By M. F. Atiyah (auth.), W.-Y. Hsiang, S. Kobayashi, I. M. Singer, J. Wolf, H.-H. Wu, A. Weinstein (eds.)

This quantity attests to the power of differential geometry because it probes deeper into its inner constitution and explores ever widening connections with different topics in arithmetic and physics. To so much folks Professor S. S. Chern is sleek differential geometry, and we, his scholars, are thankful to him for prime us to this fertile panorama. The goals of the symposium have been to study fresh advancements in geometry and to show and discover new components of study. It used to be our approach of honoring Professor Chern upon the party of his reliable retirement as Professor of arithmetic on the collage of California. This publication is a list of the clinical occasions of the symposium and displays Professor Chern's broad curiosity and effect. The convention additionally mirrored Professor Chern's character. It used to be a major social gathering, energetic but secure, combined with gentleness and reliable humor. we want him strong future health, a longevity, happiness, and a continuation of his terribly deep and unique contributions to arithmetic. I. M. Singer Contents genuine and complicated Geometry in 4 Dimensions M. F. ATIYAH. . . . . . . . . . . . . Equivariant Morse idea and the Yang-Mills Equation on Riemann Surfaces RAOUL BaTT .. eleven Isometric households of Kahler buildings EUGENIO CALABI. . 23 functions of Algebraic Geometry to complete Holomorphic Mappings MARK eco-friendly AND PHILLIP GRIFFITHS. • . . . • . . forty-one The Canonical Map for yes Hilbert Modular Surfaces F. HIRZEBRUCH . . . . . • . . . . . . . . . seventy five Tight Embeddings and Maps. Submanifolds of Geometrical category 3 in EN NICOLAAS H. KUIPER .

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This quantity attests to the power of differential geometry because it probes deeper into its inner constitution and explores ever widening connections with different matters in arithmetic and physics. To so much folks Professor S. S. Chern is smooth differential geometry, and we, his scholars, are thankful to him for best us to this fertile panorama.

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**Extra resources for The Chern Symposium 1979: Proceedings of the International Symposium on Differential Geometry in honor of S.-S. Chern, held in Berkeley, California, June 1979**

**Sample text**

J 2"1)c I .. i -:-: Ij i .. k I - i .. k 2 > 0, then cf>m : Pk(X)~ IPN is birational to its image for m sufficiently large. Less specifically, the hypothesis on the Chern classes always holds for k sufficiently large. This follows because c;(Qi) > 0 for a minimal surface of general type. (d) Examples (1) Smooth hypersurfaces in IP n' Let X be a smooth hypersurface in IP n of degree d. 22) HO(X, Sym k QI) = 0 for all k > 1 if n ;;;. 3. 23) cf>'h. m is a birational embedding for m sufficiently large, for X a surface and d> 16.

Thus since the only way (log kr can occur is when all of the j's are I. ) «k + I)n - I)! (logkf 2: ql+'" +qn=n A'll ... Aq. q's integers ;;. 0 ql! qn! m(k+ I)n-I f«k + (k! X ( (~~ f I)n - I)! )n«k X + I)n - I)! 16) for Fk(qp ... , qn)' we can calculate the leading term explicitly for low dimensions. )(2k+l)! )3(3k+2)! ' 1,1,1 (k) + S 2,1 (k) + S3(k»c 3I In particular, for surfaces the leading term is k = 1, ;! (c; - c k = 2, _1_ (7 c; - 5c 2 ), 43 ,3! 1S) If a section of HO( x, Symi\ ex ® Symi2ex ® ...

2) -00 From the spectral decomposition of Kp, one deduces, for either value of v, that the integral above converges, that J p commutes with Kp, and that J p maps Isometric Families of Kimler Structures 29 T' and T" each bijectively onto the other, so that Furthermore (v = 1,2), J~=~-IJ;, and both J I and J 2 have vanishing covariant derivatives with respect to the metric, so that they define integrable complex analytic structures with respect to the metric, and the metric is Kahlerian with respect to either of the two resulting new complex structures.