Grothendieck's 1966 algebraic de Rham questions: the open core — algebraic independence of periods and quasi-periods, Gauss–Manin horizontality of algebraic cycle classes, and E_2-finiteness of the Cousin spectral sequence

A. Grothendieck's 1966 paper "On the de Rham cohomology of algebraic varieties" (Publ. Math. IHÉS 29, 95-103) raises six sub-questions on algebraic de Rham theory: an algebraic Leray spectral sequence; finite-dimensionality of $E_2$ of a Cousin/residue spectral sequence; purely algebraic singular homology; local freeness and base change of relative de Rham sheaves; arithmetic of periods of differentials of the second kind; and an algebraic Gauss–Manin connection with a horizontality conjecture for algebraic classes. Three components were later settled (algebraic Leray sequence: Katz–Oda, Deligne, Hartshorne; algebraic singular homology: Hartshorne; characteristic-zero relative local freeness: Deligne, Deligne–Illusie; characteristic p refuted by Serre's example already cited by Grothendieck). The surviving open core is: (i) the Grothendieck period conjecture — algebraic independence of the $4g^2$ periods and quasi-periods of an abelian variety of dimension $g$ (minimal case: $\omega_1,\omega_2$ algebraically independent for an elliptic curve without complex multiplication), beyond the essentially complete linear theory (Schneider, Wüstholz) and Chudnovsky's partial results; (ii) the horizontality conjecture of footnote (13), i.e. Grothendieck's variational Hodge conjecture, open in general though its Hodge-class shadow is a theorem of Deligne; (iii) $E_2$-finiteness of the Cousin/residue spectral sequence, verified by Grothendieck only in dimension at most 2.

Background

For a smooth algebraic variety $X$ over a field $k$ of characteristic zero, the algebraic de Rham cohomology $H^*_{dR}(X/k)$ is the hypercohomology of the complex of algebraic differential forms $\Omega^\bullet_{X/k}$ — hypercohomology being the extension of sheaf cohomology to a whole complex of sheaves, computing the cohomology of the total complex by resolving all terms at once — a purely algebraic object needing no analytic or topological input. Grothendieck's comparison theorem (1966) states that for $k\subseteq\mathbb{C}$ there is a canonical isomorphism $H^*_{dR}(X/k)\otimes_k\mathbb{C}\simeq H^*(X^{an},\mathbb{C})$ with the singular cohomology of the associated complex-analytic space, proved via Hironaka resolution of singularities and a GAGA-type comparison (GAGA — Géométrie Algébrique et Géométrie Analytique — being the classical dictionary between projective algebraic varieties and their complex-analytic counterparts). The 1966 paper (a 1963 letter to Atiyah, with footnotes added in July 1965) raises six loosely linked problems about the purely algebraic reach of this theory.

