The Quillen–Lichtenbaum dimension
of complex varieties

It is easy to see that a smooth complex projective curve $X$ has $\dim_{\mathrm{QL}}(X)=0$ if and only if its genus is 0: we always have $K_{-1}(X,\mathbb{Z}/m)=0$, so the comparison map $\eta_{-1}$ cannot be an isomorphism unless $KU^{1}(X,\mathbb{Z}/m)=H^{1}(X,\mathbb{Z}/m)$ vanishes. But when $\dim X\geq 2$, it looks at first glance as if $\dim_{\mathrm{QL}}(X)$ might be totally uncomputable. In §1.3 we introduce a spectral sequence, related to the motivic spectral sequence, that computes the cone of the Bott maps $\tau_{n}$.¹¹1It is interesting to note that the cofiber of $\tau$ has another life in motivic homotopy theory [37], where it has been used to compute homotopy groups of spheres [52]. Its $E_{2}$ page is the mirror image of the $E_{2}$ page of the Bloch–Ogus–Leray spectral sequence that abuts to the coniveau filtration on $H^{*}_{\text{sing}}(X,\mathbb{Z}/m)$, so our computations end up involving unramified cohomology and some related birational invariants studied by Colliot-Thélène and Voisin in [25, Props. 3.3 and 3.4]. We obtain the following concrete statements about Quillen–Lichtenbaum dimension.

In general we will see that if $X$ is a $d$-dimensional variety with $h^{0,d}(X)\neq 0$, then $\dim_{\mathrm{QL}}(X)=d$. But starting in dimension 3 we will restrict our attention to rationally connected varieties. Recall that a smooth complex projective variety is rationally connected if any two points can be joined by a chain of rational curves [29, Cor. 4.28]. This includes all unirational varieties, as well as all smooth Fano varieties [55], and thus all $d$-dimensional smooth hypersurfaces of degree $\leq d$. It implies that $CH_{0}(X)=\mathbb{Z}$, but is strictly stronger. It implies that $h^{0,i}(X)=0$ for all $i>0$ [29, Cor. 4.18(a)].

The invariants appearing in Theorem 4(a) and (b) are more or less familiar birational invariants. The integral Hodge conjecture on $H^{4}(X,\mathbb{Z})$ and $H^{2d-2}(X,\mathbb{Z})$ has been studied in the context of birational geometry by Voisin and her coauthors in [84, Lem. 1], [93, Lem. 15], and especially [25]; note that for any Fano 4-fold, $H^{6}(X,\mathbb{Z})$ is generated by algebraic classes by [45, Thm. 1.7(ii)]. The Griffiths group $\operatorname{Griff}_{1}(X):=CH_{1}(X)_{\text{hom}}/CH_{1}(X)_{\text{alg}}$ is discussed in [95, §2.1] and [96, §1.3.2], and the quotient $H^{5}/N^{2}H^{5}$ figures prominently in [94].

We will apply Theorem 4 to the two families of Fano 4-folds shown by Hassett, Tschinkel, and Pirutka to contain both rational and irrational members, and will see that all members of both families have $\dim_{\mathrm{QL}}(X)=2$. We will see that all cubic 4-folds and Gushel–Mukai 4-folds have $\dim_{\mathrm{QL}}(X)=2$, so higher K-theory gives no obstruction to their rationality. We will see that quartic 4-folds that are very general in the Noether–Lefschetz sense have $\dim_{\mathrm{QL}}(X)=2$; Totaro [90] proved that a very general quartic 4-fold is irrational, but we do not know how the latter “very general” interacts with the former.

On the other hand, we will see that the unirational 4-fold constructed by Schreieder in [80, Cor. 1.4] has $\dim_{\mathrm{QL}}(X)=3$.

