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 problem in algebraic geometry and computational topology: providing explicit degree bounds for polynomial differential forms that represent cohomology classes in the de Rham cohomology of smooth affine complex varieties. Grothendieck’s classical theorem guarantees the existence of such polynomial representatives, but it offers no quantitative control over their degrees. Earlier work by the authors established a single‑exponential bound for the special case of hypersurfaces (codimension 1), showing that any class in Hⁿ_{dR}(X) can be represented by a form of degree at most (p·D)^{O(m)} where m is the dimension of the variety and D its degree. The present work extends this result to arbitrary codimension, i.e., to varieties defined by any regular sequence of polynomials.

The authors begin by recalling the Grothendieck‑Deligne framework for algebraic de Rham cohomology and the known effective bounds for hypersurfaces. They then formulate the general setting: let X⊂ℂⁿ be a smooth affine variety of dimension m, defined by a regular sequence f₁,…,f_k of polynomials of total degree D (so codimension k = n−m). The main theorem states that for each p (0≤p≤m) the p‑th de Rham cohomology group Hⁿ_{dR}(X) is generated by differential p‑forms whose coefficients are polynomials of degree at most (p·D)^{C·p·m}, where C is an absolute constant independent of X. In asymptotic notation this is written as degree (pD)^{O(p m)}.

The proof proceeds in three technical stages. First, the authors construct a Koszul complex K·(f) associated with the defining regular sequence. This complex resolves the structure sheaf of X and provides a concrete chain model for de Rham cohomology via the algebraic de Rham complex Ω·(ℂⁿ). By carefully choosing a generic linear change of coordinates, they ensure that the sequence is in “general position,” which simplifies the analysis of syzygies and eliminates pathological degree growth.

Second, they develop a systematic degree‑tracking mechanism for the lifting process that takes a closed algebraic p‑form on ℂⁿ and produces a representative on X. The key observation is that each step of the Koszul differential introduces at most a linear increase in degree, while the de Rham differential can be bounded by the degrees of the defining equations. To control the cumulative effect, the authors introduce two auxiliary operators: a “degree‑inflation” operator that captures the worst‑case increase when multiplying by the f_i, and a “degree‑reduction” operator that exploits the regularity of the sequence to cancel higher‑degree terms. By iterating these operators, they derive a recurrence relation for the degree bound, which solves to the claimed single‑exponential expression.

Third, the authors translate the algebraic bound into an algorithmic procedure. Given the explicit degree bound, one can compute a Gröbner basis for the ideal (f₁,…,f_k) and then use standard linear‑algebraic techniques to solve the resulting system of equations for the coefficients of the representative forms. The bound guarantees that the size of the linear system grows at most exponentially in p·m, making the computation feasible for moderate values of p and m. The paper includes illustrative examples: a smooth surface in ℂ³ defined by two quadrics (D=2, m=2, k=1) and a three‑dimensional complete intersection of degree 5 (D=5, m=3, k=2). In both cases the computed representatives respect the theoretical degree limits, confirming the practicality of the method.

Beyond computational aspects, the authors motivate their work through connections to the infinitesimal Hilbert 16th problem, which concerns the number of limit cycles that can bifurcate from a Hamiltonian system under small perturbations. The de Rham cohomology classes of the underlying algebraic curve encode the periods of the system; having explicit polynomial representatives with controlled degree enables effective estimates of the number of zeros of Abelian integrals, a central object in the Hilbert problem. Thus the degree bound not only advances algorithmic algebraic geometry but also provides a new tool for quantitative studies in dynamical systems.

In summary, the paper delivers a comprehensive and effective extension of earlier hypersurface results to arbitrary codimension, establishing that the p‑th de Rham cohomology of a smooth affine variety of dimension m and degree D can always be represented by polynomial differential forms of degree bounded by (p·D)^{O(p m)}. The proof combines Koszul resolutions, precise degree bookkeeping, and generic position arguments, and it yields an algorithmic framework with concrete applications to both computational cohomology and the theory of differential equations.