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Books on the topic 'K3 surfaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 October 2024

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1

Kondō, Shigeyuki. K3 surfaces. Berlin, Germany: European Mathematical Society, 2020.

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2

V, Nikulin V., ed. Del Pezzo and K3 surfaces. Tokyo: Mathematical Society of Japan, 2006.

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3

Faber, Carel, Gavril Farkas, and Gerard van der Geer, eds. K3 Surfaces and Their Moduli. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29959-4.

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4

Johnsen, Trygve. K3 Projective models in scrolls. Berlin: Springer, 2004.

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5

Johnsen, Trygve. K3 Projective models in scrolls. Berlin: Springer, 2004.

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6

Nikoloudakis, Nikolaos. Special K3 surfaces and Fano 3-folds. [s.l.]: typescript, 1986.

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7

France, Société mathématique de, ed. Géométrie des surfaces K3: Modules et périodes. Paris: Société mathématique de France, 1985.

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8

Scattone, Francesco. On the compactification of moduli spaces for algebraic K3 surfaces. Providence, R.I: American Mathematical Society, 1987.

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9

Laza, Radu, Matthias Schütt, and Noriko Yui, eds. Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-6403-7.

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10

Odaka, Yūji. Collapsing K3 surfaces, tropical geometry and moduli compactifications of Satake, Morgan-Shalen type. Tokyo, Japan: The Mathematical Society of Japan, 2021.

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11

1982), Séminaire Palaiseau (Octobre 1981-Janvier. Géométrie des surfaces K3: Modules et périodes : Séminaire Palaiseau, Octobre 1981-Janvier 1982. Paris: Société Mathématique de France, 1985.

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12

Pal, Vivek. Simultaneous twists of elliptic curves and the Hasse principle for certain K3 surfaces. [New York, N.Y.?]: [publisher not identified], 2016.

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13

Alta.) WIN (Conference) (2nd 2011 Banff. Women in Numbers 2: Research directions in number theory : BIRS Workshop, WIN2 - Women in Numbers 2, November 6-11, 2011, Banff International Research Station, Banff, Alberta, Canada. Edited by David Chantal 1964-, Lalín Matilde 1977-, and Manes Michelle 1970-. Providence, Rhode Island: American Mathematical Society, 2013.

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14

Huybrechts, D. K3 Surfaces. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0010.

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After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.

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15

Huybrechts, Daniel. Lectures on K3 Surfaces. Cambridge University Press, 2016.

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16

Lectures on K3 Surfaces. Cambridge University Press, 2016.

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17

Lectures on K3 Surfaces. 2016.

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18

K3 surfaces of high rank. 2006.

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19

Geer, Gerard van der, Gavril Farkas, and Carel Faber. K3 Surfaces and Their Moduli. Birkhauser Verlag, 2016.

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20

Del Pezzo and K3 Surfaces. Tokyo, Japan: The Mathematical Society of Japan, 2006. http://dx.doi.org/10.2969/msjmemoirs/015010000.

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21

Pandharipande, Rahul. Maps, Sheaves and K3 Surfaces. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198784913.003.0005.

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The conjectural equivalence of curve counting on Calami- Yau 3-folds via stable maps and stable pairs is discussed. By considering Cali-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The proof of the complete Yau-Zaslow conjecture is surveyed.

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22

K3 Surfaces and Their Moduli. Birkhäuser, 2016.

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23

Geer, Gerard van der, Gavril Farkas, and Carel Faber. K3 Surfaces and Their Moduli. Birkhäuser, 2018.

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24

Yu, Jeng-Daw. On ordinary K3 surfaces over Fp. 2006.

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25

Géométrie des surfaces K3: Modules et périodes. [Paris]: Société mathématique de France, 1985.

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26

Arithmetic And Geometry Of K3 Surfaces And Calabiyau Threefolds. Springer-Verlag New York Inc., 2013.

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27

Yui, Noriko, Radu Laza, and Matthias Schütt. Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Springer London, Limited, 2013.

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28

Voisin, Claire. On the Chow ring of K3 surfaces and hyper-Kahler manifolds. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160504.003.0005.

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This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ‎ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π‎ : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.

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29

Zaytman, Yevgeny K. K3 surfaces of high Picard number and arithmetic applications. 2010.

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30

Huybrechts, D. Derived Categories of Surfaces. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0012.

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This chapter completes the classification of algebraic surfaces from the point of view of their derived categories. Abelian, K3, and elliptic surfaces play a special role. For all other surfaces, the derived category determines the isomorphism type. The reduction to minimal surfaces is due to Kawamata, and the case of elliptic surfaces was dealt with by Bridgeland and Maciocia.

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31

Ge ome trie des surfaces K3: Modules et pe riodes. [Paris]: Socie te mathe matique de France, 1985.

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32

Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Springer New York, 2015.

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33

On the compactification of moduli spaces for algebraic k3 surfaces. 1985.

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34

Huybrechts, D. Where to Go from Here. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0013.

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This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.

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35

Huybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001.

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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

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36

Voisin, Claire. Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160504.001.0001.

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This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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37

A Celebration of Algebraic Geometry (Clay Mathematics Proceedings). American Mathematical Society, 2013.

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