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On classification of higher-dimensional algebraic varieties with ample vector bundles. (Chinese. English summary) Zbl 1289.14005

Summary: Let \(X\) be a smooth projective variety of dimension \(n(n \geq 3)\) and \(\mathcal{E}\) be an ample vector bundle with rank \(r=n-k(k\geq 1)\) over \(X\). We denote \(\Lambda(\mathcal{E}, K_X)=\max\{(-K_X-c_1(\mathcal{E}))\cdot C|R=\mathbb R_+[C]\in \Omega\), and \(l(R)=-K_X\cdot C\}\), where \(K_X\) is the canonical bundle of \(X,\;c_1(\mathcal{E})\) means the first Chern class of \(\mathcal{E},\;\Omega\) denotes the set of extremal rays \(R\) such that \((K_X+c_1(\mathcal{E}))\cdot C\leq 0,\;\mathbb R_+\) is the set of positive real number, and \(l(R)\) is the length of \(R\). The classification of \((X,\mathcal{E})\) is given when \(\Lambda(\mathcal{E}, K_X)\geq k-1\).

MSC:

14J45 Fano varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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