By Mike Field

This self-contained and comparatively simple creation to features of numerous advanced variables and intricate (especially compact) manifolds is meant to be a synthesis of these themes and a huge creation to the sphere. half I is acceptable for complex undergraduates and starting postgraduates when half II is written extra for the graduate scholar. The paintings as a complete should be beneficial to expert mathematicians or mathematical physicists who desire to gather a operating wisdom of this sector of arithmetic. Many workouts were integrated and certainly they shape a vital part of the textual content. the necessities for figuring out half i might be met through any arithmetic pupil with a primary measure and jointly the 2 elements offer an advent to the extra complex works within the topic.

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**Extra resources for Several Complex Variables and Complex Manifolds II**

**Sample text**

Relations between the dimension of the space of holomorphic 2. sections of a holomorphic line bundle L on H and other invariants of L and H. Conditions for the existence of sufficiently many holomorphic 3. sections of a holomorphic line bundle L on H for it to determine an embedding of H in projective space. Geometric genus. Let H be a compact complex manifold of dimension m and let K(M) denote the canonical bundle A'5T11* of H. We define the geometric genus Pg(M) of M to be The geometric genus is obviously a biholomorphic invariant.

And With respect to the self—conjugate basis that we have constructevery X ed on cE may be written uniquely in the form m X 1. ,z ) are called self—conjugate coordinates on Suppose that F is another complex vector space with complex basis C and associated bases C, bases B and C. with respect to the Then [A*] [Ai C* as described above for the have matrix Let A basis B of E. [aji]; [au]; [A] [iji];[A] where the matrices are computed relative to the appropriate bases associated to B and C. ,j5) ir) satisfying satisfying 25.

2. + Prove 5(fs) — f More generally, if Cr,s(M), + prove that The Dolbeault-Grothendieck Lensiia. §8. This section is devoted to the proof of an important result that plays the same r6le in the theory of complex manifolds as the lemma does in the cohomology of differential manifolds. 1. open polydisc in (p,q 0). (Dotheault-Grothendleck lemma). and suppose that f Let D be an satisfies 0 Then if W is any relatively compact open subset of D there such that au exists u s Proof. theorem The f on W. is proved inductively.