Vector bundles on algebraic curves: holomorphic triples in low genus
Let X be a smooth and projective curve over an algebraically closed field K
of characteristic 0. A holomorphic triple on X is a triple (E1, E2, φ), where
The study of holomorphic triples has been started by Bradlow and García-Prada in [11, 7] where the authors deal with the search for solutions to some gauge theoretic equations on X obtained by dimensional reduction of the Hermitian-Einstein equation on X x P1.
For those objects it is possible to introduce a notion of stability, depending on a real parameter α, and consequently to deal with the moduli spaces of α-stable triples. These spaces depends on some parameters, as the ranks and degrees of the vector bundles involved in the definition of holomorphic triple, besides of course α itself. The main properties of these triples and of their moduli spaces have been further investigated by the same authors with Gothen in  and, more recently, in  and in  by García-Prada, Hernández Ruipérez, Pioli and Tejero Prieto. Many different problems have been faced and solved in the papers cited above, in particular the authors have shown that some constraints on the parameter α must exist in order for a triple to be α-stable. Moreover, when for some fixed values of the parameters the moduli space is non empty, irreducibility and smoothness is proved and a computation for the dimension is provided. Several techniques are used to achieve these results, but probably one of the most valuable is that of flips. This method, which takes advantage of the theory of deformations and extensions for triples, takes care of how the α-stability of a holomorphic triple varies for fixed ranks and degrees as α varies in the admissible range, hence with this tool it is possible to prove non-emptiness of the moduli spaces for particular and convenient values of α (usually for "large" values of α), and then to extend this result also to the remaining cases.
However, although the general theory developed in the aforementioned papers is mainly independent of the genus g of the curve X, in fact for some particular results it is necessary to require that g>1, since the cases of the projective line (g = 0) and of elliptic curves (g = 1) deserve some special treatment. This is due mainly to the fact that in these last two cases stable and semistable vector bundles are rare to appear, hence the would-be solver has to face the problem of the lack of "good" objects to use for building holomorphic triples for which it is easier to prove &alpha-stability. Moreover in low genus the technique of flips described above, even if it is still sensible, unfortunately fails to provide useful information, thus problems like non-emptiness and irreducibility of moduli spaces in these cases are still open.
The aim of this work is then to try to answer some of the questions here mentioned precisely in the two cases of the projective line and of elliptic curves. While we emerge victorious from the challenge with elliptic curves (and in fact from the results we obtain we are also able to prove analogous results for bielliptic curves) we can obtain only partial results for the projective line, which hence deserves some more attention to be rendered in future works.
More in details in Chapter 1 we collect the preliminary definitions and results concerning holomorphic triples and we present the main results on their moduli spaces. Here we discuss the constraints on the range of admissible α in order for α-stable triples to exist, we note that even if α is a real parameter, in fact only a finite number of different moduli spaces exists, and we introduce the general technique of flips. Also using this technique it is possible to prove several results on the moduli spaces of holomorphic triples, such as irreducibility, smoothness and dimension computation.
Chapter 2 introduces the notion of coherent system (namely a pair (E,V) where E is a vector bundle on X and V is a vector subspace of the vector space of global sections of E), presents the main theory of these objects and stresses the relationships between holomorphic triples and coherent systems.
The motivations for this introduction are that the situation for those new objects is analogous to that of triples: the general theory has been developed since about 1993 in several papers, but the cases g = 0 and g = 1 deserve a special treatment. For coherent systems the theory has been improved recently to cover also these last two cases by Lange and Newstead in the two papers [15, 16] (and in fact further developed by the same authors in [17, 18] in the case of curves of genus 0, since the results so far known are not yet exhaustive). The results we can obtain for holomorphic triples are analogous to the results obtained in the previously cited papers, and summarized in an appropriate section of this chapter, hence our interest in them.
