Degeneracy theorems for holomorphic mappings between algebraic varieties
HTML articles powered by AMS MathViewer
- by Robert Molzon PDF
- Trans. Amer. Math. Soc. 270 (1982), 183-192 Request permission
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 $\kappa$-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 $\kappa$-plane sections than Hausdorff measure and arises naturally by considering the singular integrals in the First Main Theorem of Nevanlinna.References
- 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 185607, DOI 10.1007/BF02391818
- 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
- James A. Carlson and Phillip A. Griffiths, The order functions for entire holomorphic mappings, Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972-1973) Dekker, New York, 1974, pp. 225–248. MR 0404699
- James A. Carlson, A moving lemma for the transcendental Bezout problem, Ann. of Math. (2) 103 (1976), no. 2, 305–330. MR 409901, DOI 10.2307/1971008
- Mark Lee Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43–75. MR 367302, DOI 10.2307/2373660
- Phillip Griffiths and James King, Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220. MR 427690, DOI 10.1007/BF02392265
- John J. Hirschfelder, The first main theorem of value distribution in several variables, Invent. Math. 8 (1969), 1–33. MR 245840, DOI 10.1007/BF01418868
- Robert E. Molzon, Sets omitted by equidimensional holomorphic mappings, Amer. J. Math. 101 (1979), no. 6, 1271–1283. MR 548881, DOI 10.2307/2374140
- Wilhelm Stoll, Invariant forms on Grassmann manifolds, Annals of Mathematics Studies, No. 89, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. MR 0481089
- H. Wu, Remarks on the first main theorem in equidistribution theory. I, J. Differential Geometry 2 (1968), 197–202. MR 276500
- 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
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 270 (1982), 183-192
- MSC: Primary 32H30
- DOI: https://doi.org/10.1090/S0002-9947-1982-0642337-0
- MathSciNet review: 642337