Kodaira-Saito vanishing and applications

The first part of the paper contains a detailed proof of M. Saito's generalization of the Kodaira vanishing theorem, following the original argument and with ample background, based on a lecture given at a Clay workshop on mixed Hodge modules. The second part contains some recent applications, and a Kawamata-Viehweg-type statement in the setting of mixed Hodge modules.Comment: 33 pages; final version, with expository improvements, updates, and substantial additions in the background section

Rethinking Representation: the Challenge of Non-humans

This article argues that the standard model of political representation mischaracterises the structure of representation. After surveying the classical types of representation and their application to non-humans, the basic nature of representation is shown to have been unduly centred on interests, responsiveness and unidirectional protocols. It proposes a different structure by drawing inspiration from recent scholarship and developments in political philosophy, as well as the representation of non-human actors. It proposes an ontological grounding of representation in ‘irreducible multiplicity’, and a structural analysis based on the concepts of claim and relation. This abstract form of representation can take into account both human and non-human cases, and works to ground different typologies. The relational structure of representation creates interests and preferences, subjects and actors, power dynamics and seemingly immutable identities

Stable maps and Quot schemes

In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable maps to the relative Grassmannian. We establish an essentially optimal upper bound on the dimension of the two compactifications. Based on that, we prove that for an arbitrary vector bundle, the Quot schemes of quotients of large degree are irreducible and generically smooth. We precisely describe all the vector bundles for which the same thing holds in the case of the moduli spaces of stable maps. We show that there are in general no natural morphisms between the two compactifications. Finally, as an application, we obtain new cases of a conjecture on effective base point freeness for pluritheta linear series on moduli spaces of vector bundles.Comment: 39 pages, 1 figure; final version with a few expository changes suggested by the refere

Viehweg's hyperbolicity conjecture for families with maximal variation

We use the theory of Hodge modules to construct Viehweg-Zuo sheaves on the base spaces of families with maximal variation and fibers of general type, or more generally whose geometric generic fiber has a good minimal model. We deduce Viehweg's hyperbolicity conjecture in this context, namely the fact that the base spaces of such families are of log general type. This is approached as part of a general problem of identifying what spaces can support Hodge theoretic objects with certain positivity properties.Comment: 28 pages; final version with a few expository improvements, to appear in Invent. Mat