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Moduli spaces of pfaffian representations of cubic three-folds and instanton bundles

Grant number: 19/21140-1
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): February 01, 2020
Effective date (End): January 31, 2022
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal Investigator:Marcos Benevenuto Jardim
Grantee:Gaia Comaschi
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:18/21391-1 - Gauge theory and algebraic geometry, AP.TEM

Abstract

In 1999, Dimitri Markushevich and Alexander Tikhomirov described a family of rank 2 vector bundles with "natural" Cohomology, referred to as instantons, on a smooth cubic threefold X. These locally free sheaves share indeed common features with instantons on P3, vector bundles originally studied in the context of Yang-Mills gauge theory by Atiyah-Drinfeld-Hitchin-Manin. In particular, instantons F on X with c2(F)=2 satisfy a remarkable property: similar to what happens forinstantons on the projective space, F can be represented as the Cohomology of a very simple complex. F is indeed a twist of a skew-symmetric Ulrich sheaf, a coherent sheaf on X whose minimal resolution in P4 is a complex determined by a skew-symmetric matrix M of linear forms such that X is defined by Pf (M) = 0. This fact draw attention to the relation between instantons on X and Pfaffian representations of X; this relation was the starting point of my PhD thesis. In thethesis, replacing instantons by their minimal resolution, we constructed the moduli space P of Pfaffian representations of cubic threefolds and we inspected how the results about P apply to the study of moduli of instantons. First of all we proved that every 3-dimensional cubic is Pfaffian, a result previously known only for general cubics. The moduli space P is obtained as the GIT quotient of P74, the projective space of 6X6 skew-symmetric matrices of linear forms, for the action of SL(6). Our first step in describing P was formulating a (semi)stability criterion; applying it we proved at first that every Pfaffian representation of a cubic threefold is semistable and that moreover every Pfaffian representation of a smooth one is stable. Secondly we presented a complete classification of stable matrices having Pfaffian identically equal to zero. Finally we considered M the moduli space of torsion sheaves on P4 which are instantons on smooth cubic threefolds, and M, its closure in the moduli space of sheaves on P4. We located two divisors B', B''in the boundary of M; we showed that at the generic point of B', M is the blowup of P along a smooth subvariety and conjectured that the same holds for B''. These results pave the way for new interesting research themes. For instance, it would be relevant to present a complete description of the locus of semistable matrices in the space P74. To this aim we might firstly try to classify all Pfaffian representations of 3-dimensional cubics (for the moment we have proved that every cubic threefold admits at least one Pfaffian representation but this might not be unique) and consequently to classify those that are stable and strictly semistable. This first goal naturally extends to a deeper study of skew-symmetric Ulrich sheaves together with their geometric properties (such as their singularities). In the second place we might finalize the classification of semistable matrices having Pfaffian identically equal to zero, determining those that are strictly semistable. A better understanding of P has useful applications to the study of moduli of instantons; P is indeed a new compactification of M and it would then be significant to describe its boundary. Finally, a proof of our conjecture on B'' would establish a relation between P and M similar to the one between the Donaldson-Uhlenbeck and the Gieseker-Maruyama moduli spaces of instantons on algebraic surfaces, where the latter is obtained from the former by a blowup. This will allow us to consider P as a sort of Donaldson-Uhlenbeck compactification of M. (AU)