×

zbMATH — the first resource for mathematics

Holomorphic chains and the support hypothesis conjecture. (English) Zbl 0870.32003
Let \(\Omega\) be a complex manifold. The space of all locally rectifiable \((h, h)\)-currents on \(\Omega\) is denoted by \(R^{\text{loc}}_{(h,h)} (\Omega)\).
The author proves the following conjecture of Marvey and Shiffman:
Theorem. Let \(T\in R^{\text{loc}}_{(h,h)} (\Omega)\) with \(dT=0\). Then \(T\) is a holomorphic \(h\)-chain.
Holomorphic \(h\)-chains are linear combinations with integer coefficients of locally finite families of \(h\)-dimensional varieties in \(\Omega\).

MSC:
32C30 Integration on analytic sets and spaces, currents
32C25 Analytic subsets and submanifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math. 93 (1971), 65 – 74. · Zbl 0221.32011
[2] H. Alexander, Ends of varieties, Bull. Soc. Math. France 120 (1992), no. 4, 523 – 547 (English, with English and French summaries). · Zbl 0783.32004
[3] H. Alexander, B. A. Taylor, and J. L. Ullman, Areas of projections of analytic sets, Invent. Math. 16 (1972), 335 – 341. · Zbl 0238.32007
[4] Errett Bishop, Conditions for the analyticity of certain sets, Michigan Math. J. 11 (1964), 289 – 304. · Zbl 0143.30302
[5] A. M. Davie and B. K. Øksendal, Peak interpolation sets for some algebras of analytic functions, Pacific J. Math. 41 (1972), 81 – 87. · Zbl 0218.46050
[6] H. Federer, Geometric Integration Theory, Springer- Verlag, New York, 1969. · Zbl 0176.00801
[7] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. · Zbl 0213.40401
[8] Reese Harvey, Holomorphic chains and their boundaries, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R. I., 1977, pp. 309 – 382.
[9] F.R. Harvey and H.B.Lawson,Jr., Complex analytic geometry and measure theory, Proc. Sym. Pure Math. 44 (1986), 261-274.
[10] Reese Harvey and Bernard Shiffman, A characterization of holomorphic chains, Ann. of Math. (2) 99 (1974), 553 – 587. · Zbl 0287.32008
[11] James R. King, The currents defined by analytic varieties, Acta Math. 127 (1971), no. 3-4, 185 – 220. · Zbl 0224.32008
[12] Walter Rudin, Function theory in the unit ball of \?\(^{n}\), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. · Zbl 0495.32001
[13] Bernard Shiffman, Complete characterization of holomorphic chains of codimension one, Math. Ann. 274 (1986), no. 2, 233 – 256. · Zbl 0572.32005
[14] Bernard Shiffman, On the removal of singularities of analytic sets, Michigan Math. J. 15 (1968), 111 – 120. · Zbl 0165.40503
[15] John Wermer, Banach algebras and several complex variables, 2nd ed., Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 35. · Zbl 0336.46055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.