VBAC 2022: Moduli spaces and vector bundles new trends

webpage of the venue (warwick university)

Webpage of the event

Notes on some of my favourite talks, don't forget to check the VBAC's webpage for the recordings.

Some of my favourite posters

• Leonardo Roa-Leguizamón, Los Andes, Colombia, ON-LINE ONLY 

On the Segre Invariant for Rank Two Vector Bundles on P 2 . We extend the concept of the Segre Invariant for vector bundles on a curve to vector bundles on a surface X. For a vector bundle 5 E of rank 2 on X, the Segre invariant is defined as the minimum of the differences between the slope of E and the slope of all line subbundles of E. This invariant defines a semicontinuous function on the families of vector bundles on X. Thus, the Segre invariant gives a stratification of the moduli space MX,H(2; c1, c2) of H-stable vector bundles of rank 2 and fixed Chern classes c1 and c2 on the surface X into locally closed subvarieties MX,H(2; c1, c2; s) according to the value of s. For X = P 2 we determine what numbers can appear as the Segre Invariant of a rank 2 vector bundle with given Chern’s classes. The irreducibility of strata with fixed Segre invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether Theory for rank 2 vector bundles on P 2 . This is a Joint work with H. Torres-López and A.G. Zamora (Universidad Autónoma de Zacatecas, Mexico) DOI:10.1515/advgeom-2021-0003. 

• Victor do Valle Pretti, Université de Bourgogne, France. 

Determinants, even instantons and Bridgeland stability. The moduli of instantons bundles over a Fano threefold have been under investigation by many authors during the last 50 years. Their relation with exceptional collections and monads is already proven to be useful in many situations, such as the ADHM equations and D.Faenzi’s work, for example. We will present how to prove they’re stable in the sense of Bridgeland and obtain their moduli space as an open subset of the moduli of Bridgeland stable objects. This is done by finding a region where Bridgeland and quiver stability coincide, hence also obtaining general information about these moduli spaces. The general theory behind this association was proven by E.Macr`ı, here we provide a systematic way of describing the quiver regions for any smooth projective variety, if they exist. With this in hand, we can prove that instantons are stable in the sense of Bridgeland and describe them as a subvariety of a quiver moduli space. 

• Andrés Ibáñez Núñez, University of Oxford, UK. 

The HKKP filtration in GIT. Haiden-Katzarkov-KonsevichPandit have defined a canonical filtration (the HKKP filtration) for any semistable quiver representation that describes the asymptotics of a natural gradient flow on its space of hermitian metrics. Roughly speaking, the HKKP filtration measures how far is a semistable representation from being polystable. Moduli-theoretically, this filtration defines a stratification of the semistable locus in the moduli of quiver representations. We show how an analogue of the HKKP filtration can be defined for general GIT set-ups, and how it is related to gradient flows and Kirwan blow-ups. 

• Richard Haburcak, Dartmouth College, USA. 

Maximal Brill–Noether Loci via K3 Surfaces The Brill– Noether loci Mr g,d parameterize curves of genus g admitting a linear system of rank r and degree d; when the Brill–Noether number is negative, they sit as proper subvarieties of the moduli space of genus g curves. We explain a strategy for distinguishing Brill–Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill– Noether loci. Via an analysis of the stability of Lazarsfeld–Mukai bundles, we obtain new lifting results for line bundles of type g 3 d which suffice to prove the maximal Brill–Noether loci conjecture in genus 9–19, 22, and 23 

• Andres Fernandez Herrero, Cornell University, USA. 

Harder-Narasimhan stratifications, Verlinde formulas and gauged Gromov-Witten invariants. Let C be a smooth projective curve, G a reductive group, and X an affine smooth variety equipped with an action of G. The moduli stack M ap(C, X/G) parametrizes pairs (E, s), where E is a G-bundle on C and s is a section of the associated fiber bundle E(X) over C. Many moduli spaces of interest arise from equipping this stack with a stability condition. We would like to explain work in progress joint with Eduardo Gonzalez and Daniel Halpern-Leistner where we use a compactification of M ap(C, X/G) via Kontsevich stable maps (known as the moduli stack of gauged maps) in order to equip M ap(C, X/G) with a Harder-Narasimhan stratification (each pair admits a canonical parabolic reduction compatible with the extra data of the section). As an application, we provide formulas for the indexes of certain tautological K-theory classes on the semistable locus.