Here we discuss briefly also a way of seeing both holomorphic triples and coherent systems as particular augmented bundles, that is as particular objects made up with one or more vector bundles on X with some extra data of some kind (prescribed sections, a map,...). A more comprehensive introduction to this point of view can be seen in , where also other classes of augmented bundles are taken into consideration.
Chapter 3 is devoted to study the particular case of holomorphic triples on the projective line. This is probably the most difficult case we take into consideration, an evidence of this being the fact that the results here obtained are not completely exhaustive, hence not as good as one would dream. In fact we are able to obtain some necessary conditions for non-emptiness, but in general not sufficient conditions. Some particular cases are considered (namely those corresponding to some particular values for the ranks and degrees of the two vector bundles) and more precise answers are provided with these extra hypotheses. The cases n2 = 1 and n2 = 2 are in fact completely solved and reveal that the necessary conditions previously proved for the projective line are in fact also sufficient, but, so far, it is not known whether this is true in general. The results presented in this Chapter have been obtained in collaboration with Francesco Prantil of the University of Trento; our main results are the existence of some stronger constraints on the values of the parameter α in order for α-stable triples to exist (see Propositions 3.2.1 and 3.2.2), the proof of some properties of the general α-stable triple (Theorems 3.3.6 and 3.3.12) and the existence results already mentioned (see Section 3.4).
Chapter 4 deals with the study of the particular case of holomorphic triples on elliptic and bielliptic curves. The results here obtained are in some sense more interesting, since for g = 1 it is possible to provide necessary and sufficient conditions for the moduli spaces of α-stable triples to be non empty, and in these cases some further properties of the moduli spaces, such as irreducibility, can be shown. The next natural step is to consider bielliptic curves, that is curves X which are a double covering through a map f of an elliptic curve C, to extend (hopefully!) the results on the moduli spaces also to this case. In fact it turns out that α-stability behaves well in respect with the double covering map, and hence the main properties of the moduli spaces of elliptic curves are still true for bielliptic ones. Here we consider also elementary transformations and investigate how these transformations make the α-stability of a triple worse. Also in this case one finds out that the elementary transformation of a sufficiently general α-stable triple is still α-stable. The results presented in this Chapter have been obtained in collaboration with Edoardo Ballico and Francesco Prantil of the University of Trento; the main results are summarized in the following Theorems.
Theorem 1 (Thms 4.4.6 and 4.4.8). Let E1, E2 be polystable vector bundles with rank E2 <= rank E1, and μ(E2) <= μ(E1). Then there exists a homomorphism φ in Hom(E2, E1) such that the triple T = (E1, E2, φ) is α-stable for any α in (αm, αM). Moreover Nα(n1, n2, d1, d2) is irreducible, smooth of dimension -n1d2 + n2d1 + 1, for every α in (αm, αM).
Theorem 2 (Thms 4.5.4, 4.5.5 and 4.5.10). Let α in R, C be an elliptic curve, f: X --> C a double covering with X a smooth curve of genus g >= 2, σ: X --> X the involution and (E1, E2, φ) an α-stable triple on C with E1and E2 polystable vector bundles with pairwise non-isomorphic indecomposable direct factors. Then the triples (f*(E1 ), f*(E2), f*(φ)), (F1, f*(E2 ), f*(φ)) and (F1 , F2 , ψ) are 2α-stable, where F1 and F2 are obtained from f*(E1) and f*(E2 ) making a general positive elementary transformation supported in a point p of X where f is not ramified.
Chapter 5, in the end, presents some results on coherent systems, in particular on elliptic and bielliptic curves and on the projective line. Using as main tools the results of Lange and Newstead [15, 16] some spannedness-like properties are proved for sufficiently general σ-stable coherent systems. An existence result is also proved for curves of any genus g using a dimensional estimation provided by the analysis of rational curves in Grassmannians and of their Plücker embeddings performed by Ballico in . This can be seen as a first step of a longer path in the direction of those values of the parameters which are not yet covered by the general theory summarized in Chapter 2.