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Mosc. Math. J., 2004, Volume 4, Number 2, Pages 377–440 (Mi mmj154)  

This article is cited in 25 scientific papers (total in 25 papers)

Helix theory

A. L. Gorodentsevab, S. A. Kuleshovbc

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
b Independent University of Moscow
c N. E. Zhukovskii Military Aviation Engineering University

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.


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MSC: 14F05, 14J60, 18F30, 32L10
Received: May 30, 2003

Citation: A. L. Gorodentsev, S. A. Kuleshov, “Helix theory”, Mosc. Math. J., 4:2 (2004), 377–440

Citation in format AMSBIB
\by A.~L.~Gorodentsev, S.~A.~Kuleshov
\paper Helix theory
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 2
\pages 377--440

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    This publication is cited in the following articles:
    1. Costa L., Miró-Roig R.M., “Cohomological characterization of vector bundles on multiprojective spaces”, J. Algebra, 294:1 (2005), 73–96  crossref  mathscinet  zmath  isi  scopus
    2. Rouquier R., “Catégories dérivées et géométrie birationnelle (d'après Bondal, Orlov, Bridgeland, Kawamata et al.) [Derived categories and birational geometry (after Bondal, Orlov, Bridgeland, Kawamata et al.)]”, Séminaire Bourbaki, Vol. 2004/2005, Exposés 938–951, Astérisque, 307, Exp. No. 946, Société Mathématique de France, Paris, 2006, 283–307  mathscinet  zmath  isi
    3. Fukaya K., Seidel P., Smith I., “Exact Lagrangian submanifolds in simply-connected cotangent bundles”, Invent. Math., 172:1 (2008), 1–27  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Miró-Roig R.M., Soares H., “Cohomological characterisation of Steiner bundles”, Forum Math., 21:5 (2009), 871–891  crossref  mathscinet  zmath  isi  scopus
    5. Simpson C., “Katz's Middle Convolution Algorithm”, Pure Appl. Math. Q., 5:2, Special Issue: In honor of Friedrich Hirzebruch, Part 1 (2009), 781–852  crossref  mathscinet  zmath  isi  elib
    6. Seidel P., “Symplectic homology as Hochschild homology”, Proceedings of Symposia in Pure Mathematics: Algebraic Geometry Seattle 2005, Proceedings of Symposia in Pure Mathematics, 80, no. 1- 2, 2009, 415–434  crossref  mathscinet  zmath  isi
    7. Fukaya K., Seidel P., Smith I., “The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint”, Homological Mirror Symmetry: New Developments and Perspectives, Lecture Notes in Physics, 757, 2009, 1–26  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Bridgeland T., Stern D., “Helices on del Pezzo surfaces and tilting Calabi-Yau algebras”, Adv Math, 224:4 (2010), 1672–1716  crossref  mathscinet  zmath  isi  scopus
    9. Polishchuk A., “K-theoretic exceptional collections at roots of unity”, J K Theory, 7:1 (2011), 169–201  crossref  mathscinet  zmath  isi  elib  scopus
    10. De Deken O., Lowen W., “Abelian and derived deformations in the presence of Z-generating geometric helices”, J Noncommut Geom, 5:4 (2011), 477–505  crossref  mathscinet  zmath  isi  scopus
    11. Polishchuk A., “Simple Helices on Fano Threefolds”, Canad Math Bull, 54:3 (2011), 520–526  crossref  mathscinet  zmath  isi
    12. Buan A.B., Reiten I., Thomas H., “From M-Clusters to M-Noncrossing Partitions via Exceptional Sequences”, Math. Z., 271:3-4 (2012), 1117–1139  crossref  mathscinet  zmath  isi  scopus
    13. Costa L., Maria Miro-Roig R., “Derived Category of Toric Varieties with Small Picard Number”, Cent. Eur. J. Math., 10:4 (2012), 1280–1291  crossref  mathscinet  zmath  isi  elib  scopus
    14. Okawa Sh., Uehara H., “Exceptional Sheaves on the Hirzebruch Surface F-2”, Int. Math. Res. Notices, 2015, no. 23, 12781–12803  crossref  mathscinet  zmath  isi  scopus
    15. Galkin S., Golyshev V., Iritani H., “Gamma Classes and Quantum Cohomology of Fano Manifolds: Gamma Conjectures”, Duke Math. J., 165:11 (2016), 2005–2077  crossref  mathscinet  zmath  isi  scopus
    16. Maria Miro-Roig R., Soares H., “Exceptional Bundles of Homological Dimension K”, Forum Math., 29:3 (2017), 701–715  crossref  mathscinet  zmath  isi  scopus
    17. Faenzi D., Malaspina F., “Surfaces of Minimal Degree of Tame Representation Type and Mutations of Cohen-Macaulay Modules”, Adv. Math., 310 (2017), 663–695  crossref  mathscinet  zmath  isi  scopus
    18. Chunyi Li, “Deformations of the Hilbert scheme of points on a del Pezzo surface”, Mosc. Math. J., 17:2 (2017), 291–321  mathnet  crossref  mathscinet
    19. Sanna G., “Small Charge Instantons and Jumping Lines on the Quintic Del Pezzo Threefold”, Int. Math. Res. Notices, 2017, no. 21, 6523–6583  crossref  isi  scopus
    20. Cotti G. Guzzetti D., “Analytic Geometry of Semisimple Coalescent Frobenius Structures”, Random Matrices-Theor. Appl., 6:4, SI (2017), 1740004  crossref  zmath  isi  scopus
    21. Abe T., “Semistable Sheaves With Symmetric C(1) on a Quadric Surface”, Nagoya Math. J., 227 (2017), 86–159  crossref  isi  scopus
    22. Arcara D., Miles E., “Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2”, Int. Math. Res. Notices, 2017, no. 11, 3426–3462  crossref  isi
    23. Aprodu M., Huh S., Malaspina F., Pons-Llopis J., “Ulrich Bundles on Smooth Projective Varieties of Minimal Degree”, Proc. Amer. Math. Soc., 147:12 (2019), 5117–5129  crossref  mathscinet  zmath  isi  scopus
    24. Dimitrov G., Katzarkov L., “Some New Categorical Invariants”, Sel. Math.-New Ser., 25:3 (2019), UNSP 45  crossref  mathscinet  isi  scopus
    25. Giordano Cotti, Boris Dubrovin, Davide Guzzetti, “Local Moduli of Semisimple Frobenius Coalescent Structures”, SIGMA, 16 (2020), 040, 105 pp.  mathnet  crossref
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