In dimension 5, we will analyze one example, relying on calculations of Fu and and Tian [35]:

The algebraic and topological K-theory of $X$ depend only on the derived category of coherent sheaves $D^{b}_{\mathrm{coh}}(X)=\operatorname{Perf}(X)$, so the same is true of $\dim_{\mathrm{QL}}(X)$. In fact one can extend Definition 1 to any $\mathbb{C}$-linear dg-category or stable $\infty$-category, using Blanc’s topological K-theory [16]. When the derived category of a variety admits a semi-orthogonal decomposition

$$D^{b}_{\mathrm{coh}}(X)=\langle\mathcal{A}_{1},\dotsc,\mathcal{A}_{k}\rangle,$$

we will see that

$$\dim_{\mathrm{QL}}(X)=\max(\dim_{\mathrm{QL}}(\mathcal{A}_{1}),\,\dotsc,\,\dim_{\mathrm{QL}}(\mathcal{A}_{k})).$$

We extract two kinds of results from this observation.

First, we use homological projective duality to produce some interesting Fano 4-folds with $\dim_{\mathrm{QL}}(X)=2$, which therefore satisfy all the conditions of Theorem 4(a) and (b):

In each case, $D^{b}_{\mathrm{coh}}(X)$ admits a semi-orthogonal decomposition consisting of an exceptional collection and the derived category of a surface $S$ with $h^{0,2}\neq 0$. In the first example, where $H^{*}(X,\mathbb{Z})$ is torsion-free, it is easier to deduce the integral Hodge conjecture using Perry’s [74, Prop. 5.16(2)], but the result on $\operatorname{Griff}_{1}(X)$ is new. In the second example, we do not know whether there is torsion in $H^{*}(X,\mathbb{Z})$. In all three examples, the fully faithful functors $D^{b}_{\mathrm{coh}}(S)\to D^{b}_{\mathrm{coh}}(X)$ are induced by ideal sheaves of explicit subschemes of $S\times X$, and it should be possible to prove Theorem 6 more directly by analyzing the geometry of those correspondences; but it is also gratifying to get integral information about $H^{4}$ and $CH_{1}$ from derived categories, not just rational information.

Second, we compute the higher K-theory of some Kuznetsov components of derived categories. For many Fano varieties $X$, Kuznetsov has identified a semi-orthogonal decomposition of $D^{b}_{\mathrm{coh}}(X)$ consisting of an exceptional collection and an interesting piece $\mathcal{A}_{X}$, which often behaves like the derived category of a lower-dimensional variety. Most famously, if $X$ is a cubic 4-fold [60] or Gushel–Mukai 4-fold [64], then the Kuznetsov component $\mathcal{A}_{X}$ behaves like the derived category of a K3 surface in terms of Serre duality, Hochschild homology and cohomology, and the Euler pairing on $KU^{0}$. We find that its algebraic K-theory behaves like that of a K3 surface as well:

We will also describe the K-theory of Kuznetsov components of derived categories of Fano 3-folds in §3.3, and of cubic 5-folds in §5.

We thank Dan Dugger and Ben Antieau for extensive discussions. Claire Voisin was generous with her advice, as were Jean-Louis Colliot-Thélène, Lie Fu, James Hotchkiss, John Christian Ottem, Jørgen Rennemo, Stefan Schreieder, and Zhiyu Tian. N.A. thanks the Max Planck Institute for Mathematics in Bonn for their hospitality, and was supported by NSF grant no. DMS-1902213. E.E. was supported by the NSERC grant RGPIN-2023-04233 “Reimagining motivic cohomology.”

For spectra, we use the notation and syntax of higher algebra [66]; we write $\otimes$ and $\oplus$ and $[1]$ where some authors would write $\wedge$ and $\vee$ and $\Sigma$. Algebraic K-theory for us is connective algebraic K-theory, that is, the universal additive invariant in the sense of [20].

Let $X$ be a variety over the complex numbers. In this section we explain the Bott maps

$$\tau_{n}\colon K_{n}(X,\mathbb{Z}/m)\to K_{n+2}(X,\mathbb{Z}/m),$$

(1)

their relation to the comparison maps

$$\eta_{n}\colon K_{n}(X,\mathbb{Z}/m)\to KU^{-n}(X,\mathbb{Z}/m),$$

(2)

and the equivalence between formulating Definition 1 with (1) or (2). Then we upgrade the Bott maps (1) to a map of spectra, take its cofiber, and give another equivalent formulation of Definition 1.

