An elementary invariant definition of the functions of bidegree $(p,q)$
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- by Michael Freeman PDF
- Proc. Amer. Math. Soc. 50 (1975), 265-272 Request permission
Abstract:
The alternating $r$-linear complex-valued functions of bidegree $(p,q),p + q = r$, are usually defined on a complex vector space $V$ as the span of the elements ${g_{{i_1}}} \wedge \ldots \wedge {g_{{i_p}}} \wedge {\bar g_{{j_1}}} \wedge \ldots \wedge {\bar g_{{j_q}}}$, where $\{ {g_i}:i \in I\}$ is a basis for ${V^ \ast }$ , or by means of a representation of the exterior power of a direct sum. The former definition is not a priori invariant under coordinate changes and not easily adaptable to analysis on infinite-dimensional spaces, and the latter one rests on a rather involved abstract construction. Here it is shown how to give a new coordinate-free definition of the $(p,q)$ functions by means of a simple identity which characterizes them by their action as $r$-linear maps on $V$. It seems well adapted for analysis on infinite-dimensional spaces.References
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N. Bourbaki, Éléments de mathématique. Fasc. XXXVI. Fascicule de résultats (Paragraphes 8 à 15), Actualités Sci. Indust., no. 1347, Hermann, Paris, 1971. MR 43 #6834.
- W. H. Greub, Multilinear algebra, Die Grundlehren der mathematischen Wissenschaften, Band 136, Springer-Verlag New York, Inc., New York, 1967. MR 0224623
- H. Blaine Lawson Jr. and James Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. (2) 98 (1973), 427–450. MR 324529, DOI 10.2307/1970913
- H. K. Nickerson, D. C. Spencer, and N. E. Steenrod, Advanced calculus, D. Van Nostrand Co., Inc., Toronto-Princeton, N.J.-New York-London, 1959. MR 0123651 M. Spivak, A comprehensive introduction to differential geometry. Vol. 1, Publish or Perish, Boston, Mass., 1970. MR 42 #2369. A. Weil, Variétés Kählériennes, Hermann, Paris, 1957.
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 50 (1975), 265-272
- MSC: Primary 32A99; Secondary 15A75
- DOI: https://doi.org/10.1090/S0002-9939-1975-0374469-0
- MathSciNet review: 0374469