Koppelman formulas on Grassmannians

The $${\bar{\partial }}$$-Equation, Duality, and Holomorphic Forms on a Reduced Complex Space

“…Such a weight should satisfy the same properties but with the condition g 0,0 D = 1 replaced by g 0,0 D = Id V , cf. [22] and [3]. If g is a weight with values in…”

Koppelman formulas on flag manifolds and harmonic forms

“…As a linear operator Γ(X ×X, E⊗H i,ζ ) → Γ(X ×X, H i,z ⊗T * 0,1 ) is C ∞ (X ×X)-linear (cf. [3]), and hence defines a section of the appropriate bundle. Proof.…”

“…Weights for line bundles. We recall from [3] that we can form exterior powers, tensor powers, and duals of weights. The following proposition is an immediate consequence of this construction.…”

“…The key ingredient is to have a manifold X admitting the Diagonal Property, i.e., that X ×X admits a holomorphic vector bundle E → X ×X of rank equal to the dimension of X, and a holomorphic section η of E which vanishes to the first order on the diagonal ∆ ⊆ X × X and is nonzero elsewhere. We will therefore be very brief and refer to [3] and [2] for the framework in which our contructions take place, as well as for a historical background on integral formulas.…”

“…The approach of this paper is a straightforward generalization of the one in [3] which deals with Grassmannians. That paper in turn is an application of (a slight generalization of) the method developed in [2] for constructing integral formulas on complex manifold.…”