Topological K-theory satisfies Bott periodicity: if $\beta$ is a generator of $KU^{-2}(\text{point})\cong\mathbb{Z}$, then multiplication by $\beta$ gives isomorphisms

and

for all $n$ and $m$.

In algebraic K-theory with $\mathbb{Z}$ coefficients, there is no Bott periodicity, but with $\mathbb{Z}/m$ coefficients we can proceed as follows. Suslin’s rigidity theorem [85, Cor. 4.7] implies that $\eta_{n}\colon K_{n}(\mathbb{C},\mathbb{Z}/m)\to KU^{-n}(\text{point},\mathbb{Z}/m)$ is an isomorphism for all $n\geq 0$, so we would like to take the same generator $\beta\in KU^{-2}(\text{point})$, map it to a generator of $KU^{-2}(\text{point},\mathbb{Z}/m)\cong\mathbb{Z}/m$, lift it to $K_{2}(\mathbb{C},\mathbb{Z}/m)$, and let it act on $K_{*}(X,\mathbb{Z}/m)$ to give the maps (1) above.

But there are well-known difficulties when $m\equiv 2\pmod{4}$, which we cannot ignore because $\mathbb{Z}/2$ coefficients are the most interesting ones for our applications. In that case the Moore spectrum $\mathbb{S}/m$ does not admit a unital multiplication $\mathbb{S}/m\otimes\mathbb{S}/m\to\mathbb{S}/m$, so $K(\mathbb{C},\mathbb{Z}/m)$ and $K(X,\mathbb{Z}/m)$, which are obtained from $K(\mathbb{C})$ and $K(X)$ by tensoring with $\mathbb{S}/m$, may also fail to be ring spectra.⁴⁴4Pedrini and Weibel [71] cite Araki and Toda [7, Thm. 10.7] to say that the homotopy groups $K_{*}(\mathbb{C},\mathbb{Z}/m)$ and $K_{*}(X,\mathbb{Z}/m)$ are nonetheless graded rings because $\sqrt{-1}\in\mathbb{C}$, but soon we will really need to work with spectra. But the action $\mathbb{S}/2m\otimes\mathbb{S}/m\to\mathbb{S}/m$ is better behaved, so we define (1) by letting the image of $\beta$ in $K_{2}(\mathbb{C},\mathbb{Z}/2m)$ act on $K_{*}(X,\mathbb{Z}/m)$. There are further difficulties with associativity of this action when $m$ is divisible by 4 but not 16, or by 3 but not 9, but these do not affect us. For further discussion of these issues, see Thomason’s [86, App. A], or for a more recent perspective see [10, §3 and App. A] or [23].

Thus we have defined the Bott maps (1) for all $m$, and we have a commutative square

Knowing that $\eta_{n}$ is an isomorphism for $n\gg 0$, we see for any given $N$ that $\eta_{n}$ is an isomorphism (or injective) for all $n\geq N$ if and only if $\tau_{n}$ is an isomorphism (or injective) for all $n\geq N$: thus the two formulations of Definition 1 are equivalent, as promised.

Next we upgrade the Bott maps (1) to a map of spectra

$$\tau\colon K(X,\mathbb{Z}/m)[2]\to K(X,\mathbb{Z}/m)$$

(3)

When $m\equiv 2\pmod{4}$, we start with our generator of $K_{2}(\mathbb{C},\mathbb{Z}/2m)$ and take the corresponding map

$$S^{2}\to K(\mathbb{C},\mathbb{Z}/2m).$$

Then we tensor with $K(X,\mathbb{Z}/m)$ to get

$$S^{2}\otimes K(X,\mathbb{Z}/m)\to K(\mathbb{C},\mathbb{Z}/2m)\otimes K(X,\mathbb{Z}/m),$$

and post-compose with the map

$$K(\mathbb{C},\mathbb{Z}/2m)\otimes K(X,\mathbb{Z}/m)\to K(X,\mathbb{Z}/m)$$

obtained by tensoring the multiplication map

$$K(\mathbb{C})\otimes K(X)\to K(X)$$

with the action

$$\mathbb{S}/2m\otimes\mathbb{S}/m\to\mathbb{S}/m.$$

The case $m\not\equiv 2\pmod{4}$ is more straightforward: we do the same with $m$ in place of $2m$.

