Variation of moduli of parabolic Higgs bundles

A moduli problem in algebraic geometry is the problem of constructing a space parametrizing all objects of some kind modulo some equivalence. If the equivalence is anything but equality, one usually has to impose some sort of stability condition on the objects represented. In many cases, however, this stability condition is not canonical, but depends on a parameter, which typically varies in a finite-dimensional rational vector space. The moduli spaces obtained for different values of the parameter are birational (at least if there are any stable points), and for several moduli problems the birational transformations between the different moduli spaces have been well characterized. Without exception, it has been found that the space of parameters contains a finite number of hyperplanes called walls on whose complement the stability condition is locally constant, so that the moduli space undergoes a birational transformation when a wall is crossed. If the moduli spaces are smooth, the birational transformation typically has the following very special form. A subvariety of the moduli space, isomorphic to the total space of a P -bundle, is blown up; the resulting exceptional divisor is a P × P -bundle over the same base; and it is blown down along the other ruling to yield the new moduli space, which therefore contains the total space of a P -bundle. The object of this paper is to extend the narrative above to the moduli spaces of parabolic Higgs bundles on a curve. However, there is a twist in the tale. The walls, to be sure, still exist and play their usual role. But when a wall is crossed, the birational transformation undergone by the moduli space is not of the form described above. Rather, it is a socalled elementary transformation. These are birational transformations defined on varieties admitting a holomorphic symplectic form, in which the exceptional divisor is a partial flag bundle, indeed a PT P -bundle. They were discovered by Mukai [9] in the early 1980s, on moduli spaces of sheaves on abelian and K3 surfaces; since then they have been observed in several settings. See Huybrechts [6, 7] for an informative discussion. Because of their symplectic nature, the appearance of these transformations on the Higgs moduli spaces is quite natural. It also seems to be related to the non-triviality of the obstruction space to the moduli problem, a hint that would be worth pursuing. The variation of moduli of ordinary parabolic bundles, without a Higgs field, was studied by Boden and Hu [1]. They described the projective bundles that are the exceptional loci of the birational transformations. A paper of the author [16, §7] used geometric invariant theory to show that the moduli spaces on either side of the wall become isomorphic after these exceptional loci are blown up. The present work, although it describes a similar result for parabolic Higgs bundles, does not use geometric invariant theory. Rather, it resembles another work of the author [15] which studied similar phenomena in the moduli spaces of