Some families of local systems over smooth projective varieties
One can try to study the fundamental group of a smooth, complex projective variety \(S\) by looking at spaces of representations of \(\pi_ 1 (S)\). If the space of representations is a union of isolated points, then this structure reduces to the discrete structure of the set of representations. So a natural problem is to try to find examples of varieties \(S\) where there exist nontrivial continuous families of representations (or, equivalently, local systems on \(S) \). It is not too hard to see that if \(S\) is a projective algebraic curve of genus \(g \geq 2\), then the moduli space of representations of rank \(r \geq 1\) has dimension \((2g - 2) r^ 2 + 2\); or, for example, if \(S\) is a variety with \(\dim H^ 1(S, \mathbb{C}) = a\), then the space of representations of rank 1 has dimension \(a\). Taking tensor products of pullbacks of families of local systems arising in these ways, we obtain some more families. To pursue the problem we ask: Do families other than these exist? The idea in this article is to use the next natural construction, taking higher direct images of local systems. Suppose that \(f:X \to S\) is a smooth projective morphism and \(\{W_ t\}\) is a family of local systems on \(X\), chosen in a simple way. Let \(V_ t = R^ i f_ * (W_ t)\). This is a collection of local systems, and if the ranks are constant, then it is a continuous family. We can hope that \(\{V_ t\}\) will be an interesting family of local systems on \(S\). The principal question that needs to be addressed is whether, if the family \(\{W_ t\}\) varies nontrivially, the family of direct images \(\{V_ t\}\) varies nontrivially. We are also interested in constructing examples where we can show that the family \(\{V_ t\}\) does not, by some miracle, arise from a simpler construction such as the one described before. This article consists essentially of two parts. The first (sections 1-5) is devoted to answering our principal question in a fairly general situation. For this we develop the technique of taking the direct image of a harmonic bundle and its associated Higgs bundle. We give a way to calculate the spectral varieties of the Higgs bundles associated to \(V_ t\), as a way of verifying that the \(V_ t\) vary nontrivially. -- The second part (sections 6-8) is concerned with the construction of a particular class of examples and the verification of some additional properties about monodromy groups and possible factorization through morphisms to algebraic curves; these properties serve to show that our examples do not come from tensor products of pullbacks. The methods used in this part are all fairly well known.
local systems on complex projective variety
scientific article; zbMATH DE number 490545