Finally we define $K/\tau(X,\mathbb{Z}/m)$ as the cofiber of the spectrum-level Bott maps (3). Its homotopy groups $(K/\tau)_{*}(X,\mathbb{Z}/m)$ fit into a long exact sequence

$$\dotsb\to K_{n-2}(X,\mathbb{Z}/m)\xrightarrow{\tau_{n-2}}K_{n}(X,\mathbb{Z}/m)\to(K/\tau)_{n}(X,\mathbb{Z}/m)\\ \to K_{n-3}(X,\mathbb{Z}/m)\xrightarrow{\tau_{n-3}}K_{n-1}(X,\mathbb{Z}/m)\to\dotsb,$$

giving another equivalent formulation of Definition 1:

If $X$ and $Y$ are smooth complex projective varieties then every equivalence $D^{b}_{\mathrm{coh}}(X)\to D^{b}_{\mathrm{coh}}(Y)$ is induced by a Fourier–Mukai kernel, which induces weak equivlaences $K(X)\to K(Y)$ and $KU(X)\to KU(Y)$, and similarly with finite coefficients, compatible with all comparison and Bott maps. Thus $\dim_{\mathrm{QL}}(X)$ depends only on $D^{b}_{\mathrm{coh}}(X)$.

In fact, if $\mathcal{C}$ is any dg-category or stable $\infty$-category over $\mathbb{C}$, we can define the spectrum $K(\mathcal{C})$ using Waldhausen’s S-construction as in [20], and $KU(\mathcal{C})$ following Blanc [16], and with finite coefficients we can define Bott maps just as in the previous section, so we can extend Definition 1:

The equivalence between (a) and (b) depends on a result of Antieau and Heller [5, Thm. 3.3]: they proved Blanc’s conjecture [16, Conj. 4.27] that the natural map from to (connective) algebraic K-theory with finite coefficients to semitopological K-theory with finite coefficients is an equivalence, and this implies that $KU(\mathcal{C},\mathbb{Z}/m)$ is the direct limit of the Bott maps on $K(\mathcal{C},\mathbb{Z}/m)$. When $\mathcal{C}=D^{b}_{\mathrm{coh}}(X)=\operatorname{Perf}(X)$ we recover our earlier definitions.

In general we do not know whether $\dim_{\mathrm{QL}}(\mathcal{C})$ is finite, even when $\mathcal{C}$ is smooth and proper; cf. [5, Question 4.2]. But in practice we are only interested in admissible subcategories $\mathcal{C}\subset D^{b}_{\mathrm{coh}}(X)$, in which case we will see that $\dim_{\mathrm{QL}}(\mathcal{C})\leq\dim_{\mathrm{QL}}(X)$. Recall that a full triangulated subcategory of $D^{b}_{\mathrm{coh}}(X)$ is called admissible if the inclusion functor has left and right adjoints; it inherits its dg or stable $\infty$ enhancement from $D^{b}_{\mathrm{coh}}(X)$.

A semi-orthogonal decomposition

$$D^{b}_{\mathrm{coh}}(X)=\langle\mathcal{A}_{1},\dotsc,\mathcal{A}_{k}\rangle$$

is a sequence of admissible subcategories, with no Homs or Exts from right to left, that generate $D^{b}_{\mathrm{coh}}(X)$ as a triangulated category.

A semi-orthogonal decomposition on $D^{b}_{\mathrm{coh}}(X)$ induces a direct sum decomposition on $K(X)$ and $KU(X)$: one approach is to take the projection kernels of [61, Thm. 7.1] and let them induce idempotents on $K(X)$ and $KU(X)$. In fact one could take this as the definition of $KU$ of an admissible subcategory rather than invoking Blanc’s machinery. The idempotents and decompositions are compatible with all comparison and Bott maps, so we also get a direct sum decomposition of $K/\tau(X)$, and we find:

This is as far we can get with semi-orthogonal decompositions alone; to proceed further, we construct a motivic spectral sequence for $K/\tau$, which allows us to make more subtle connections between Quillen–Lichtenbaum dimension and geometry.

