Trace formulae for curvature of Jet Bundles over planar domain

For a domain \Omega in \mathbb{C} and an operator T in \mathcal{B}_n(\Omega), Cowen and Douglas construct a Hermitian holomorphic vector bundle E_T over \Omega corresponding to T. The Hermitian holomorphic vector bundle E_T is obtained as a pull-back of the tautological bundle S(n,\mathcal{H}) defined over \mathcal{G}r(n,\mathcal{H}) by a nondegenerate holomorphic map z\mapsto {\rm{ker}}(T-z) for z\in\Omega. To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are somewhat difficult to comprehend. They have given a set of invariants to determine if two rank n Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle \mathcal{J}_k(\mathcal{L}_f), we have shown that the curvature of the line bundle \mathcal{L}_f completely determines the class of \mathcal{J}_k(\mathcal{L}_f). In case of rank n Hermitian Holomorphic vector bundle E_f, We have calculated the curvature of jet bundle \mathcal{J}_k(E_f) and also have generalized the trace formula for jet bundle \mathcal{J}_k(E_f).Comment: 14 page

A note on projective modules over real affine algebras

Let A be an affine algebra over the field of real numbers of dimension d. Let f \in A be an element not belonging to any real maximal ideal of A. Let P be a projective A-module of rank \geq d-1. Let (a,p) \in A_f \oplus P_f be a unimodular element. Then the projective A_f module Q=A_f \oplus P_f/(a,p)A_f is extended from A.Comment: 11 page

Stability result for projective modules over blowup rings

Let R be an affine algebra of dimension n \geq 3 over an algebraically closed field k. Suppose char k =0 or char k =p \geq n. Let g,f_1,...,f_r be a R-regular sequence and A=R[f_1/g,...,f_r/g]. Let P be a projective A-module of rank n-1 which is extended from R. Let (a,p) \in (A \op P) be a unimodular element and Q=A\op P/(a,p)A. Then, Q is extended from R. A similar result for affine algebras over reals are also proved.Comment: 15 page

Flag structure for operators in the Cowen-Douglas class

The explicit description of homogeneous operators and localization of a Hilbert module naturally leads to the definition of a class of Cowen-Douglas operators possessing a flag structure. These operators are irreducible. We show that the flag structure is rigid in the sense that the unitary equivalence class of the operator and the flag structure determine each other. We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.Comment: Announcement, 6 page