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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.

##### MSC:
 14D15 Formal methods and deformations in algebraic geometry 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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##### References:
 [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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