Location: tbd (Mathematical Institute, Endenicher Allee 60)
Current Time:Tuesday, 4pm
Organizers: Georg Oberdieck
Archive (of the Algebraic Geometry and Gromov-Witten Theory Seminar, 2021):
Archive (of the Algebraic Geometry, Physics, and Gromov-Witten theory Seminar):
Abstracts:
Denis Nesterov - Enumerative mirror symmetry of moduli spaces of Higgs bundles and S-duality: We propose a bunch of conjectures, called “Enumerative mirror symmetry”, which relate curve-counting invariants of moduli spaces of Higgs SL-bundles and Higgs PGL-bundles. These conjectures lie somewhere between Topological mirror symmetry (Hausel-Thaddeus conjectures) and Categorical mirror symmetry (Geometric Langlands correspondence). I will (try to) explain how Enumerative mirror symmetry is related to so-called S-duality (conjectures formulated mathematically by Jiang-Kool). This relation is the source of the main evidence for Enumerative mirror symmetry.
Weite Pi: The moduli spaces of one-dimensional sheaves on \mathbb{P}^2 are first studied by Carlos Simpson and Le Potier, and they admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one should be able to obtain certain BPS invariants from this morphism. In this talk, we investigate the cohomology ring structure of these moduli spaces. We will derive a minimal set of tautological generators for the cohomology ring, and discuss how they are related, through the key notion of perversity, to curve counting invariants for local \mathbb{P}^2 . Based on joint work with Junliang Shen, and with Junliang Shen and Yakov Kononov in progress.
Nikolas Kuhn: The Atiyah class is the basic tool to construct perfect obstruction theories on moduli spaces of sheaves and perfect complexes. There are important variants for related problems: The reduced Atiyah class governs the deformation theory for Quot-schemes, and one has a deformation-obstruction class for spaces of morphisms between to given sheaves. Moreover, these classes have compatibilities which allow for construction of compatible relative obstruction theories. I will discuss these properties and additionally address how one can extend the Atiyah class and its variants to the setting of algebraic (Artin-) stacks. The results obtained are possibly implied by constructions in derived algebraic geometry, but do not rely on derived methods beyond the use of simplicial resolutions and allow for a self-contained treatment. We hope this makes them more accessible to algebraic geometers.
Martijn Kool: Hilbert schemes of points on affine space of dimension d>2 are in general very singular. For d=3 they can be realized as critical loci of regular functions on smooth ambient spaces. For d=4 they can be realized as zero loci of isotropic sections of quadratic bundles on smooth ambient spaces. These serve as the “local models” of Donaldson-Thomas theory of Calabi-Yau 3- and 4-folds and can be used to virtually enumerate 3- and 4-dimensional piles of boxes.
Sergej Monavari: Classically, Donaldson-Thomas invariants are integer valued invariants that virtually count stable coherent sheaves on Calabi-Yau threefolds. On a Calabi-Yau fourfold, higher obstructions prevent the existence of virtual fundamental classes in the sense of Behrend-Fantechi. Nevertheless, Borisov-Joyce (via derived differential geometry) and Oh-Thomas (via deformation theory) constructed virtual fundamental classes in this setting, modulo choices of orientations. We review their constructions and explain how to define naturally numerical and torus-equivariant invariants. Finally we discuss how, conjecturally, DT/PT/GW/GV invariants are related to each other and show instances where the conjectures have been checked. This is based on joint work with Y. Cao and M. Kool.
Rosa Schwarz: The most illustrative case of a Gromov-Witten invariant is the number of degree d curves in a projective plane through a number of points. There one considers the moduli space of stable maps of degree d from curves to the projective plane, and represent the passing-through-points conditions as intersecting classes of pullbacks via evaluation maps. Finally, the numerical invariants are computed by taking an integral, i.e. pushing forward to a point. However, one may also have enumerative questions not about curves in a certain variety X, but about line bundles on families of curves. Frenkel, Teleman and Tolland in their article [Gromov-Witten Gauge Theory] construct similar GW-invariants for the moduli space of stable curves together with line bundles, i.e. maps to the quotient stack BGm. Even after choosing a suitable compactification, it is not trivial to show the pushforward to a point actually exists and yields numbers. In this talk, I will illustrate the construction of these Gromov-Witten invariants in the case of genus zero 3-marked curves, and hopefully towards the end of the talk actually compute some of these numbers.
Luca Battistella: The study of curves in projective space is the basis of classical enumerative geometry. Kontsevich's space of stable maps provides a well-behaved compactification for rational curves, but in positive genus the locus of smooth curves is usually not even dense. When the degree is higher than the Riemann-Roch bound, we define the "main component" to be the closure of this locus. It is still Vakil-Murphy singular, and its modular interpretation is mysterious (which stable maps are smoothable?). One way to desingularise the main component is by blowing up its boundary according to local equations in a smooth ambient space (Hu, Li, Niu, Vakil, Zinger...). A different approach - championed by Ranganathan--Santos-Parker--Wise in genus one - highlights the modular interpretation via logarithmic geometry: the exceptional divisors represent contractions to Gorenstein curves such that the stable map factors through them in a non-special way. In joint work with Francesca Carocci, we extend the construction to curves of genus two, displaying a wide array of new phenomena. Firstly, the geometry of Gorenstein singularities appears to be intimately related to the canonical map of their semistable tails, both at an algebro-geometric and at a tropical level. Indeed, we construct a logarithmic modification of Mochizuki's space of admissible covers over which the universal contraction lives. Secondly, non-reduced curves appear naturally all over the place. As a consequence, we describe new criteria for a genus two stable map to be smoothable.
Oliver Leigh: A stable map is said to have "divisible ramification" if the order of every ramification locus is divisible by r (a fixed positive integer). In this talk I'll review the theory of stable maps with divisible ramification and discuss how this leads to a geometric framework from which to view and prove Zvonkine's r-ELSV formula. I will also discuss recent results within this framework.