Terminology. A differential of the second kind on a smooth variety is a closed algebraic differential form whose local primitives have at worst poles with zero residues (equivalently, a form locally exact in the Zariski topology modulo exact forms); such forms, not just the everywhere-regular ones, represent all of de Rham cohomology. On an elliptic curve $E/\mathbb{C}$ with algebraic moduli, a nonzero holomorphic differential $\omega$ has two periods $\omega_1=\int_{\gamma_1}\omega$, $\omega_2=\int_{\gamma_2}\omega$ (integrals over a basis of singular homology), and the associated differential of the second kind has two quasi-periods $\eta_1,\eta_2$; the only universally known algebraic relation among the four is the Legendre relation $\omega_1\eta_2-\omega_2\eta_1=\pm 2\pi i$. For an abelian variety of dimension $g$ there are analogously $4g^2$ periods and quasi-periods — the entries of the $2g\times 2g$ matrix pairing a basis of the first de Rham cohomology (dimension $2g$) with a basis of the first singular homology (rank $2g$). An elliptic curve has complex multiplication (CM) when its endomorphism ring is strictly larger than $\mathbb{Z}$; the non-CM case is generic. The Gauss–Manin connection is the integrable algebraic connection (a rule for differentiating sections along tangent directions on the base, satisfying the Leibniz rule; integrable means its curvature vanishes, so flat sections exist locally) on the relative de Rham sheaves $R^qf_*(\Omega^\bullet_{X/S})$ of a smooth proper family $f:X\to S$ that differentiates fiberwise cohomology classes with respect to parameters on the base; its purely algebraic construction was one of the 1966 questions. On a smooth projective complex variety, a rational cohomology class of Hodge type $(p,p)$ (that is, lying wholly in the $(p,p)$ summand of the Hodge decomposition of $H^{2p}(X^{an},\mathbb{C})$) is a Hodge class; it is algebraic if it is a rational multiple of the class of an algebraic subvariety. Grothendieck's variational (infinitesimal) Hodge conjecture predicts that in a family, a class flat for the Gauss–Manin connection that is algebraic at one point is algebraic at every point. Finally, for smooth $X/k$ the Cousin (or residue) spectral sequence has initial terms $E_1^{pq}=\bigoplus_{x\in X^{(p)}}H^{q-p}(k(x)/k)$, where $X^{(p)}$ is the set of codimension-$p$ points and $H^*(K/k)$ is the de Rham cohomology of the finitely generated field extension $K/k$ (the direct limit of de Rham cohomologies of smooth affine models of $K$); it abuts to $H^*_{dR}(X/k)$, and in the induced filtration the last graded piece of $H^p$ is its "algebraic" part, generated by classes of codimension-$p$ cycles.

Known results delimiting the open core: Schneider (1937) proved that each period and each quasi-period of the second kind of an algebraically defined elliptic curve is either zero or transcendental; Wüstholz's analytic subgroup theorem (1989) — the statement that an algebraic subgroup of a commutative algebraic group over $\overline{\mathbb{Q}}$ containing no nonzero algebraic point contains no nonzero analytic point, from which all linear relations among periods follow — determines all linear relations over $\overline{\mathbb{Q}}$ among periods and quasi-periods of abelian varieties, making the linear theory essentially complete; Chudnovsky (1976) proved that for an elliptic curve with algebraic invariants — i.e. algebraic coefficients $g_2,g_3$ of its Weierstrass equation — at least two of the four numbers $g_2,g_3,\omega/\pi,\eta/\pi$ are algebraically independent (where $\omega$ is a period of the curve and $\eta$ a quasi-period); Deligne's theorem of the fixed part and absolute Hodge theory prove the Hodge-class shadow of the variational conjecture (an absolute Hodge class is a cohomology class that behaves as an algebraic class under all the canonical comparison isomorphisms; the theorem of the fixed part states that a Hodge class on one fiber of a smooth projective family that extends as a flat section of the Gauss–Manin connection is a Hodge class on every fiber; Principle B is the analogous statement for absolute Hodge classes on abelian schemes); Hartshorne (1975) constructed the algebraic singular homology and the general Leray-type spectral sequence asked for in 1966, and proved finite-dimensionality of de Rham cohomology for all finite-type schemes in characteristic zero.

Problem Statement

Resolve the surviving open components of the questions raised in A. Grothendieck, "On the de Rham cohomology of algebraic varieties", Publ. Math. IHÉS 29 (1966), 95-103. The components settled by later work (algebraic Leray spectral sequence; purely algebraic singular homology; characteristic-zero local freeness and base change of relative de Rham sheaves) are excluded; the open core is the following three components, all three of which are to be resolved; component (i) is primary in significance.

