Doubling Construction of Calabi–Yau Fourfolds from Toric Fano Fourfolds

Abstract

We give a differential-geometric construction of Calabi–Yau fourfolds by the ‘doubling’ method, which was introduced in Doi and Yotsutani (N Y J Math 20:1203–1235, 2014) to construct Calabi–Yau threefolds. We also give examples of Calabi–Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are admissible pairs, which were first dealt with by Kovalev (J Reine Angew Math 565:125–160, 2003). Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact Kähler manifold $\overline{X}$ and a smooth anticanonical divisor D on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ with $\dim _{\mathbb {C}}\overline{X}_i=4$ satisfy the gluing condition, we can glue $\overline{X}_1\setminus D_1$ and $\overline{X}_2\setminus D_2$ together to obtain a compact Riemannian 8-manifold (M, g) whose holonomy group $\mathrm {Hol}(g)$ is contained in $\mathrm {Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of M equals 2, then M is a Calabi–Yau fourfold, i.e., a compact Ricci-flat Kähler fourfold with holonomy $\mathrm {SU}(4)$. In particular, if $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ are identical to an admissible pair $(\overline{X},D)$, then the gluing condition holds automatically, so that we obtain a compact Riemannian 8-manifold M with holonomy contained in $\mathrm {Spin}(7)$. Moreover, we show that if the admissible pair is obtained from any of the toric Fano fourfolds, then the resulting manifold M is a Calabi–Yau fourfold by computing $\widehat{A}(M)=2$.

Keywords

Ricci-flat metrics / Calabi–Yau manifolds / Spin(7)-structures / Gluing / Doubling / Toric geometry

References

[1] Batyrev VV. On the classification of toric Fano $4$-folds. Algebraic geometry, $9$. J. Math. Sci. (New York). 1999, 94 1021-1050

[2] Cox D. The homogeneous coordinate ring of a toric variety. J. Algebr. Geom. 1995, 4 15-50

[3] Cox D, Little J, Schenck H. Toric Varieties (Graduate Studies in Mathematics). 2011 Providence: AMS

[4] Doi M. Gluing construction of compact complex surface with trivial canonical bundle. J. Math. Soc. Jpn.. 2009, 61 853-884

[5] Doi M, Yotsutani N. Doubling construction of Calabi–Yau threefolds. N. Y. J. Math.. 2014, 20 1203-1235

[6] Dolgachev I. Lectures on Invariant Theory. 2003 Cambridge: Cambridge University Press

[7] Fulton W. Introduction to Toric Varieties, Annals of Mathematics Studies 131. 1993 Princeton: Princeton University Press

[8] Graded Ring Database. http://grdb.lboro.ac.uk/forms/toricsmooth

[9] Harvey R. Spinors and Calibrations, Perspectives in Mathematics 9. 1990 San Diego: Academic Press

[10] Hein, H.-J.: Complete Calabi–Yau metrics from ${\mathbb{P}}^2$ #9 $\overline{\mathbb{P}}^2$.arXiv:1003.2646

[11] Joyce DD. Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs. 2000 Oxford: Oxford University Press

[12] Kovalev A. Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math.. 2003, 565 125-160

[13] Salamon SM. Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics 201. 1989 Harlow: Longman

[14] Sato, H.: Studies on Toric Fano Varieties, pp 1–99. Tohoku Mathematical Publications 23. Tohoku University, Math Institute, Sendai (2002)

[15] Tian G, Yau S-T. Complete Kähler manifolds with zero Ricci curvature. I. J. Am. Math. Soc.. 1990, 3 579-609