實係數全純鏈的兩種刻劃方法與在代數圈上的應用

Two Characterizations of Real Holomorphic Chains and Applications in the study of Algebraic Cycles

楊進瑞(Chin-Jui Yang)

指導教授 ： 鄭志豪

國立清華大學/理學院/數學系/碩士(2020年)

摘要

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。

關鍵字

全純鏈 ； 代數圈

並列摘要

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.

並列關鍵字

holomorphic ； chains ； algebraic ； cycles

參考文獻

[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.