论文标题
卡拉比（Calabi-Yau）三倍，皮卡德（Picard）
Calabi-Yau threefolds with Picard number three
论文作者
论文摘要
在本文中，我们继续研究（简单地连接）平滑的calabi-yau三倍的界限问题：1706.01268。已知这种三倍的差异类别被确定为有限的多种可能性，由积分中间的共同体和积分第二个共同体学上的两种整体形式确定，即Cubic Cuppoducts形式和Cupp-prododuct fie Second Chern类给出的线性形式。两篇论文中解决的问题是，这些立方体和线性形式的知识是否决定了三倍以有限的家庭，这就是这种三倍的模量是有限的。如果这是真的，那么特别是中间积分的共同体学将通过了解这两种形式而受到限制。这个问题至关重要的是对三倍的刚性不可移动的表面进行研究，这是不可约束的表面，它们对三倍的复杂结构的任何小变形变形，但对于三倍没有多次移动。我们在上一篇论文中表明，如果没有这样的表面，那么上述问题的答案是肯定的。此外，如果PICARD编号为2，则答案被证明是是的，没有对Calabi-Yau三倍的任何进一步的假设。本文的主要结果是Picard编号3，在最多有一个刚性不可移动的表面的情况下，我们证明了有界性，假设立方体形式是光滑的（从而定义了真正的椭圆形曲线）；在两种情况下，Hessian曲线是奇异的，我们还假设由第二个Chern类定义的线不会在反射点上与此曲线相交。除了上一篇论文中描述的方法外，证明中的进一步至关重要的工具还将是椭圆曲线的Hessian上的经典Steinian涉及。
In this paper, we continue the study of boundedness questions for (simply connected) smooth Calabi-Yau threefolds commenced in arXiv:1706.01268. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second chern class. The question addressed in both papers is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowing these two forms. Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. We showed in the previous paper that if there are no such surfaces, then the answer to the above question is yes. Moreover if the Picard number is 2, the answer was shown to be yes without any further assumptions on the Calabi-Yau threefold. The main results of this paper are for Picard number 3, where we prove boundedness in the case where there is at most one rigid non-movable surface, assuming the cubic form is smooth (thereby defining a real elliptic curve); in the two cases where the Hessian curve is singular, we also assume that the line defined by the second chern class does not intersect this curve at an inflexion point. In addition to the methods described in the previous paper, a further crucial tool in the proofs will be the classical Steinian involution on the Hessian of an elliptic curve.
