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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2006 Volume 70, Issue 5, Pages 13–30 (Mi im616)

This article is cited in 2 papers

Holomorphic bundles on diagonal Hopf manifolds

M. S. Verbitsky

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We show that every stable holomorphic bundle on the Hopf manifold $M=(\mathbb C^n\setminus0)/\langle A\rangle$ with $\dim M\geqslant 3$, where $A\in\operatorname{GL}(n,\mathbb C)$ is a diagonal linear operator with all eigenvalues satisfying $|\alpha_i|<1$, can be lifted to a $\widetilde G_F$-equivariant coherent sheaf on $\mathbb C^n$, where $\widetilde G_F\cong(\mathbb C^*)^l$ is a commutative Lie group acting on $\mathbb C^n$ and containing $A$. This is used to show that all bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves with all subquotients $F_i/F_{i-1}$ of rank $1$.

UDC: 515.171.3+515.174.5

MSC: 53C55, 14J60

Received: 30.08.2005
Revised: 16.06.2006

DOI: 10.4213/im616


 English version:
Izvestiya: Mathematics, 2006, 70:5, 867–882

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