论文标题
卡拉比（Calabi-Yau）的拓扑三倍，皮卡德（Picard）
The topology of Calabi-Yau threefolds with Picard number three
论文作者
论文摘要
我们询问简单连接的紧凑型平滑的6个manifolds，可以支持Calabi-yau三倍的结构。 In particular, we study the interesting case of Calabi-Yau threefolds $X$ with second betti number 3. We have a cup-product cubic form on the second integral cohomology, a linear form given by the second Chern class, and the integral middle cohomology, and if $X$ is simply connected with torsion free homology this information determines precisely the diffeomorphism class of the underlying 6-manifold by a result of 墙。为简单起见，我们假设立方形式定义了一条光滑的真实椭圆曲线，其黑森不可还原。在进一步的相对温和的假设下，$ x $上没有可移动的表面$ e $，$ 1 \ le e e^3 \ le 8 $，我们证明，真正的椭圆曲线必须具有两个连接的组件，而不是一个，而kähller圆锥则包含在开放的圆锥锥中，在边界组件上;此外，我们还表明，第二个Chern班对此开放式锥体也很积极。使用Wall的结果，对于任何给定的第三个Betti编号，我们都有大量的平滑紧凑定向的6个manifolds的示例，这些示例不支持Calabi-Yau结构，这两种情况下都在Cutic定义具有一个或两个连接组件的真实椭圆曲线的情况下。此外，除非$ C_2 $在椭圆曲线的真实反射点上消失，即使在上述条件下确实发生了卡拉比YAU结构，它们的界限也只有一个有界的家族，这些家族不是椭圆形的。
We ask about the simply connected compact smooth 6-manifolds which can support structures of Calabi-Yau threefolds. In particular, we study the interesting case of Calabi-Yau threefolds $X$ with second betti number 3. We have a cup-product cubic form on the second integral cohomology, a linear form given by the second Chern class, and the integral middle cohomology, and if $X$ is simply connected with torsion free homology this information determines precisely the diffeomorphism class of the underlying 6-manifold by a result of Wall. For simplicity, we assume that the cubic form defines a smooth real elliptic curve whose Hessian is irreducible. Under a further relatively mild assumption that there are no non-movable surfaces $E$ on $X$ with $1 \le E^3 \le 8$, we prove that the real elliptic curve must have two connected components rather than one, and that the Kähler cone is contained in the open positive cone on the bounded component; we show moreover that the second Chern class is also positive on this open cone. Using Wall's result, for any given third Betti number we therefore have an abundance of examples of smooth compact oriented 6-manifolds which support no Calabi-Yau structures, both in the cases when the cubic defines a real elliptic curve with one or two connected components. Moreover, except possibly if $c_2$ vanishes at a real inflexion point of the elliptic curve, even when Calabi-Yau structures do occur under the above conditions, there will be only a bounded family of them which are not birationally elliptic.
