In recent work by physicists, positive geometries are defined as certain semi-algebraic sets together with a meromorphic differential form called the canonical form: calculating scattering amplitudes reduces to determining the canonical form. Examples of planar positive geometries are polygons and, more generally, “curved” polygons, so-called polypols. The latter were proposed by Wachspress as...

A cohomology class of a smooth complex variety of dimension n is said to be of coniveau at least c if it vanishes on the complement of a closed subvariety of codimension at least c, and of strong coniveau at least c if it comes about by proper pushforward from the cohomology of a smooth variety of dimension at most n–c. The notions of coniveau and strong coniveau each define a filtration on...

We state results from noncommutative deformation theory of modules over an associative k-algebra A necessary for this work. We define a set of A-modules containing the simple modules whose elements we call spectral, aSpec A, for which there exists a topology where the simple modules are the closed points. Applying results from deformation theory we prove that there exists a sheaf of...

We prove a blow-up formula for the generating series of virtual χy-genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula for topological χy-genera of Göttsche. Our formula is a refinement of one by Vafa-Witten relating to S-duality. We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the...