We provide an algorithm to compute an effective description of the homology of complex projective hypersurfaces relying on Picard-Lefschetz theory. Next, we use this description to compute high-precision numerical approximations of the periods of the hypersurface. This is an improvement over existing algorithms as this new method allows for the computation of periods of smooth quartic surfaces in an hour on a laptop, which could not be attained with previous methods. The general theory presented in this paper can be generalised to varieties other than just hypersurfaces, such as elliptic fibrations as showcased on an example coming from Feynman graphs. Our algorithm comes with a SageMath implementation.

翻译：我们提出一种基于Picard-Lefschetz理论计算复杂射影超曲面同调有效描述的算法。进而利用该描述对超曲面周期进行高精度数值近似计算。这一方法改进了现有算法，能够在笔记本电脑上于一小时内完成光滑四次曲面的周期计算，这是以往方法无法达到的。文中提出的通用理论可推广至超曲面以外的其他簇，例如椭圆纤维化——这在来自费曼图的一个示例中得到了展示。我们的算法附带SageMath实现。