Hyperbolic localization of the Donaldson-Thomas sheaf

Abstract

In this paper we prove a toric localization formula in cohomological Donaldson Thomas theory. Consider a −1-shifted symplectic scheme or stack with a C * action leaving the −1-shifted symplectic form invariant. This includes the moduli stack of sheaves or complexes of sheaves on a Calabi-Yau threefold with a C * -invariant Calabi-Yau form, or the intersection of two C * -invariant Lagrangians in a symplectic space with a C * -invariant symplectic form. In this case we express the restriction of the Donaldson-Thomas perverse sheaf defined by Joyce et al. to the attracting variety as a sum of cohomological shifts of the DT perverse sheaves on the C * fixed components. This result can be seen as a −1-shifted version of the Bia lynicki-Birula decomposition for smooth schemes.5 Hyperbolic localization on a d-critical algebraic stack 43 5.1 Hyperbolic localization on an algebraic stack .