Grothendieck's standard conjectures of type $C^{+}$ (sign) and $D$ for smooth projective schemes and their noncommutative extension to smooth proper dg categories

Grothendieck's standard conjectures of type $C^{+}$ and $D$ assert, for every smooth projective scheme, the algebraicity of the even Künneth projector (the sign conjecture) and the coincidence of homological with numerical equivalence of algebraic cycles. They are proved only in restricted families: dimension bounds, abelian varieties, complete intersections, and the quadric fibrations, intersections of quadrics, determinantal linear sections, and intersections of bilinear divisors recently reached through homological projective duality. Building on the framework of noncommutative numerical motives of Marcolli and Tabuada, Tabuada (2018) proved the noncommutative analogues of both conjectures for smooth proper dg categories equivalent to Grothendieck's conjectures on scheme categories, so the general commutative problem and its noncommutative extension form one linked question. The conjectures remain open in general, and the noncommutative conjectures for Lefschetz primitive components that do not admit full exceptional collections remain the concrete open frontier.

Background

Let $k$ be a field of characteristic zero and $X$ a smooth projective $k$-scheme of dimension $d$. A Weil cohomology theory (for example Betti cohomology when $k=\mathbb{C}$, algebraic de Rham cohomology, or $\ell$-adic cohomology) assigns to $X$ finite-dimensional graded vector spaces $H^{*}(X)$ with a cup product and a cycle class map from algebraic cycles — finite $\mathbb{Q}$-linear combinations of closed subvarieties — to cohomology. The Künneth decomposition of the diagonal splits $H^{*}(X\times X)$ and yields projectors $\pi^{i}_{X}$; the even Künneth projector is $\pi^{+}_{X}=\sum_{i}\pi^{2i}_{X}$. A projector is called algebraic if it is induced by an algebraic cycle with rational coefficients on $X\times X$ (a correspondence). Two adequate equivalence relations on cycles are central: a cycle is homologically trivial if its class vanishes in $H^{*}(X)$, and numerically trivial if all its intersection pairings with cycles of complementary dimension vanish; homological triviality implies numerical triviality, and the converse is the open direction.

In the late 1960s Grothendieck conjectured, among his standard conjectures on algebraic cycles, that $\pi^{+}_{X}$ is algebraic for every $X$ (conjecture $C^{+}$, also called the sign conjecture, since it asserts algebraicity of the projector onto the subspace where the Lefschetz involution acts by $+1$), and that homological and numerical equivalence coincide on the $\mathbb{Q}$-vector space $Z^{*}(X)_{\mathbb{Q}}$ of cycles (conjecture $D$). These conjectures are the foundation of Grothendieck's theory of pure motives: a Chow motive is built from a smooth projective scheme and an algebraic projector, and $D$ for all $X$ is equivalent, by a theorem of Jannsen, to the category of numerical motives being abelian and semisimple, which would give motivic Galois groups and a Tannakian theory of pure motives. Known cases: $C^{+}$ holds in dimension at most $2$ and $D$ in dimension at most $4$ (in characteristic zero; the latter via the Hodge index theorem, work of Kleiman and Lieberman); both hold for abelian varieties; and for smooth complete intersections the Lefschetz-type conjecture $B$ holds (Grothendieck), which implies $C^{+}$ and $D$. One also knows the implication $D(X\times X)\Rightarrow C^{+}(X)$. Beyond such cases the conjectures remain open, and no counterexample is known.

The source paper extends the conjectures to noncommutative geometry. A dg category is a category enriched over complexes of $k$-vector spaces; every scheme $X$ gives the dg category $\mathrm{perf}^{\mathrm{dg}}(X)$ enhancing the category of perfect complexes. Following Kontsevich's terminology, a dg category is smooth and proper when it is perfect as a bimodule over itself and has finite-dimensional total cohomology of all Hom complexes; these are the noncommutative analogues of smooth proper schemes. Periodic cyclic homology gives a functor $HP^{\pm}$ on smooth proper dg categories with values in finite-dimensional $\mathbb{Z}/2$-graded vector spaces, and for a scheme it identifies with the even/odd part of de Rham cohomology, so $HP^{+}$ plays the role of a noncommutative cohomology. On the Grothendieck group $K_{0}(\mathcal{A})_{\mathbb{Q}}$ one has the Euler pairing $\chi([M],[N])=\sum_{n}(-1)^{n}\dim\mathrm{Hom}(M,N[n])$ and a cycle class map $K_{0}(\mathcal{A})_{\mathbb{Q}}\to HP^{+}(\mathcal{A})$. In the framework of noncommutative motives of Marcolli and Tabuada, the noncommutative conjectures are: $C^{+}_{nc}(\mathcal{A})$, the even Künneth projector of $HP^{\pm}(\mathcal{A})$ is algebraic, i.e. lifts to an endomorphism of the noncommutative Chow motive of $\mathcal{A}$; and $D_{nc}(\mathcal{A})$, the kernel of the cycle class map equals the radical of the Euler pairing. Tabuada proved that for every smooth projective scheme $X$ one has equivalences of conjectures $C^{+}(X)\Leftrightarrow C^{+}_{nc}(\mathrm{perf}^{\mathrm{dg}}(X))$ and $D(X)\Leftrightarrow D_{nc}(\mathrm{perf}^{\mathrm{dg}}(X))$, so the classical conjectures are exactly the noncommutative ones restricted to scheme categories, and the noncommutative conjectures for all smooth proper dg categories are the natural simultaneous extension.

