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# Nilpotent orbits of mixed Hodge structure

#### Abstract

We discuss a class of variations of mixed Hodge structure that are admissible in the sense of J. Steenbrink and S. Zucker, (29). For these variations the presence of graded polarization forms determines mixed Hodge norms on the vector bundles underlying the variations which have the same degeneration properties as the Hodge norms on variations of pure polarized Hodge structure described by W. Schmid in (26). Another result of this dissertation is a discussion of sufficient conditions for the existence of a relative weight filtration of a cone of monodromy logarithms. In particular, our conditions do not require the presence of Hodge data. Our approach to working with relative weight filtrations $W(N, W\prime)$ and mixed Hodge structures $(W\prime, \Psi)$ is to split the weight filtrations $W\prime$ with canonical splittings introduced by P. Deligne in (11). These splittings agree with the splittings that occur in the $\rm sl\sb2$-theory of variations of pure Hodge structure of (26) and (4), but are more general, as they don't require filtrations $W\prime$ to be monodromy weight filtrations. We also include a proof of and a discussion of Deligne's unpublished result. ^

Mathematics

#### Recommended Citation

Christine Schwarz, "Nilpotent orbits of mixed Hodge structure" (January 1, 1998). Doctoral Dissertations Available from Proquest. Paper AAI9823774.
http://scholarworks.umass.edu/dissertations/AAI9823774

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