The existence of elliptic fibre space structures on Calabi-Yau threefolds
The existence of elliptic fibre space structures on Calabi-Yau threefolds
复制标题
Calabi-Yau三重体上椭圆纤维空间结构的存在性
DOI：
10.1017/s030500419700220x
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发表时间：
1994
影响因子：
1.4
通讯作者：
P. Wilson
中科院分区：
文献类型：
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作者：
P. Wilson
在[11]中，我们考虑了在光滑的卡拉比 - 丘三流形\(X\)上椭圆纤维空间结构存在性的问题。必要条件是在\(X\)上存在一个数值有效除子\(D\)，满足\(D^3 = 0\)；\(D^2\neq0\)以及\(D\cdot c_2\geq0\)。在\(D\cdot c_2\geq0\)的情形下，据观察[9]中的一个结果意味着存在一个由\(D\)确定的椭圆纤维空间结构，所以[11]关注的是\(D\cdot c_2 = 0\)的情形。在这种情形下，我们一般无法推断出椭圆纤维空间结构的存在性，而只能在存在进一步条件的情况下才能做到。特别地，由[11]中的定理可知，如果\(r\)表示\(X\)上满足\(D|_E\equiv0\)的有理曲面\(E\)的个数，并且如果欧拉示性数\(e(X)\neq2r\)，那么\(D\)的某个正倍数确定了\(X\)上的一个椭圆纤维空间结构（该定理实际上证明了一个稍强的结果，即如果\(X\)上存在任何满足\(D|_E\equiv0\)的非有理曲面\(E\)，则给出椭圆纤维空间结构）。本文的目的是阐明上述关于\(e(X)/2 - r\)非零的相当神秘的条件的含义。
In [11], we considered the question of existence of elliptic fibre space structures on a smooth Calabi–Yau threefold X. Necessary conditions are that there exists a nef divisor D on X with D3=0; D2[nequiv ]0 and D·c2[ges ]0. In the case when D·c2[ges ]0, it was observed that a result from [9] implies that there is an elliptic fibre space structure determined by D, and so [11] concerned itself with the case when D·c2=0. In this case, we were not able to deduce in general the existence of an elliptic fibre space structure, but only in the presence of a further condition. In particular, it follows from the Theorem in [11] that, if r denotes the number of rational surfaces E on X with D[mid ]E≡0, and if the Euler characteristic e(X)≠2r, then some positive multiple of D determines an elliptic fibre space structure on X (the theorem in fact proves a slightly stronger result, which gives the elliptic fibre space structure if there is any non-rational surface E on X with D[mid ]E≡0). The purpose of this note is to clarify the meaning of the above rather mysterious condition on the non-vanishing of e(X)/2−r.