복소 그라스만 다양체 사이의 매장

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. 김홍종.이 논문에서는 복소 그라스만 다양체 Gr(2,m) 에서 복소 그라스만 다양체 Gr(2,n) 으로 가는 복소해석적 매장의 선형성에 대해 규명하고자 한다. 복소 그라스만 다양체 Gr(2,n) 위의 보편벡터다발의 쌍대벡터다발을, 주어진 매장으로 끌어당긴 벡터다발을 E 라고 하면, 가능한 모든 E 의 전 천 특성류들을 조사함으로써 이러한 매장들에 대한 성질을 파악할 수 있다. 먼저, 앞으로의 논리를 전개하는데 유용하게 사용될 Gr(2,m) 의 코호몰로지환의 Z-가군 기저를 하나 잡고, 모든 코호몰로지류들은 이 기저에 대한 선형결합으로 표현한다. 복소 그라스만 다양체 Gr(2,m) 에서 Gr(2,n) 으로 가는 각각의 복소해석적 매장에 대해, 벡터다발 E 의 전 천 특성류를 이 기저로 표현하면, 세 개의 정수계수를 가지는 선형결합으로 유일하게 쓸 수 있는데, 매장의 선형성은 이러한 정수들에 의해 완전히 결정된다. 주어진 매장으로부터 유도된 법벡터다발의 천 특성류와 오일러 특성류로부터, 3-변수 디오판토스 방정식들을 얻을 수 있고, 해석적벡터다발의 천 특성류가 음이 아닐 판정 기준과 함께 이 방정식들을 풀면, 특정 정수의 상계를 비롯한, 세 개의 정수들이 만족해야 하는 몇 가지 조건들을 얻을 수 있다. 이 정수 상계로부터 W. Barth 와 A. Van de Ven 이 증명한 결과들을 벡터다발 E 에 적용할 수 있고, 복소 그라스만 다양체 Gr(2,m)에서 Gr(2,n) 으로 가는 복소해석적 매장이 항상 선형이 되기 위한 m 과 n 의 조건들을 구할 수 있다.In this thesis, we characterize the linearity of holomorphic embeddings of the complex Grassmannian Gr(2,m) into the complex Grassmannian Gr(2,n). We study such embeddings by finding all possible total Chern classes of the pullback bundles E of the dual bundles of the universal bundles on Gr(2,n) under these embeddings. We first take a Z-module basis of the cohomology ring of Gr(2,m) which is useful for further works, and express every cohomology classes as a linear combination with respect to this basis. For each holomorphic embedding of Gr(2,m) into Gr(2,n), the total Chern class of E is written uniquely as a linear combination of basis elements with three integral coefficients, the linearity of the embedding is determined completely by these integers. We obtain several conditions on the three integers, including a upper bound of the specific integer by solving 3-variate Diophantine equations which are constructed from the Chern classes and the Euler class of the normal bundle induced by the embedding, together with a criterion of the numerical non-negativity of Chern classes of holomorphic vector bundles. This upper bound enables us to apply W. Barth and A. Van de Ven's results to E, and we find conditions on m and n for which any holomorphic embedding of Gr(2,m) into Gr(2,n) is linear.1. Introduction 1 2. Preliminaries 12 2.1 Schubert cycles on Grassmannians 12 2.2 Vector bundles on Grassmannians 19 3. Embeddings of Gr(2,m) into Gr(2,n) 22 3.1 Linear embeddings 23 3.2 Equivalent conditions for the linearity 26 3.3 Equations in a, b and c 30 3.3.1 Euler class of N_{\mathbb{R}} 30 3.3.2 Total Chern class of N 32 3.3.3 Refined equations in \omega_{1,0} and \omega_{1,1} 36 3.4 Inequalities in a, b and c 38 3.4.1 Numerical non-negativity of Chern classes 38 3.4.2 Refined equation in \omega_{1,1} 40 3.4.3 Refined equation in \omega_{1,0} 44 4. Characterization of linear embeddings 48 4.1 General case 49 4.2 Special case 58 Bibliography 64 Abstract (in Korean) 66Docto

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