The paper's engine is homological projective duality (HPD), introduced by Kuznetsov: if $X$ admits a Lefschetz decomposition of $\mathrm{perf}(X)$ with primitive component $\mathcal{A}_{0}$ and $Y$ is the HP-dual of $X$, then for mutually dual generic linear sections $X_{L}$ and $Y_{L}$ the conjectures are equivalent, $C^{+}(X_{L})\Leftrightarrow C^{+}(Y_{L})$ and $D(X_{L})\Leftrightarrow D(Y_{L})$, provided the noncommutative conjectures $C^{+}_{nc}(\mathcal{A}_{0}^{\mathrm{dg}})$ resp. $D_{nc}(\mathcal{A}_{0}^{\mathrm{dg}})$ hold for the dg enhancement of the primitive component. These hold whenever $\mathcal{A}_{0}$ admits a full exceptional collection — a sequence of exceptional objects (objects $E$ with $\mathrm{Hom}^{*}(E,E)=k\cdot\mathrm{id}$) that generate the category — but no method is known without one. This yields the paper's new families: quadric fibrations with smooth discriminant over low-dimensional bases, intersections of few quadrics, linear sections of determinantal varieties in numerical ranges, and intersections of bilinear divisors, all under stated dimension, parity, and discriminant-smoothness hypotheses. Outside these families — in particular for primitive components without full exceptional collections, singular HP-duals, or singular discriminant loci — both the commutative and the noncommutative conjectures remain open.

Problem Statement

Over a base field $k$ of characteristic zero, decide the following linked conjectures in full generality.

(1) Standard conjecture of type $C^{+}$ (sign conjecture): for every smooth projective $k$-scheme $X$ and every Weil cohomology theory $H^{*}$, the even Künneth projector $\pi^{+}_{X}=\sum_{i}\pi^{2i}_{X}$ is algebraic, i.e. induced by an algebraic cycle with rational coefficients on $X\times X$.

(2) Standard conjecture of type $D$: for every smooth projective $k$-scheme $X$, homological and numerical equivalence coincide on rational algebraic cycles, $Z^{*}(X)_{\mathbb{Q}}/\sim_{\mathrm{hom}}=Z^{*}(X)_{\mathbb{Q}}/\sim_{\mathrm{num}}$.

(3) Noncommutative extension: for every smooth proper $k$-linear dg category $\mathcal{A}$, the conjectures $C^{+}_{nc}(\mathcal{A})$ — the even Künneth projector of periodic cyclic homology $HP^{\pm}(\mathcal{A})$ is algebraic, i.e. lifts through the noncommutative Chow motive of $\mathcal{A}$ — and $D_{nc}(\mathcal{A})$ — the kernel of the cycle class map $K_{0}(\mathcal{A})_{\mathbb{Q}}\to HP^{+}(\mathcal{A})$ equals the radical of the Euler pairing $\chi$ on $K_{0}(\mathcal{A})_{\mathbb{Q}}$.

Parts (1)-(2) and (3) form one coherent objective: by the equivalence theorem, (3) restricted to scheme categories is exactly (1)-(2), so the problem is to settle the standard conjectures on the commutative side for all smooth projective schemes and simultaneously on the noncommutative side for all smooth proper dg categories. The derived sub-problem highlighted by the source — establishing $C^{+}_{nc}$ and $D_{nc}$ for Lefschetz primitive components $\mathcal{A}_{0}$ of homological projective duality pairs that do not admit full exceptional collections, which by HPD-invariance would settle the conjectures for the corresponding linear sections — is part of this objective and is where the known reduction strategy currently stops.

Known results confined to restricted families — the dimension bounds, abelian varieties, complete intersections, matrix factorization categories, or the families settled in the source paper and its successors — fall short of the generality required by (1)-(3).

References

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