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 exa...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(X,D)consists of a compact Kähler manifold X and a smooth anticanonical divisor D on X.If two admissible pairs(X_(1),D_(1))and(X_(2),D_(2))with dimC X_(i)=4 satisfy the gluing condition,we can glue X_(1)\D_(1)and X_(2)\D_(2)together to obtain a compact Riemannian 8-manifold(M,g)whose holonomy group Hol(g)is contained in Spin(7).Furthermore,if theA-genus of M equals 2,then M is a Calabi-Yau fourfold,i.e.,a compact Ricci-flat Kähler fourfold with holonomy SU(4).In particular,if(X_(1),D_(1))and(X_(2),D_(2))are identical to an admissible pair(X,D),then the gluing condition holds automatically,so that we obtain a compact Riemannian 8-manifold M with holonomy contained in 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^A(M)=2.展开更多
文摘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(X,D)consists of a compact Kähler manifold X and a smooth anticanonical divisor D on X.If two admissible pairs(X_(1),D_(1))and(X_(2),D_(2))with dimC X_(i)=4 satisfy the gluing condition,we can glue X_(1)\D_(1)and X_(2)\D_(2)together to obtain a compact Riemannian 8-manifold(M,g)whose holonomy group Hol(g)is contained in Spin(7).Furthermore,if theA-genus of M equals 2,then M is a Calabi-Yau fourfold,i.e.,a compact Ricci-flat Kähler fourfold with holonomy SU(4).In particular,if(X_(1),D_(1))and(X_(2),D_(2))are identical to an admissible pair(X,D),then the gluing condition holds automatically,so that we obtain a compact Riemannian 8-manifold M with holonomy contained in 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^A(M)=2.
