Coleff-Herrera products on a complex manifold is
one of the key properties of the Coleff-Herrera product On a singular variety,
the duality theorem for Coleff-Herrera products is in general false In Paper II,
we discuss necessary and sufficient conditions for when the duality theorem
holds, and in particular we show that on any singular variety, one can find
examples where the duality principle fails Another important property of the Coleff-Herrera product is the transformation law In Paper III, we describe a comparison formula for Andersson-Wulcan currents, generalizing the transformation law Applications of this formula include giving a proof by means of residue currents of a theorem of Hickel
related to the Jacobian of a holomorphic mapping, and constructing a current
on a singular variety satisfying the duality principle The failure of the duality theorem for Coleff-Herrera products leads to the
search for an alternative In Paper IV, we elaborate on the construction in Paper III, of a current satisfying the duality principle for an arbitrary ideal In
particular, using the comparison formula, we explain how we can view this
construction as an intrinsic construction on the variety, generalizing the construction of Andersson and Wulcan

http://publications.lib.chalmers.se/records/fulltext/167866/167866.pdf

Residue currents on singular varieties

Courants résidus et opérateurs de Monge–Ampère

According to a classical result of Weil [15,1, a divisor ~ of a smooth n-dimensional projective variety X is homologous to zero if and only if it is the residue of a closed meromorphic 1-form on X. Griffiths proved recently [9, pp. 3-8,1 that a 0-cycle ~t of X is homologous to zero if and onty if it is the Grothendieck residue of a meromorphic n-form & on X having poles in the union of a family of complex hypersurfaces Yl . . . . . II, of X, such that (~ Y~ is 0-dimensional and contains the support of ~. We show in this paper (Theorem 3.7) that, in fact, any q-dimensional algebraic cycle ~ of X, 0< q < n, is the analytic residue of a semimeromorphic (n-q)-form on X, having poles in the union of a family ~ = {Y1 . . . . . Yn-q} of hypersurfaces in X such that N ~ contains the support of ~. The form & is not closed, in general, but its differential verifies

D. Lieberman

Papers

Algebraic Cycles as Residues of Meromorphic Forms