Vector bundles on elliptic curves and factors of automorphy (Q2914231)

The aim of the article is to interpret results of M. Atiyah on classification of vector bundles on an elliptic curve into the language of factors of automorphy. The paper provides some results which are used without proofs by authors, in particular in \textit{I. Burban} and \textit{B. Kreussler} [``Vector bundles on degenerations of elliptic curves and Yang-Baxter equations'', \url{arxiv:0708.1685}] and \textit{A. Polishchuk} and \textit{E. Zaslow} [Adv. Theor. Math. Phys. 2, No. 2, 443--470 (1998; Zbl 0947.14017)], with proofs.NEWLINENEWLINELet \(\Gamma\) be a group acting upon a complex manifold \(Y\). The \(r\)-dimensional factor of automorphy is a holomorphic function \(f: \Gamma \times Y \to \mathrm{GL}_r(\mathbb C)\) such that \(f(\lambda \mu, y)=f(\lambda, \mu y)f(\mu,y)\). Two factors of automorphy \(f\) and \(f'\) are said to be equivalent if there is a holomorphic function \(h: Y \to \mathrm{GL}_r(\mathbb C)\) such that \(h(\lambda y ) f(\lambda,y)=f'(\lambda,y) h(y).\)NEWLINENEWLINELet \(X\) be a complex manifold, \(p: Y\to X\) its universal covering and \(\Gamma\) be the fundamental group of \(X\) acting on \(Y\) by deck transformations. Then there is a bijection between the set of equivalence classes of \(r\)-dimensional factors of automorphy and the set of isomorphism classes of vector bundles on \(X\) with trivial pull-back along \(p\).NEWLINENEWLINEThe correspondence between 1-dimensional complex tori and elliptic curves and between classifications of holomorphic vector bundles on complex projective variety and of algebraic vector bundles on the same variety yield in the reformulation of Atiyah's results in terms of factors of automorphy.NEWLINENEWLINEThe article consists of 5 sections. Section 1 is an introduction. Section 2 contains the construction of the correspondence between vector bundles of rank \(r\) and \(r\)-dimensional factors of automorphy. Section 3 is devoted to properties of factors of automorphy. In particular, their relation to theta functions is investigated as well as relation to operations on vector bundles. Section 4 is focused on study of complex tori. Also, the author proves a criterion when the factor of automorphy \(f\) defines a trivial bundle and criteria for two factors of automorphy to be equivalent. Section 5 deals with a classification of vector bundles over a complex torus.NEWLINENEWLINEThe author works with factors of automorphy depending only on direction \(\tau\) of lattice \({\mathbb Z}+\tau{\mathbb Z}\) of a torus, i.e., with holomorphic functions \({\mathbb C}^{\ast}\to \mathrm{GL}_r({\mathbb C})\).