Let \(X\) be a smooth projective variety. In recent years a great deal of attention was given by different authors to vector bundles on \(X\), equipped with some extra structure [A. Bertram, J. Algebr. Geom. 3, No. 4, 703-724 (1994; Zbl 0826.14017), S. B. Bradlow and G. D. Daskalopoulos, Int. J. Math. 2, No. 5, 477-513 (1991; Zbl 0759.32013) and ibid. 4, No. 6, 903-925 (1993; Zbl 0798.32020), O. Garcia-Prada, ibid. 5, No. 1, 1-52 (1994; Zbl 0799.32022), M. Thaddeus, Invent. Math. 117, No. 2, 317-353 (1994)]. The authors here take a more general point of view, considering on \(X\) pairs \((E, \alpha)\) consisting of a coherent \({\mathcal O}_X\) module \(E\), with a nontrivial homomorphism \(\alpha : E \to E_0\) where \(E_0\) is a fixed coherent \({\mathcal O}_X\) module. [In a successive paper, Int. J. Math. 6, No. 2, 297-324 (1995), the authors propose the name “framed modules” for these objects.] A suitable notion of (semi)-stability for framed modules is defined in terms of Hilbert polynomials of the modules involved. Such notion is given with respect to a polynomial \(\delta\) with rational coefficients and positive leading coefficient. A corresponding notion of slope-(semi)-stability is also introduced with the usual chain of implications: slope-stable implies stable implies semistable implies slope-semistable. The authors consider a moduli problem for framed modules showing that when the dimension of \(X\) is one or two there exists a fine moduli space of stable framed modules, with a natural compactification obtained with equivalence classes of semistable modules. This result is obtained using geometric invariant theory techniques and it is generalized to higher dimensions by the same authors in their paper cited above.
Particular choices of the fixed module \(E_0\) give rise to previously investigated objects, shedding some new light on their moduli problems. If \(E_0 = {\mathcal O}_X\) then the situation is dual to having a fixed global holomorphic section of \(E\) (Higgs pairs), with the appropriate notion of (semi)stability. If \(E_0\) is chosen to be a vector bundle over a divisor of \(X\) then framed bundles on surfaces [M. Lehn, “Modulräume gerahmter Vektorbündel”, Dissertation, Bonner Math. Schriften 241 (1992; Zbl 0841.14012)] or a slightly generalized version of level structures on curves [C. S. Seshadri, “Fibrés vectoriels sur les courbes algébriques”, Astérisque 96 (1982; Zbl 0517.14008)] are found.
A generalization of a result of Bogomolov on surfaces is proved, showing that the restriction of a slope-stable pair to a smooth curve of high degree is still slope-stable. Generalizing Thaddeus’ approach for curves to surfaces, the whole series of moduli spaces obtained letting the polynomial \(\delta\) vary is analyzed.