(i) Grothendieck period conjecture component (footnote (10) of the source). Minimal case: let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ without complex multiplication, let $\omega$ be a nonzero holomorphic differential on $E$ defined over $\overline{\mathbb{Q}}$ (such differentials form a one-dimensional $\overline{\mathbb{Q}}$-vector space, and the claim is invariant under scaling $\omega$ by an element of $\overline{\mathbb{Q}}^\times$), with periods $\omega_1,\omega_2$, and let $\eta_1,\eta_2$ be the quasi-periods of the associated differential of the second kind. Prove that $\omega_1$ and $\omega_2$ are algebraically independent over $\mathbb{Q}$. General case: prove that for an abelian variety $A$ over $\overline{\mathbb{Q}}$ of dimension $g$, every polynomial relation over $\overline{\mathbb{Q}}$ among the $4g^2$ periods and quasi-periods of $A$ together with the number $2\pi i$ (equivalently, among the entries of a period matrix of $A$ together with $2\pi i$) is a consequence of the standard relations, namely the bilinear (Riemann/Legendre) relations — in the elliptic case, of $\omega_1\eta_2-\omega_2\eta_1=\pm 2\pi i$ — together with the $\overline{\mathbb{Q}}$-linear relations among periods and quasi-periods imposed by the endomorphisms of $A$. A disproof would be an explicit abelian variety over $\overline{\mathbb{Q}}$ with an algebraic relation among its periods (and $2\pi i$) not generated by the standard ones. (This is the source-faithful form of the conjecture now called the Grothendieck period conjecture; the linear theory is complete by Wüstholz's theorem, so the open content is precisely algebraic independence.) The General case contains the Minimal case as its $g=1$ non-CM instance: a counterexample to the Minimal case is also a counterexample to the General case, whereas a proof of the Minimal case alone does not imply the General case.

(ii) Gauss–Manin horizontality / variational Hodge component (footnote (13) of the source). Let $f:X\to S$ be a proper smooth morphism of schemes of characteristic zero with $S$ connected and reduced, let $\nabla$ be the canonical (Gauss–Manin) integrable connection on the relative de Rham sheaf $R^qf_*(\Omega^\bullet_{X/S})$, and let $\theta$ be a global section of $R^qf_*(\Omega^\bullet_{X/S})$ that is horizontal (flat) for $\nabla$. Prove or disprove: if for one point $s_0\in S$ the class $\theta_{s_0}$ on the fiber $X_{s_0}$ is the cohomology class of an algebraic cycle with rational coefficients, then the same holds for $\theta_s$ for every $s\in S$. (Grothendieck's footnote (13) states this as an equivalence for arbitrary sections $\theta$ — algebraic on every fiber if and only if constant for the canonical connection and algebraic at one point — but the converse direction, read with the footnote's own fiberwise definition of "algebraic on every fiber", is false: for a family carrying a relative divisor $D$ and a nonconstant function $g\in\Gamma(S,\mathcal{O}_S)$, the section $\theta=(1+g(s))\,[D_s]$ restricts on every fiber to a rational multiple of an algebraic cycle class, yet $\nabla\theta=dg\otimes[D]\neq 0$, so $\theta$ is not horizontal. The open content is therefore the implication stated, for horizontal sections. This is Grothendieck's variational Hodge conjecture. The Hodge-class shadow — horizontality plus Hodge type at one point forces Hodge type everywhere — is known; the cycle-theoretic statement is open in general.)

(iii) Cousin $E_2$-finiteness component (footnote (8) of the source). Let $X$ be a smooth variety over a field $k$ of characteristic zero, and consider the Cousin (residue) spectral sequence with $E_1^{pq}=\bigoplus_{x\in X^{(p)}}H^{q-p}(k(x)/k)$ converging to $H^*_{dR}(X/k)$, where $X^{(p)}$ is the set of codimension-$p$ points of $X$ and $H^*(K/k)$ is the de Rham cohomology of the finitely generated field extension $K/k$, i.e. the direct limit of the de Rham cohomologies of smooth affine models of $K$ over $k$. Determine whether $E_2^{pq}$ is finite-dimensional over $k$ for all $(p,q)$ when $\dim X\geq 3$; Grothendieck verified this for $\dim X\leq 2$ and for small $(p,q)$.

References

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