03842nam a22006135i 4500 978-3-031-69067-9 DE-He213 20251030113250.0 cr nn 008mamaa 241029s2024 sz | s |||| 0|eng d 9783031690679 978-3-031-69067-9 10.1007/978-3-031-69067-9 doi QA564-609 PBMW bicssc MAT012010 bisacsh PBMW thema Cotti, Giordano. author. aut http://id.loc.gov/vocabulary/relators/aut 86777 Helix Structures in Quantum Cohomology of Fano Varieties [electronic resource] / by Giordano Cotti, Boris A. Dubrovin, Davide Guzzetti. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Springer, 2024. XIII, 236 p. 18 illus., 15 illus. in color. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Lecture Notes in Mathematics, 1617-9692 ; 2356 - Introduction -- Gromov–Witten Theory and Quantum Cohomology -- Helix Theory in Triangulated Categories -- Non-Symmetric Orthogonal Geometry of Mukai Lattices -- The Main Conjecture -- Proof of the Main Conjecture for Projective Spaces -- Proof of the Main Conjecture for Grassmannians. This research monograph provides a comprehensive study of a conjecture initially proposed by the second author at the 1998 International Congress of Mathematicians (ICM). This conjecture asserts the equivalence, for a Fano variety, between the semisimplicity condition of its quantum cohomology and the existence of full exceptional collections in its derived category of coherent sheaves. Additionally, in its quantitative form, the conjecture specifies an explicit relation between the monodromy data of the quantum cohomology, characteristic classes, and exceptional collections. A refined version of the conjecture is introduced, with a particular focus on the central connection matrix, and a precise link is established between this refined conjecture and Γ-conjecture II, as proposed by S. Galkin, V. Golyshev, and H. Iritani. By performing explicit calculations of the monodromy data, the validity of the refined conjecture for all complex Grassmannians G(r,k) is demonstrated. Intended for students and researchers, the book serves as an introduction to quantum cohomology and its isomonodromic approach, along with its algebraic counterpart in the derived category of coherent sheaves. Algebraic geometry. Mathematical physics. Differential equations. Geometry, Differential. 69684 Algebra, Homological. 70150 Algebraic Geometry. Mathematical Physics. Differential Equations. Differential Geometry. 69686 Category Theory, Homological Algebra. 70155 Dubrovin, Boris A. author. aut http://id.loc.gov/vocabulary/relators/aut 86778 Guzzetti, Davide. author. aut http://id.loc.gov/vocabulary/relators/aut 86779 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031690662 Printed edition: 9783031690686 Lecture Notes in Mathematics, 1617-9692 ; 2356 75434 https://doi.org/10.1007/978-3-031-69067-9 ZDB-2-SMA ZDB-2-SXMS ZDB-2-LNM 64576 64576