New construction of complex manifold via conifold transition
type: Item
instance of: scholarly article
title: New construction of complex manifold via conifold transition
zbMATH Open document ID: 1256.14040
author: Jinxing Xu
publication date: 2012-11-07T00:00:00Z
review text: 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--3…
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32C18
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 6102793
zbMATH Keywords: Calabi-Yau threefolds
zbMATH Keywords: conifold transitions
zbMATH Keywords: complex structures on connected sums of \(S^3 \times S^3\)
zbMATH Keywords: hypersurface in toric Fano \(4\)-folds
reviewed by: Daniel Guan
MaRDI profile type: Publication
full work available at URL: s10114-012-0666-2
OpenAlex ID: W2099372332
cites work: The moduli space of 3-folds with \(K=0\) may nevertheless be irreducible
cites work: Geometric transitions
cites work: Q4840444
cites work: Simultaneous resolution of threefold double points
cites work: Double solids
cites work: Classification problems in differential topology. V: On certain 6- manifolds
cites work: Connecting moduli spaces of Calabi-Yau threefolds
cites work: Complete classification of reflexive polyhedra in four dimensions
cites work: On the classification of smooth projective toric varieties
cites work: Q4885281
cites work: Q4011172
cites work: Rational Curves in Calabi-Yau Threefolds
cites work: Rational curves of degree at most 9 on a general quintic threefold
cites work: Q4324883
cites work: Homological equivalence, modulo algebraic equivalence, is not finitely generated
cites work: Introduction to Toric Varieties. (AM-131)
cites work: The functor of a smooth toric variety
DOI: 10.1007/S10114-012-0666-2
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6102793