Abstract:
This is a detailed review of helix theory, which describes exceptional sheaves and exceptional bases for derived categories of coherent sheaves on Fano varieties. We explain systematically all basic ideas and constructions related to exceptional objects. Projective spaces and Del Pezzo surfaces are considered especially extensively. Some arithmetic relationships with the mirror symmetry phenomenon are discussed as well. This paper may be considered as a necessary supplement to the book [HuLe], which completely ignores rich structures beyond the zero-dimensional moduli spaces.
Key words and phrases:
Exceptional collections, mutations, semiorthogonal decompositions in triangulated categories.
This publication is cited in the following 47 articles:
Filip Gawron, Ozhan Genc, “$\ell $-away ACM bundles on Fano surfaces”, Boll Unione Mat Ital, 17:1 (2024), 75
Tomoki Yoshida, “Full exceptional collections of line bundles on the blow-up of ${\mathbb {P}}^{5}$ along Segre threefold”, manuscripta math., 2024
Johannes Krah, “Mutations of numerically exceptional collections on surfaces”, Math. Z., 307:4 (2024)
Giordano Cotti, Boris A. Dubrovin, Davide Guzzetti, Lecture Notes in Mathematics, 2356, Helix Structures in Quantum Cohomology of Fano Varieties, 2024, 125
Giordano Cotti, Boris A. Dubrovin, Davide Guzzetti, Lecture Notes in Mathematics, 2356, Helix Structures in Quantum Cohomology of Fano Varieties, 2024, 53
Giordano Cotti, Boris A. Dubrovin, Davide Guzzetti, Lecture Notes in Mathematics, 2356, Helix Structures in Quantum Cohomology of Fano Varieties, 2024, 1
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Dmitry Galakhov, “On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry”, J. High Energ. Phys., 2022:10 (2022)
V. Antonelli, F. Malaspina, “H-instanton bundles on three-dimensional polarized projective varieties”, Journal of Algebra, 598 (2022), 570
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