In this section we introduce a spectral sequence to compute $(K/\tau)_{*}(X,\mathbb{Z}/m)$. To describe the $E_{2}$ page, let

$$\pi\colon X_{\text{an}}\to X_{\text{Zar}}$$

be the identity map from $X$ in the analytic topology to the Zariski topology, and for any Abelian group $A$, let

$$\mathcal{H}^{i}_{X}(A):=R^{i}\pi_{*}\,\underline{A},$$

where $\underline{A}$ is the constant sheaf on $X$ in the analytic topology; in other words, $\mathcal{H}^{i}_{X}(A)$ is the sheaf associated to the presheaf which maps a Zariski open set $U\subset X$ to $H^{i}_{\text{sing}}(U,A)$.

Bloch and Ogus [18, eq. (0.3)] proved that $H^{i}(\mathcal{H}^{j}_{X}(A))$ vanishes for $i>j$ and any $A$, so we can write out the $E_{2}$ page of the spectral sequence from Theorem 1.6 as follows, supressing the $\mathbb{Z}/m$ coefficients:⁵⁵5It would be interesting to understand the differentials, which should be some incarnation of Steenrod squares or powers. We suspect that many of them vanish.

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(6)

If $\dim X=d$ then we also have $\mathcal{H}^{j}_{X}(A)=0$ for $j>d$, because $X$ is smooth. From this we see again that

$$(K/\tau)_{n}(X,\mathbb{Z}/m)=0\qquad\text{for }n>d,$$

which is exactly the statement of the Quillen–Lichtenbaum conjecture for $X$; but we note that the proof of Theorem 1.6 will rely on Rost and Voevodsky’s resolution of the Bloch–Kato conjecture.

We also get

$$(K/\tau)_{d}(X,\mathbb{Z}/m)=H^{0}(\mathcal{H}^{d}_{X}(\mathbb{Z}/m))$$

(7)

and an exact sequence of low-degree terms

$$0\to H^{1}(\mathcal{H}^{d}_{X}(\mathbb{Z}/m))\to(K/\tau)_{d-1}(X,\mathbb{Z}/m)\\ \to H^{0}(\mathcal{H}^{d-1}_{X}(\mathbb{Z}/m))\to H^{2}(\mathcal{H}^{d}_{X}(\mathbb{Z}/m))$$

(8)

which we will use often.

The groups $H^{0}(\mathcal{H}^{j}_{X}(A))$ are sometimes called unramified cohomology and denoted $H^{j}_{\text{nr}}(X,A)$; they are well-known birational invariants, and if $X=\mathbb{P}^{d}$ then they vanish for $j>0$. Colliot-Thélène and Voisin have shown that $H^{i}(\mathcal{H}^{d}_{X}(A))$ are birational invariants as well [25, Prop. 3.4], and if $X=\mathbb{P}^{d}$ then they vanish for $i<d$ [25, Prop. 3.3(iii)].

It may be that all rationally connected varieties satisfy $H^{0}(\mathcal{H}^{d}_{X}(\mathbb{Z}/m))=0$ for all $m$, or even $H^{i}(\mathcal{H}^{d}_{X}(\mathbb{Z}))=0$ for all $i<d$. For discussion along these lines, see [96, §1.3.2], after [81, Thm. 1.4], and [88, §3].⁶⁶6The “Kato homology” $KH_{a}(X,A)$ that appears in [88] is isomorphic to $H^{d-a}(\mathcal{H}^{d}_{X}(A))$, so the conjeture about $H^{<d}(\mathcal{H}^{d}_{X})$ is Tian’s statement $\mathbf{RC}(n,k)$ for all $n$ and $k$. If this were true, it would simplify and strengthen our Theorem 4, and allow us to say something about rationally connected 5-folds in general.

We also make the following observation, which is immediate from (7) but seems hard to prove more directly:

Lastly, we repackage a result of Farb, Kisin, and Wolfson and use it to relate Quillen–Lichtenbaum to the outermost Hodge number.⁷⁷7We thank James Hotchkiss for pointing this out.