Abstract
I give a thorough introduction to the global theory of, possibly singular, symplectic complex spaces. The spaces are assumed to be of Kähler type for the most part. My presentation focuses on the study of proper and flat deformations. The corresponding local theory of symplectic singularities is hardly touched upon. The key results are a local Torelli theorem on the second cohomology, which generalizes the local Torelli theorem for irreducible symplectic complex manifolds to potentially singular spaces, as well as an analogous generalization of the Fujiki relation. Moreover, an entire section is devoted to developing the Beauville-Bogomolov quadratic form in the generalized context.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Less frequently the Beauville-Bogomolov-Fujiki form.
- 2.
I slightly deviate from Beauville’s original formula [2, p. 772] by writing \(w^{r-1}\bar{w}^{r-1}\) instead of \((w\bar{w})^{r-1}\) as the former has a more natural feel in calculations.
- 3.
I should note that Grauert [12] calls “versell” what we call “semi-universal.” This might be confusing given that nowadays people use the English word “versal” as a synonym for “complete,” which is a condition strictly weaker than that of semi-universality. In other words, Grauert’s (German) “versell” is not equivalent to, but strictly stronger than the contemporary (English) “versal.”
References
C. Bănică, O. Stănăşilă, Algebraic Methods in the Global Theory of Complex Spaces. Editura Academiei, Bucharest (Wiley, London, 1976), p. 296
A. Beauville, Variétés kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983)
A. Beauville, Symplectic singularities. Invent. Math. 139(3), 541–549 (2000)
J. Bingener, On deformations of Kähler spaces I. Math. Z. 182(4), 505–535 (1983). doi:10.1007/BF01215480
F.A. Bogomolov, On the decomposition of Kähler manifolds with trivial canonical class. Math. USSR. Sb. 22(4), 580–583 (1974)
F.A. Bogomolov, Hamiltonian Kähler manifolds. Sov. Math. Dokl. 19, 1462–1465 (1978)
B. Conrad, Grothendieck Duality and Base Change. Lecture Notes in Mathematics, vol. 1750 (Springer, Heidelberg, 2000), pp. vi+296. doi:10.1007/b75857
R. Elkik, Singularités rationnelles et déformations. Invent. Math. 47(2) 139–147 (1978)
G. Fischer, Complex Analytic Geometry. Lecture Notes in Mathematics, vol. 538 (Springer, Heidelberg, 1976), pp. vii+201
H. Flenner, Extendability of differential forms on non-isolated singularities. Invent. Math. 94(2), 317–326 (1988)
A. Fujiki, On the de Rham Cohomology Group of a Compact Kähler smplectic manifold, in Algebraic Geometry, Sendai, 1985. Adv. Stud. Pure Math., vol. 10 (Amsterdam, North-Holland, 1987), pp. 105–165
H. Grauert, Der Satz von Kuranishi für kompakte komplexe Räume Invent. Math. 25 107–142 (1974)
H. Grauert, R. Remmert, Coherent Analytic Seaves. Grundlehren der mathematischen Wissenschaften, vol. 265 (Springer, Heidelberg, 1984)
G.-M. Greuel, C. Lossen, E. Shustin, Introduction to Singularities and Deformations. Springer Monographs in Mathematics (Springer, Heidelberg, 2007)
D. Huybrechts, Compact hyperkähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999)
D. Huybrechts, Complex Geometry: An Introduction (Universitext, Springer, Heidelberg, 2005)
L. Illusie, Complexe Cotangent et Déformations I. Lecture Notes in Mathematics, vol. 239 (Springer, Heidelberg 1971), pp. xv+355
M. Kneser, R. Scharlau, Quadratische Formen (Springer, Heidelberg, 2002)
J. Kollár, S. Mori, Classification of three-dimensional flips. J. Am. Math. Soc. 5(3), 533–703 (1992). doi:10.2307/2152704
S. Lang, Algebra, 3rd rev. edn. Graduate Texts in Mathematics, vol. 211 (Springer, New York, 2002), pp. xvi+914. doi:10.1007/978-1-4613-0041-0
D. Matsushita, Fujiki relation on symplectic varieties (2001). arXiv: math.AG/0109165
B.G. Moishezon, Singular Kählerian spaces, in Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) (University of Tokyo Press, Tokyo, 1975), pp. 343–351
T. Muir, A Treatise on the Theory of Determinants (Dover, New York 1960), pp. vii+766
Y. Namikawa, Deformation theory of singular symplectic n-folds. Math. Ann. 319, 597–623 (2001)
Y. Namikawa, Extension of 2-forms and symplectic varieties. J. Reine Angew. Math. 539, 123–147 (2001)
Y. Namikawa, On deformations of \(\mathbb{Q}\)-factorial symplectic varieties. J. Reine Angew. Math. 599, 97–110 (2006)
V.P. Palamodov, Deformations of complex spaces. Uspehi Mat. Nauk 31(189), 3, 129–194 (1976)
V.P. Palamodov, Deformations of complex spaces, in Several Complex Variables IV: Algebraic Aspects of Complex Analysis. Encyclopaedia of Mathematical Sciences (Springer, Heidelberg, 1990), pp. 105–194
E. Sernesi, Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, vol. 334 (Springer, Heidelberg 2006), pp. xii+339
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kirschner, T. (2015). Symplectic Complex Spaces. In: Period Mappings with Applications to Symplectic Complex Spaces. Lecture Notes in Mathematics, vol 2140. Springer, Cham. https://doi.org/10.1007/978-3-319-17521-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-17521-8_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17520-1
Online ISBN: 978-3-319-17521-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)