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By Riaz A Usmani

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3. The dual space to Tx is denoted Tx∗ . Note. The dimension of Tx∗ over C is N . 4. A 1-form ω on of Tx∗ . Example. Let f ∈ C ∞ . For x ∈ (df )x (v) is a map ω assigning to each x in an element , put v(f ), all v ∈ Tx . 23 24 4. Differential Forms Then (df )x ∈ Tx∗ . df is the 1-form on assigning to each x in the element (df )x . Note. dx1 , . . , dxN are particular 1-forms. In a natural way 1-forms may be added and multiplied by scalar functions. 2. Every 1-form ω admits a unique representation N ω Cj dxj , 1 the Cj being scalar functions on .

Z) n 0 We put N (2) an x n . (x) n 0 Note that (1) holds. Let be a rational function holomorphic on σ (x), P (z) , Q(z) (z) P and Q being polynomials and Q(z) 0 for z ∈ σ (x). ) and we define (3) (x) P (x) · Q(x)−1 We again verify (1). Now let be an open set with σ (x) ⊂ and fix ∈ H ( ). 9 that we can choose a sequence {fn } of rational functions holomorphic in such that fn → uniformly on compact subsets of . ) For each n, fn (x) was defined above. We want to define (x) lim fn (x). 2. limn→∞ fn (x) exist in A and depends only on x and choice of {fn }.

Then ω ∈ ∧p,1 ( ) for some p and we have ∈ C∞( ) aI d z¯ 1 ∧ dzI , aI ω for each I. I 0 ¯ ∂ω I,k ∂aI d z¯ k ∧ d z¯ I ∧ dzI . ∂ z¯ k Hence (∂aI /∂ z¯ k )d z¯ k ∧ d z¯ 1 ∧ dzI ∂aI ∂ z¯ k 0 for each k and I . It follows that 0, k ≥ 2, all I. 3 there exists for every I, AI in C ∞ ( neighborhood of n , such that ∂AI ∂ z¯ 1 Put ω˜ I aI and AI dzI ∈ ∧p,0 ( ∂AI ∂ z¯ k 0, k 1 ), 1 being some 2, . . , n. 1 ). ∂¯ ω˜ I,k ∂AI d z¯ k ∧ dzI ∂ z¯ k ω. We proceed by induction. Assume that the assertion of the theorem holds whenever ω is of level ≤ ν − 1 and consider ω of level ν.

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