Mixed twistor structures

The purpose of this paper is to introduce the notion of mixed twistor structure as a generalization of the notion of mixed Hodge structure. Recall that a mixed Hodge structure is a vector space V with three filtrations F , F ′ and W (the first two decreasing, the last increasing) such that the two filtrations F and F ′ induce i-opposed filtrations on Gr i (V ). (Generally (V,W ) is required to have a real structure and F ′ is the complex conjugate of F but that is not very relevant for us here.) Given a MHS (V,W, F, F ) we can form the Rees bundle E := ξ(V, F, F ) over P [Si4] [Si7]. In brief this is obtained from the trivial bundle V ×P by using F to make an elementary transformation over 0 and F ′ to make an elementary transformation over ∞. The bundle E is graded by strict subbundles which we denote WiE and the condition of opposedness of the filtrations is equivalent to the condition that Gr i (E) be semistable of slope i on P 1 (in other words a direct sum of copies of OP1(i)). The notion of mixed twistor structure is simply obtained by abstracting this situation: an MTS is a pair consisting of a bundle E over P and a filtration by strict subbundles WiE such that the Gr W i (E) are semistable of slope i. In the construction starting with a mixed Hodge structure the resulting (E,W ) has an action of Gm covering the standard action on P 1 and in fact the mixed Hodge structures are simply the Gm-equivariant mixed twistor structures. Thus, in some sense, the passage from “Hodge” to “Twistor” is simply forgetting to have an action of Gm. This principle occured already, in a primitive way, in the passage from systems of Hodge bundles (cf [Si1]) to Higgs bundles in [Si2]. We will give some generalizations of basic classical results for mixed Hodge structures, to the mixed twistor setting. The process of making these generalizations is relatively direct although some work must be done to develop the appropriate notion of variation of mixed twistor structure. The overall idea is that we have the following Meta-theorem If the words “mixed Hodge structure” (resp. “variation of mixed Hodge structure”) are replaced by the words “mixed twistor structure” (resp. “variation of mixed twistor structure”) in the hypotheses and conclusions of any theorem in Hodge theory, then one obtains a true statement. The proof of the new statement will be analogous to the proof of the old statement. We don’t prove this meta-theorem but support it with several examples using the basic theorems of mixed Hodge theory. The utility of the notion of mixed twistor structure, and of the above meta-theorem, is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety, where up until now