Danil Gubarevich soutient sa thèse, intitulée « On Gromov-Witten invariants and the Hall induction » encadrée par Dimitri Zvonkine, le jeudi 21 mai, 13h30, bâtiment Fermat, salle 2201.
Résumé :
The first two chapters of the manuscript provide the motivation for the subjects we were studying in this thesis. First we discuss the relation between the Gromov-Witten and Donaldson-Thomas enumerative theories and explain how the DT-theory was categorified by means of perverse sheaves of vanishing cycles. This part aims to show the link between Chapters 3 and 4.
Further we review the generalities on genus 0 Gromov-Witten theory with the focus on smooth complete intersections, relevant to Chapter 3.
To motivate Chapter 4, we review the notions of the classical Hall algebra of a finitary hereditary category and its cohomological enhancement by the cohomological Hall algebra, using quiver representations as a key example. These algebras serve as the primary inspiration for the constructions and theorems discussed in Chapter 4.
Finally, we state the results of our thesis.
Chapter 3 discusses GW-theory of even dimensional complete intersections X of two quadrics in CP^{m+2}. These varieties are exceptional from the point of view of the Gromov-Witten theory: they are (together with surfaces of degrees 2 and 3) the only complete intersections whose Gromov-Witten theory is not invariant under the full orthogonal or symplectic group acting on the primitive cohomology. The genus 0 Gromov-Witten theory of X was studied by Xiaowen Hu. He used geometric arguments and the associativity of quantum cohomology to compute all genus 0 correlators except one, which cannot be determined by his methods. In our paper, we use a different method, based on Jun Li’s degeneration formula. We show [Theorem (??)] the existence of a basis in cohomology of X, such that the remaining Gromov-Witten invariant in this basis vanishes.
In Chapter 4 we study an example of the Hall induction, the natural analog of the Hall multiplication in situations where the moduli space at hand is not the moduli space of objects in some abelian category. We study the Hall induction for the moduli space T^*(V/G): the cotangent stack of the quotient stack V/G, where V is a representation of a complex reductive group G. First we indicate how to define the Hall induction map via a representation of a Levi subgroup on the level of the critical Hall induction and then conjugate this map by the dimensional reduction isomorphism. We adapt a result proven for cohomological and K-theoretical Hall algebras of a quiver: the first is the torsion freeness of the Hall induction for the stack T^*(V/G) deformed by a certain torus T_s. This result is equivalent to an embedding of the Hall induction into the space of symmetric polynomials. As a corollary, we prove that the T_s-equivariant Borel-Moore homology of T^*(V/G) is concentrated in even homological degrees.