Topics in Algebraic and Analytic Geometry. (MN-13): Notes From a Course of Phillip Griffiths

230 Pages·1974·9.208 MB·English

by  Phillip A. Griffiths, John Frank Adams

TOPICS IN ALGEBRAIC
AND ANALYTIC GEOMETRY
BY
PHILLIP GRIFFITHS
AND
JOHN ADAMS
Preliminary Informal Notes
of University Courses and Seminars
in Mathematics
MATHEMATICAL NOTES
PRINCETON UNIVERSITY PRESS

TOPICS IN ALGEBRAIC AND ANALYTIC GEOMETRY
Notes £rom a course or
PHILLIP GRIFFITHS
Written and revised by
JOHN ADAMS
PRINCETON UNIVERSITY PRESS
AND
UNIVERSITY OF TOKYO PRESS
PRINCETON, NEW JERSEY
1974
(S}
Copyright 1974 by Princeton·University Press
Published by Princeton University Press, Princeton and London
All Rights Reserved
L.C. Card: 74-2968
I.S.B.N.: 0-691-08151-4
Library of Congress Cataloging in Publication Data
will be found on the last printed page of this book
Published in Japan exclusively
by University of Tokyo Press
in other parts of the world by
Princeton University Press
Printed in the United States of America
Introduction
This is a revised version of the notes taken from a class taught
at Princeton University in 1971-1972. The table of contents gives a good
description of the material covered.
The notes focus on comparison theorems between the algebraic,
analytic, and continuous categories.

CONTENTS
Chapter One
Section 1. Sheaf theory, ringed spaces l
Section 2. Local structure of analytic and algebraic sets 9
Section 3. lPn 19
Chapter Two
Section 1. Sheaves of modules 23
Section 2. Vector bundles 33
Section 3. Sheaf cohomology and computations on lPn 45
Chapter Three
Section 1. Maximum principle and Schwarz lemma on analytic spaces 56
Section 2. Siegel's theorem 61
Section 3. Chow's theorem 69
Chapter Four
Section 1. GAGA 73
Chapter Five
Section 1. Line bundles, divisors, and maps to lPn 83
Section 2. Grassmannians and vector bundles 94
Section 3. Chern classes and curvature 112
Section 4. Analytic cocycles 128
Chapter Six
Section 1. K-theory and Bott periodicity 136
Section 2, K -theory as a generalized cohomology theory 144
Chapter Seven
Section 1. The Chern character and obstruction theory 154
Section 2. The Atiyah-Hirzebruch spectral sequence 164
Section 3. K-theory on algebraic varieties 183
Chapter Eight
Section l, Stein manifold theory 196
Section 2. Holomorphic vector bundles on polydisks 203
Chapter Nine
Concluding remarks 215
Bibliography 217
vi
1
Chapter One
§I Sheaf theory, ringed spaces
n
On a: , the space of n complex variables, there is a sequence of sheaves
defined in the usual topology:
0 :::) 0 :::) 0 :::) 0
cont diff hol alg
where for an open set Uc a:n :
r(u, 0 ) = continuous complex-valued functions defined on u .
cont
r(u, odiff ) = c"' complex-valued functions defined on u .
=
r(U, Ohol ) holomorphic functions defined on U •
T(U, 0 a1g ) = rational holomorphic functions defined on U •
(A holomorphic function cp on U is said to be rational just in case each a in
U has a neighborhood W in U such that there are two polynomials p, q
with q nowhere zero in W , with cp= p/q in W • In fact, because the
polynomial ring a: [z1, ••• , zn] is a unique factorization domain, one can always
=
take W U.)
If W is an open set prqperly contained in the open set U then there is
always an element of r(W, 0 ) which does not extend to U, and similarly
cont
for 0 diff • But Ohol and 0 alg behave differently: We will prove a little
theorem about this.
I.l.2
2
I .A THEOREM (Hartog's removable singularities theorem) In case n ::::_ 2 and
W .= U less a point, every holomorphic function on W extends to U •
We may suppose that W = U - { (O, ••• , O)}. If 6 is a small positive number
= { I
such that An (6 ) z : sup J z i ~ 6 } is contained in U we define for each
i=l, ... ,n
f holomorphic in W
J
f(s, z2, ••• , zn)
fl(zl' "., zn) = 211"l-1 de:
s-
zl
Jz J=6
1
for z in the interior of the polydisk. It will suffice to show that f = f1 in the
interior of the polydisk less its center. But for a point inside the polydisk with
z2 "I 0 the formula
f(zl' "., zn) = 211'/_l J f(s,z2, ... ,zn) de:
e:- z
- 1
Jz J=
1 Ii
is valid, so f = f1 where both are defined.
This type of behavior is more pronounced in the case of 0 a1g • Every
holomorphic rational function on W will extend to U unless there is a poly-
nomial with zeroes in U but no zeroes in W •
The sheaf 0 a1g may naturally be restricted to a coarser topology on a:n ,
the Zariski topology . A set in a:n is a Zariski closed set in case it is the locus
of zeroes of a set of polynomials- one can always take the set of polynomials to be an
ideal in the polynomial ring. The Zari ski closed set associated to an ideal ~ will
be denoted V(I) ("variety of I") • The complement of a Zariski closed set is a