Effective de Rham Cohomology - The General Case

Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. After having proved a single exponential bound for the degrees of these forms in the case of a hypersurface, here we generalize this result to arbitrary codimension. More precisely, we show that the p-th de Rham cohomology of a smooth affine variety of dimension m and degree D can be represented by differential forms of degree (pD)^{O(pm)}. This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.

💡 Research Summary

The paper addresses a fundamental gap in the effective version of Grothendieck’s algebraic de Rham theorem. While Grothendieck proved that every cohomology class of a smooth complex affine variety can be represented by a differential form with polynomial coefficients, his proof gives no quantitative bound on the degrees of those polynomials. In a previous work the author obtained a single‑exponential bound for hypersurfaces; the present work extends this to arbitrary codimension and arbitrary dimension.

The main theorem (Theorem 1.1) states that for a smooth affine variety (X) of dimension (m) and degree (D) over an algebraically closed field of characteristic zero, every class in the algebraic de Rham cohomology (H_{\mathrm{dR}}^{p}(X)) can be represented by a (p)-form whose coefficients are polynomials of degree at most ((pD)^{O(pm)}). The bound is explicit: \