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Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. (English) Zbl 0848.14004
A normal crossing variety is a reduced complex analytic space which is locally isomorphic to a normal crossing divisor on a smooth variety. Moreover, it is called a simple normal crossing variety if the irreducible components are smooth. According to R. Friedman [Ann. Math., II. Ser. 118, 75-114 (1983; Zbl 0569.14002)], we can define the concept of \(d\)-semistability for simple normal crossing varieties. For example, the central fiber of a semistable degeneration is a \(d\)-semistable simple normal crossing variety. Friedman’s theorem (loc. cit.) states that for an arbitrary \(d\)-semistable simple normal crossing compact Kähler surface \(X\) with trivial \(K_X\) and \(H^1(X, {\mathcal O}_X)\), there exists a semistable degeneration \(f'\) as above, i.e., the \(d\)-semistable K3 surface \(X\) has a smoothing. The purpose of this paper is to give an alternative easy proof of Friedman’s theorem, and generalize it to higher dimensional varieties.

14D15 Formal methods and deformations in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
Full Text: DOI EuDML
[1] [A] M. Artin: On the solutions of analytic equations. Invent. Math.5, 277-291 (1968) · Zbl 0172.05301 · doi:10.1007/BF01389777
[2] [C] C.H. Clemens: Degenerations of Kähler manifolds. Duke Math. J.44, 215-290 (1977) · Zbl 0353.14005 · doi:10.1215/S0012-7094-77-04410-6
[3] [D] P. Deligne: Théorie de Hodge III. Publ. Math. IHES.44, 5-77 (1975)
[4] [F] R. Friedman: Global smoothings of varieties with normal crossings. Ann. Math.118, 75-114 (1983) · Zbl 0569.14002 · doi:10.2307/2006955
[5] [Gra] H. Grauert: Der Satz von Kuranishi für kompakte komplexe Räume. Invent. Math.25, 107-142 (1974) · Zbl 0286.32015 · doi:10.1007/BF01390171
[6] [Gro] A. Grothendieck: Revetement Etale et Groupe Fondamental (SGA1). Lecture Notes in Math., vol. 224, Berlin: Hodelberg New York: Springer, 1971
[7] [Kat] K. Kato: Logarithmic structures of Fontaine-Illusie. In: Algebraic Analysis Geometry and Number Theory (1988), Johns Hopkins University, pp. 191-224 · Zbl 0776.14004
[8] [K] Y. Kawamata: Unobstructed deformations?a remark on a paper of Z. Ran. J. Alg. Geom.1, 183-190 (1992). · Zbl 0818.14004
[9] [Ku] V. Kulikov: Degenerations of K3 and Enriques surfaces. Math. USSR-Izv.11, 957-989 (1977) · Zbl 0387.14007 · doi:10.1070/IM1977v011n05ABEH001753
[10] [P] U. Persson: On Degeneration of Surfaces. Mem. AMS., vol. 189, 1977 · Zbl 0368.14008
[11] [PP] U. Persson, H. Pinkham: Degeneration of surfaces with trivial canonical bundle. Ann. Math.113, 45-66 (1981) · Zbl 0447.14007 · doi:10.2307/1971133
[12] [R] Z. Ran: Deformations of manifolds with torsion or negative canonical bundle. J. Alg. Geom.1, 279-291 (1992) · Zbl 0818.14003
[13] [Sch] M. Schlessinger: Functors of Artin rings. Trans. AMS130, 208-222 (1968) · Zbl 0167.49503 · doi:10.1090/S0002-9947-1968-0217093-3
[14] [St] J. Steebrink: Limits of Hodge structure. Invent. Math.31, 229-257 (1976) · Zbl 0312.14007 · doi:10.1007/BF01403146
[15] [Wa] J. Wavrick: Obstructions to the existence of a space of moduli. In: Global Analysis, papers in honor of K. Kodaira, D.C. Spencer and S. Iyanaga (eds.), Univ. Tokyo Press and Princeton University Press, 1969, pp. 403-414
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