Positive geometries: their adjoints and canonical forms

• Ragni Piene (University of Oslo)

Room: E-102 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 generalized algebraic finite elements. In his quest to define barycentric coordinates for polypols, he introduced the adjoint curve of a rational polypol. We show that a rational regular polypol gives a positive geometry and give an explicit expression for its canonical form in terms of the adjoint and boundary curves of the polypol. In the special case that the polypol is a convex polygon, we show that the adjoint curve is hyperbolic and describe its nested ovals. This talk is based on joint work with K. Kohn, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn, M.-S. Sorea, and S. Telen.

Two coniveau filtrations

• Jørgen Vold Rennemo (University of Oslo)

Room: E-102 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 the cohomology groups of a variety. These filtrations are known to coincide in many cases, but Benoist and Ottem have recently given examples to show that they differ in general. I will tell that story and explain a construction of some new examples where the filtrations differ, which are found in joint work with John Christian Ottem.

A Lecture on Associative Varieties

• Arvid Siqveland (University of South-Eastern Norway)

Room: E-102 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 associative rings O_X on the topological space X = aSpec A giving it the structure of a locally ringed space. In general, an associative variety X is a ringed space with an open covering {Ui = aSpec Ai}_{i∈I}. When A is a commutative k-algebra, aSpec A ≃ Spec A, and so the category aVark of associative varieties is an extension of the category of varieties Vark , i.e. there exists a faithfully flat functor I : Vark → aVark . Our main result says that any associative variety X is aSpec(O_X(X)) for the k-algebra O_X(X), and so any study of varieties can be reduced to the study of the associative algebra O_X(X). (We make Geometry Algebraic again.)

The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces

• Nikolas Kuhn (University of Oslo)

Room: E-102 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 setting of framed sheaves on ℙ2. The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on ℙ2, which has recently been extend to the setting of Gieseker H-stable sheaves on H-polarised surfaces by Kuhn-Tanaka.