Effective homology and periods of complex projective hypersurfaces (Q6590638)

The \(k\)-th period matrix of a smooth complex variety \(X\) is the matrix of the De Rham duality \(H_k(X)\times H^k_{\mathrm{DR}}\rightarrow \mathbb{C}\), \(([\gamma ],[\omega ])\mapsto \int _{\gamma}\omega\) between singular homology and (algebraic) De Rham cohomology. In experimental mathematics, numerical computation of periods is a useful tool to investigate some algebraic invariants. The authors introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard-Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section. The authors provide a SageMath implementation. On a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.