Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions

We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

When care needs piracy: the case for disobedience in struggles against imperial property regimes

The aim of the Pirate Care project is to put the politics back into caring and to disrupt the global property regime that is colonising public welfare services and turning them into privately traded assets. Piracy refers to all the practices of survival and solidarity that disobey unjust legal and social rules that support property at the expense of living beings. The idea of piracy enables the foregrounding of the need to expand the realm of conceivable political responses to the crisis.

Theoretical and methodological considerations for a robust evaluation of the Linguistic Proximity Model

When we think of the debates surrounding linguistic transfer in L3 acquisition, one of the most prominent discussions concerns whether transfer occurs in a wholesale fashion or whether it is property-by-property. One such model is the Linguistic Proximity Model (LPM, Mykhaylyk et al., 2015; Westergaard et al., 2017; Westergaard, 2019), which maintains that transfer is property-by-property, with what Westergaard refers to as Full Transfer Potential (FTP). Westergaard injects the notion of complexity at each stage of development and recognizes the need to determine how a range of variables drive outcomes across these different stages. With that said, there are a set of points in the proposal that we believe are short of explanatory logic and will benefit from further consideration; we focus on two here. The first regards the need to go beyond post-hoc explanations of non-facilitative transfer via a commitment to a testable, proposal for when the LPM predicts such transfer will occur. The second relates to the current trend of using existing data to support property-by-property versus wholesale transfer. We contend that this application of existing data is an unsound practice because these data are in fact compatible with multiple theoretical accounts.

Doing Things with Legal History

This chapter discusses the role of historical analysis in property law. The history of property has been used to offer support for property rights. Their long history makes the distribution of property look normal, indeed natural and something that cannot or should not be challenged. However, historically in the U.S there have been competing visions of property. From the Progressive era onward especially, the history of property has been used to show the unequal distribution of property and to offer an alternative vision that expands the rights of non-owners of property. In the late twentieth and early twenty-first century, the history of opposition to feudalism and protection of the rights of non-owners was used to protect the rights of non-owners. Thus, the history of property has been a tool of judges and legislators to support property rights and it has also been, less frequently, a tool of critique.