Gromov-Witten Theory, geometry and representations
Final Report Abstract
The project is concerned with fundamental research at the crossroad of Algebraic Geometry, Representation Theory and Algebraic Combinatorics. Algebraic Geometry studies geometric objects that can be described by polynomial equations, i.e. equations only involving the simplest mathematical operations + and ×. A very classical aspect of Algebraic Geometry is studied in so-called Enumerative Geometry which consists in counting the number of solutions in particular geometric situations (e.g., in ordinary 3-dimensional space, there is a unique line passing through 2 points). In Modern Enumerative Geometry, the above numbers are used to define a new product giving rise to the Quantum Cohomology Ring. These Quantum Cohomology Rings are still mysterious even for simple geometric objects. The objective of the founded project was to study such rings for geometric objects that posses many symmetries; they are called homogeneous spaces and spherical varieties (geometric objects built on the model of a sphere). The rich supply of symmetries (by its group action) makes the study simpler; sometimes it translates the geometric problem into an explicit problem which can be solved with the help of combinatorics and/or the aid of a computer implementation. The Principal Investigator and his collaborators made important advances on the above subject: they created an understanding of the structure of quantum cohomology for new geometric objects and developed new geometric techniques to attack harder problems in the field.
Publications
- (2019) On quantum cohomology of Grassmannians of isotropic lines, unfoldings of An -singularities, and Lefschetz exceptional collections. Ann. inst. Fourier (Annales de l’institut Fourier) 69 (3) 955–991
Cruz Morales, John Alexander; Mellit, Anton; Perrin, Nicolas; Smirnov, Maxim
(See online at https://doi.org/10.5802/aif.3263) - Algebraic rational cells, equivariant intersection theory, and Poincaré duality. Math. Zeit., 282 (2016), no. 1-2, 79–97
R. Gonzales
(See online at https://doi.org/10.1007/s00209-015-1533-5) - Fano varieties in Mori fibre spaces, IMRN (2016)
G. Codogni, A. Fanelli, R. Svaldi and L. Tasin
(See online at https://doi.org/10.1093/imrn/rnv173) - Rational connectedness implies finiteness of quantum K-theory. Asian Journal of Mathematics, 20 (2016), no. 1, 117–122
A. Buch, P.-E. Chaput, L. Mihalcea, and N. Perrin
(See online at https://doi.org/10.4310/AJM.2016.v20.n1.a5) - A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties. Algebraic Geometry, 5 (2018), no. 5, 568–595
A. Buch, P.-E. Chaput, L. Mihalcea, and N. Perrin
(See online at https://doi.org/10.14231/AG-2018-015) - Projected Gromov-Witten varieties in cominuscule spaces. Proceedings of the AMS, 146 (2018), no. 9, 3647–3660
A. Buch, P.-E. Chaput, L. Mihalcea, and N. Perrin
(See online at https://doi.org/10.1090/proc/13839)
