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Books on the topic 'Abelian surfaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

H, Lange. Complex Abelian varieties. Berlin: Springer-Verlag, 1992.

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2

1943-, Lange H., and Lange H. 1943-, eds. Complex Abelian varieties. 2nd ed. Berlin: Springer, 2004.

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3

Lange, Herbert. Complex Abelian varieties. Berlin: Springer, 1992.

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4

Faltings, Gerd. Degeneration of Abelian varieties. Berlin: Springer-Verlag, 1990.

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5

Muñoz Porras, José M., Sorin Popescu, and Rubí E. Rodríguez, eds. The Geometry of Riemann Surfaces and Abelian Varieties. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/conm/397.

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6

Hulek, Klaus. Moduli spaces of Abelian surfaces: Compactification, degenerations, and theta functions. Berlin: Walter de Gruyter, 1993.

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7

Muñoz Porras, Jose M. 1956-, Popescu Sorin 1963-, Rodríguez Rubí E. 1953-, and Recillas-Pishmish Sevín 1943-, eds. The geometery [sic] of Riemann surfaces and Abelian varieties: III Iberoamerican Congress on Geometry in honor of Professor Sevin Recillas-Pishmish's 60th birthday, June 8-12, 2004, Salamanca, Spain. Providence, RI: American Mathematical Society, 2006.

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8

Complex analysis 2: Riemann surfaces, several complex variables, abelian functions, higher modular functions. Heidelberg: Springer, 2011.

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9

Hironaka, Eriko. Abelian coverings of the complex projective plane branched along configurations of real lines. Providence, R.I: American Mathematical Society, 1993.

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10

Rodríguez, Rubí E., 1953- editor of compilation, ed. Riemann and Klein surfaces, automorphisms, symmetries and moduli spaces: Conference in honor of Emilio Bujalance on Riemann and Klein surfaces, symmetries and moduli spaces, June 24-28, 2013, Linköping University, Linköping, Sweden. Providence, Rhode Island: American Mathematical Society, 2014.

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11

Lange, Herbert, and Christina Birkenhake. Complex Abelian Varieties. Springer Berlin Heidelberg, 2010.

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12

Faltings, Gerd. Degeneration of Abelian Varieties. Springer, 2010.

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13

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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14

Vorlesungen über Riemann's theorie der Abel'schen integrale. 2nd ed. Leipzig: B. G. Teubner, 1991.

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15

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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16

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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17

The Beilinson Complex And Canonical Rings of Irregular Surfaces (Memoirs of the American Mathematical Society). American Mathematical Society, 2006.

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18

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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19

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

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