我們研究實係數可求長的流的一些基本性質並且在刻劃由實係數全純鏈定義的流方面，給了金的定裡的一個推廣以及赫維和什夫曼的定理的一個實係數版本。第一個情況的證明使用了蕭的半連續定理，再來我們採用亞歷山大的策略來證明第二個情況。最後，我們用這些成果來得到霍奇猜想的一個充分條件和一些可由代數圈表示的同調類的相關結果。

We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem and a real version of Harvey and Shiffman’s theorem in characterizing currents defined by holomorphic chains with real coefficients. The proof of the first case uses Siu’s semicontinuity theorem and we adopt the strategy of Alexander to prove the second case. These conclusions are applied to get some applications in complex geometry containing a sufficient condition for the Hodge conjecture and generalizations of some results of Harvey and Shiffman.

1 Introduction................................... 3
2 Real rectifiable currents....................... 4
3 A generalization of King’s theorem............. 11
4 The proof of the second main theorem for k = 1. 14
5 Complete the proof of the second main theorem.. 20
6 Applications................................... 22

[1] H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in C^n, Amer. J. Math. 93 (1971), 65-74.

[2] H. Alexander, Holomorphic chains and the support hypothesis conjecture, JAMS, 10, No. 1(1997), 123-138.

[3] J. P. Demailly, Complex analytic and differential geometry, lecture notes on the webpage of the author.

[4] T. C. Dinh, M. G. Lawrence, Polynomial hulls and positive currents, Ann. Fac. Sci. Toulouse Math., XII(2003), 317-334.

[5] T. C. Dinh, N. Sibony, Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., Vol. 203, No. 1(2009), 1-82.

[6] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, Amer. Math. Soc., Providence, RI, 1998.

[7] H. Federer, Some theorems on integral currents, Trans. of AMS, Vol. 117(1965), 4367.

[8] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969.

[9] H. Federer, Real flat chains, cochains and variational problem, Vol. 24, No. 4(1974), 351-407.

[10] H. Federer, W. H. Fleming, Normal and integral currents, Ann. Math., 72, No. 3(1960), 458-520.

[11] E. Friedlander, H. B. Lawson, A theory of algebraic cocycles, Annals of Math., 136(1992), 361-428.

[12] T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, 1969.

[13] R. Harvey, Holomorphic chains and their boundaries, Proc. Sym. Pure Math., 30 part 1 (1977), 309-392.

[14] R. Harvey, H. B. Lawson, On the boundaries of complex analytic varieties, Ann. of Math., 102(1975), 233-290.

[15] R. Harvey, H. B. Lawson, A introduction to potential theory in calibrated geometry, Amer. J. Math. 131, no. 4 (2009), 893-944.

[16] R. Harvey, H. B. Lawson, Duality of positive currents and plurisubharmonic functions in calibrated geometry, Amer. J. Math. 131, no. 5 (2009), 1211-1240.

[17] F. R. Harvey, J. R. King, On the Structure of Positive Current, Inventiones math. 15 (1972), 47-52.

[18] F. R. Harvey, A. W. Knapp, Positive (p,p) forms, Wirtinger’s inequality, and currents, Value distribution theory, Part A (Proc. Tulane Univ. Program on Value-Distribution Theorey in Complex Analysis and Related Topics in Differential Geometry, 1972-1973), pp. 43-62. Dekker, New York, 1974.

[19] R. Harvey, B. Shiffman, A characterization of holomorphic chains, Ann. of Math., 99, No. 3, (1974), 553-587.

[20] J. R. King, The currents defined by analytic varieties, Acta Math. 127(1971), 185-220.

[21] S. G. Krantz, H. R. Parks, Geometric integration theory, Birkhauser, 2008.

[22] H. B. Lawson, The stable homology of a flat torus, Math. Scand., 36(1975), 49-73.

[23] H. B. Lawson, Algebraic cycles and homotopy theory, Annals of Math., 129(1989), 253-291.

[24] P. Lelong, Inte´gration sur un ensemble analytique complexe, Bull. Soc. Math. France 85 (1957) 239262.

[25] L. Simon, Lectures on Geometric Measure Theory, Australian National University Cen- tre for Mathematical Analysis, Canberra, 1983.

[26] B. Shiffman, On the removal of singularities of analytic sets, Michigan Math. J. 15(1968), 111-120.

[27] Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27(1974), 1-2, 53-156.

[28] G. Stolzenberg, Volumes, limits, and extensions of analytic varieties, Lecture Notes in Math., Vol. 19, 1966.

[29] J. H. Teh, C. J. Yang, Real rectifiable currents and algebraic cycles, arXiv:1810.00355.

[30] J. H. Teh, C. J. Yang, A characterization of real holomorphic chains and applications in the study of algebraic cycles, arXiv:1901.04152

[31] J. H. Teh, C. J. Yang, Bott-Chern homology, Bott-Chern differential cohomology and the Hodge conjecture, arXiv: 1910.01780

[32] J. Wermer, Banach Algebras and Several Complex Variables, Second Edtion, Springer- Verlag, New York, 1976.