New construction of complex manifold via conifold transition
This paper is related to a question raised by \textit{R. Friedman} [Proc. Symp. Pure Math. 53, 103--134 (1991; Zbl 0753.14035)]: could one find non-Kähler Calabi-Yau (complex structure with trivial canonical line bundle) structure on \(k\) connect sum of \(S^3 \times S^3\)? Friedman proved that this is true for \(k\geq 103\) in [loc. cit.]. Friedman used an example from \textit{H. Clemens}, [Publ. Math., Inst. Hautes Étud. Sci. 58, 231--250 (1983; Zbl 0529.14002)] that there is a quintic hypersurface \(M\) in \(\mathbb{C}\mathbb{P}^4\) with infinite many disjoint rational curves with normal bundle \({\mathcal O} (-1) \times {\mathcal O}(-1)\). In his paper, Clemens first constructed a quartic hypersurface \(S\) in \(\mathbb{C}\mathbb{P}^3\), which is a \(K3\) surface, with infinite many disjoint rational curves with normal bundles \({\mathcal O}(-2)\). Using \(S\) as an intersection of \(M\) with a hyperplane, he constructed \(M\). He used a torus bundle structure of \(S\) on \(\mathbb{C}\mathbb{P}^1\) to obtain the rational curves. First he constructed \(L_0\) as the zero section, \(L_1\) was a nonzero section. He then multiply \(L_1\) by \(n\) to get the \(L_n\) sections. Friedman blowed down \(l\) curves of the rational curves, which generated the \(b_2\), and deformed the resulted variety to a smooth non-Kähler Calabi-Yau threefold. Then according to \textit{M. Rossi} [J. Geom. Phys. 56, No. 9, 1940--1983 (2006; Zbl 1106.32019)] theorem 3.3, \(k\) is \(b_3 (M)/2+l-b_2 (M)\). In [\textit{P. Lu} and \textit{G. Tian}, Symp. Math. 36, 284--293 (1996; Zbl 0854.32016)], Lu and Tian used a different (and singular) quintic hypersurface in \(\mathbb{C}\mathbb{P}^4\) to obtain the manifold needed for \(2\leq k\leq 106\), see Theorem 2.2 there. They used the same method as Friedman. The complex structure given by Calabi-Eckmann on \(S^3 \times S^3\) does not have a trivial canonical line bundle. There is an effort in [\textit{G. Grantcharov}, Adv. Math. 226, No. 4, 3136--3159 (2011; Zbl 1213.53091)] to deform the complex structure into a Calabi-Yau structure. In this paper, the author deformed the generic singular anticanonical hypersurface of smooth toric Fano 4-folds with rank two Picard group, i.e., \(b_2 =2\), other than the \(B_1\) in [\textit{V. V. Batyrev}, Tôhoku Math. J., II. Ser. 43, No.4, 569--585 (1991; Zbl 0792.14026)] to non-Kähler Calabi-Yau threefolds. Reviewer's remark: There are some typos in the paper as the reviewer mentioned the cited theorem and formula for \(k\) and the references should possibly be put into an alphabet order. Note that \(\mathbb{C}\mathbb{P}^4\), used by Friedman and Lu-Tian, is just the smooth toric Fano 4-folds with a rank one Picard group, i.e., \(b_2 =1\).
complex structures on connected sums of \(S^3 \times S^3\)
scientific article; zbMATH DE number 6102793
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