Complex Manifolds without Potential Theory: with an appendix by Shiing-shen Chern

By Shiing-shen Chern

From the studies of the second one edition: "The new tools of advanced manifold thought are very helpful instruments for investigations in algebraic geometry, advanced functionality concept, differential operators etc. The differential geometrical equipment of this concept have been built basically below the impression of Professor S.-S. Chern's works. the current ebook is a moment edition... it could actually function an creation to, and a survey of, this idea and is predicated at the author's lectures held on the collage of California and at a summer season seminar of the Canadian Mathematical Congress....
The textual content is illustrated through many examples... The booklet is warmly suggested to everybody attracted to advanced differential geometry." #Acta Scientiarum Mathematicarum, forty-one, 3-4#

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Extra info for Complex Manifolds without Potential Theory: with an appendix on the geometry of characteristic classes

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Ar ) be a symmetric invariant polynomial. PCA) = PCA, ... 46) ----- QCB,A) rPCB,A, ... ) t , we get - PCrl) A2 = ... Let 43 This proves (B). 2). 50). are two COO-sections of the bundle, H(~,n) E is a COO-function having properties corresponding to A complex vector bundle with an hermitian structure is called an hermitian vector bundle. By a partition of unity argument, it can be shown that every complex vector bundle can be given an hermitian structure. 52) Hs = = tHS (H(s i' s» j , and is in turn completely determined by this matrix.

55a) dH wH + Ht(;i 0 , 45 where the subscript s is dropped. 55a) that an admissible connection always exists in an hermitian vector bundle. , QH is skew-hermitian. A frame field unitary, if Hs = I (= s of an hermitian vector bundle is called the unit matrix). , the connection and curvature matrices wand Q are both skew-hermitian. 59) i det

Di3, M and i3 where a. y of is a given real- runs over all real-valued l-forms 52 in M. 1 -l - . 2 l It suffices to prove that As previously let COO r-forms and d. Also let and Cpq definition Cr APq i2H~1,1)(M,Z) = 0 . Ar be the sheaf of germs of complex-valued be the sub sheaf of the germs of be the sheaf of germs of be the sub sheaf of germs of CO =C and COO o. APq Ar closed under Coo-forms of type (p,q) closed under We have the diagram a. +l Or k' C - > Aorl ----+ k tive. Moreover both horizontal sequences are exact.

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