Degeneracy theorems for holomorphic mappings between algebraic varieties
Author:
Robert Molzon
Journal:
Trans. Amer. Math. Soc. 270 (1982), 183192
MSC:
Primary 32H30
MathSciNet review:
642337
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Abstract: Degeneracy theorems are proved for holomorphic mappings from affine algebraic manifolds to projective algebraic manifolds of equal dimensions. A mapping is degenerate if it satisfies a growth estimate and omits a set of plane sections of positive capacity; the capacity being defined in terms of a singular integral. The capacity is a more delicate method of measuring the size of a set of plane sections than Hausdorff measure and arises naturally by considering the singular integrals in the First Main Theorem of Nevanlinna.
 [1]
Raoul
Bott and S.
S. Chern, Hermitian vector bundles and the equidistribution of the
zeroes of their holomorphic sections, Acta Math. 114
(1965), 71–112. MR 0185607
(32 #3070)
 [2]
Lennart
Carleson, Selected problems on exceptional sets, Van Nostrand
Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton,
N.J.Toronto, Ont.London, 1967. MR 0225986
(37 #1576)
 [3]
James
A. Carlson and Phillip
A. Griffiths, The order functions for entire holomorphic
mappings, Value distribution theory, Part A (Proc. Tulane Univ.
Program, 1972–1973), Dekker, New York, 1974, pp. 225–248.
MR
0404699 (53 #8499)
 [4]
James
A. Carlson, A moving lemma for the transcendental Bezout
problem, Ann. of Math. (2) 103 (1976), no. 2,
305–330. MR 0409901
(53 #13653)
 [5]
Mark
Lee Green, Some Picard theorems for holomorphic maps to algebraic
varieties, Amer. J. Math. 97 (1975), 43–75. MR 0367302
(51 #3544)
 [6]
Phillip
Griffiths and James
King, Nevanlinna theory and holomorphic mappings between algebraic
varieties, Acta Math. 130 (1973), 145–220. MR 0427690
(55 #721)
 [7]
John
J. Hirschfelder, The first main theorem of value distribution in
several variables, Invent. Math. 8 (1969),
1–33. MR
0245840 (39 #7146)
 [8]
Robert
E. Molzon, Sets omitted by equidimensional holomorphic
mappings, Amer. J. Math. 101 (1979), no. 6,
1271–1283. MR 548881
(80m:32004), http://dx.doi.org/10.2307/2374140
 [9]
Wilhelm
Stoll, Invariant forms on Grassmann manifolds, Princeton
University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977.
Annals of Mathematics Studies, No. 89. MR 0481089
(58 #1235)
 [10]
H.
Wu, Remarks on the first main theorem in equidistribution theory.
III, J. Differential Geometry 3 (1969), 83–94.
MR
0276502 (43 #2247c)
 [11]
H.
Wu, Mappings of Riemann surfaces (Nevanlinna theory), Entire
Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La
Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968,
pp. 480–532. MR 0237772
(38 #6053)
 [1]
 R. Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections, Acta Math. 114 (1967), 71112. MR 0185607 (32:3070)
 [2]
 Lennart Carleson, Selected problems on exceptional sets, Van Nostrand, Princeton, N. J., 1967. MR 0225986 (37:1576)
 [3]
 J. Carlson and P. Griffiths, The order functions for entire holomorphic mappings, Value Distribution Theory (edited by R. Kujala and A. Vitter, III), Dekker, New York, 1974. MR 0404699 (53:8499)
 [4]
 J. Carlson, A moving lemma for the transcendental Bezout problem, Ann. of Math. (2) 130 (1976), 305330. MR 0409901 (53:13653)
 [5]
 M. Green, Holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 317322. MR 0367302 (51:3544)
 [6]
 P. Griffiths and J. King, Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145220. MR 0427690 (55:721)
 [7]
 J. Hirschfelder, First main theorem of value distribution, Invent. Math. 8 (1969), 133. MR 0245840 (39:7146)
 [8]
 R. Molzon, Sets omitted by equidimensional holomorphic mappings, Amer. J. Math. 101 (1979), 12711283. MR 548881 (80m:32004)
 [9]
 W. Stoll, Invariant forms of Grassmann manifolds, Ann. of Math. Studies, no. 89, Princeton Univ. Press, Princeton, N. J., 1977. MR 0481089 (58:1235)
 [10]
 H. Wu, Remarks on the first main theorem in equidistribution theory. III, J. Differential Geom. 3 (1968), 8394. MR 0276502 (43:2247c)
 [11]
 , Mappings of Riemann surfaces (Nevanlinna theory), Proc. Sympos. in Pure Math., vol. 11, Amer. Math. Soc., Providence, R. I., 1968, pp. 480532. MR 0237772 (38:6053)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206423370
PII:
S 00029947(1982)06423370
Article copyright:
© Copyright 1982
American Mathematical Society
