Grothendieck's standard conjecture B for smooth complex projective threefolds of general type (Kodaira dimension 3)

Grothendieck's standard conjecture $B(X)$ of Lefschetz type predicts that on a smooth projective variety the inverses of the Lefschetz operators defined by an ample divisor are induced by algebraic correspondences. It is known for curves, surfaces, abelian varieties, varieties dominated by such, and — by S. G. Tankeev's theorem — for all smooth complex projective threefolds of Kodaira dimension below 3. Tankeev's 2019 announcement (the source of this problem) and its companion paper established $B(X)$ only for special classes of general-type threefolds fibered over a surface (existence of a section, endomorphism ring $\mathbb{Z}$ of a fiber Jacobian, rank-1 Kodaira–Spencer variation) and for fiber products of elliptic surfaces. The problem is to prove $B(X)$ for all smooth complex projective threefolds of general type (Kodaira dimension 3) — the last open Kodaira-dimension range in dimension three.

Background

Let $X$ be a smooth projective variety of dimension $d$ over $\mathbb{C}$. An algebraic $\mathbb{Q}$-correspondence on $X$ is the class of an algebraic cycle with rational coefficients on the product $X\times X$; such a class acts as a linear operator on the singular cohomology $H^{*}(X,\mathbb{Q})$. If $H$ is an ample divisor (a divisor whose associated line bundle is positive) and $L=c_{1}(H)\cup-$ denotes cup product with its first Chern class, the hard Lefschetz theorem states that $L^{d-i}:H^{i}(X,\mathbb{Q})\to H^{2d-i}(X,\mathbb{Q})$ is an isomorphism for $0\le i\le d$. Grothendieck's standard conjecture of Lefschetz type, conjecture $B(X)$, asserts that the inverse of this isomorphism is induced by an algebraic $\mathbb{Q}$-correspondence on $X\times X$; equivalently, that the dual Lefschetz operator $^{c}\Lambda_{H}$, characterized by $\langle{}^{c}\Lambda_{H}a,b\rangle=\langle a,L\,b\rangle$ with respect to the cup-product pairing, is algebraic. The conjecture was formulated by Grothendieck in 1968–69 and has been open ever since. $B(X)$ implies that the Künneth components of the diagonal of $X$ are algebraic (the pieces of the diagonal class on $X\times X$ of pure cohomological degree, into which it decomposes by the Künneth formula), and $B$ for all smooth projective varieties would imply that homological and numerical equivalence of algebraic cycles coincide (that is, that two cycles whose intersection numbers with every cycle agree must carry the same cohomology class), giving the hypothesized category of motives (the conjectural abelian building blocks into which the cohomology of algebraic varieties should decompose) its expected abelian, semisimple structure; Kleiman's survey is the standard reference for this formalism.

Conjecture $B(X)$ is known for curves (essentially trivial), for surfaces (via the Hodge index theorem, which governs the signature of the intersection pairing on divisors of a surface; Lieberman), for abelian varieties (complex tori that are projective, with the group law), and for varieties dominated by products of curves or abelian varieties; moreover, if there is a dominant rational map $X\dashrightarrow Y$ and $B(X)$ holds, then $B(Y)$ holds. For threefolds, Tankeev proved that $B(X)$ holds when $X$ admits a fibration over a curve (a surjective morphism with connected fibers onto a lower-dimensional base) with a section whose generic fiber is ruled (birational to $\mathbb{P}^{1}\times C$ for a curve $C$), Enriques (a smooth projective surface whose canonical bundle is nontrivial but has order two), or $K3$ (a $K3$ surface is a simply connected smooth projective surface with trivial canonical bundle), and then that $B(X)$ holds for every smooth complex projective threefold of Kodaira dimension $\kappa(X)<3$. Here the Kodaira dimension $\kappa(X)$ measures the growth rate of pluricanonical forms and takes values $-\infty,0,\dots,d$; $\kappa(X)=d$ means $X$ is of general type. Consequently, in dimension three the conjecture is open precisely for varieties of general type.

Tankeev's 2019 short communication — the source of the present problem — and its companion paper announced the first $\kappa=3$ classes: (a) a smooth projective threefold $X$ with a projective morphism $\pi:X\to S$ onto a smooth projective surface, admitting a section, such that $\mathrm{End}(\mathrm{Pic}^{0}(X_{s}))=\mathbb{Z}$ for some closed smooth fiber (the fiber is a curve, $\mathrm{Pic}^{0}(X_{s})$ is its Jacobian, and the condition says this Jacobian has the smallest possible endomorphism ring), and such that $\pi$ is smooth over a nonempty open subset $U\subset S$ where the Kodaira–Spencer map has rank 1 at every point — the Kodaira–Spencer map $T_{S,u}\to H^{1}(X_{u},T_{X_{u}})$ measures the first-order variation of the complex structure of the fibers, so rank 1 means the family varies in essentially one direction; and (b) fiber products of three elliptic surfaces (surfaces fibered over a curve whose generic fiber is an elliptic curve, that is, a smooth projective curve of genus one with a chosen origin) over a curve with pairwise disjoint discriminant loci (the sets of base points over which the fibers degenerate). For class (a), the note additionally announces that when the generic fiber of $\pi$ has genus $2$, the condition $\mathrm{End}(\mathrm{Pic}^{0}(X_{s}))=\mathbb{Z}$ can be dropped, and when the genus is $3$ it can be replaced by triviality of the trace of the Jacobian of the generic schematic fiber (which is automatic when $\mathrm{Pic}^{0}(X_{s})$ is a simple abelian variety for some closed smooth fiber). The same author's later work treats projective compactifications of Néron models (the canonical smooth group schemes extending an abelian variety over the generic point of a degenerating family) of 3-dimensional abelian varieties. All of these results impose strong restrictions, and no later work has removed them.

Problem Statement

Prove Grothendieck's standard conjecture $B(X)$ for every smooth complex projective threefold $X$ of general type (Kodaira dimension $\kappa(X)=3$): for every ample divisor $H$ on $X$, with $L=c_{1}(H)\cup-$, the inverse of each isomorphism $L^{3-i}:H^{i}(X,\mathbb{Q})\to H^{6-i}(X,\mathbb{Q})$, $0\le i\le 3$, is induced by an algebraic $\mathbb{Q}$-correspondence on $X\times X$; equivalently, the dual Lefschetz operator ${}^{c}\Lambda_{H}$ on $H^{*}(X,\mathbb{Q})$ is algebraic. In particular, settle the cases left open by the source announcement of 2019: threefolds $\pi:X\to S$ fibered over a surface whose Kodaira–Spencer map has rank $\ge 2$ on a dense open set (genuine two-parameter variation of complex structure), fibrations without a section, fibrations whose fiber Jacobians have nontrivial endomorphism rings beyond the genus-2 and genus-3 relaxations announced in the source, and threefolds of general type admitting no fibration over a surface at all. A refutation would consist of an explicit general-type threefold, an ample divisor, and an index $i$ for which the inverse of $L^{3-i}$ is provably not induced by any algebraic $\mathbb{Q}$-correspondence.

References

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