# Comptes Rendus
Mathématique

Xi Chen, James D. Lewis and Gregory Pearlstein

Indecomposable $K_1$ classes on a Surface and Membrane Integrals

Volume 358, issue 4 (2020), p. 511-513

Published online: 28 July 2020

https://doi.org/10.5802/crmath.69

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class {$\gamma$} $\in H_1(X, \mathbb{Z}) \simeq H^3(X, \mathbb{Z}(2))$ (Poincaré duality). We now assume that $\xi \in K_{1,\text{ind}}^{(2)}(X)^{\circ}$. Then $\gamma$ bounds a real 2-chain $\zeta$. There is the *integrally defined* transcendental Abel–Jacobi map,

$$ \underline{\Phi}: K_{1,\text{ind}}^{(2)}(X)^{\circ} \to \frac{H^{2,0}(X)^{\vee}}{H_2(X, \mathbb{Z})}, \quad \underline{\Phi}(\xi)(\omega) = \int_{\zeta} \omega. $$

It is our belief that $\underline{\Phi}$ is injective. This is the subject matter of our paper where we provide some evidence in support of this. Our main results are stated in Theorem 4 and Corollary 5.

## 2. Notation

We assume that the reader is familiar with mixed Hodge structures (MHS). Let $V$ be a $\mathbb{Z}$-MHS and $\mathbb{Z}(r)$ the Tate twist. We put

$$ \Gamma V = \operatorname{hom}_{\mathrm{MHS}}(\mathbb{Z}(0), V), \text{ and } J V = \operatorname{Ext}_{\mathrm{MHS}}(\mathbb{Z}(0), V). $$

## 3. The full Abel–Jacobi map on indecomposables

Let $H_{\text{tr}}^2(X, \mathbb{Z}) := H^2(X, \mathbb{Z})/\text{NS}(X)$ be transcendental cohomology, where NS stands for the Neron–Severi group. There is a full Abel–Jacobi map [3] on indecomposables,

$$ \Phi: K_{1,\text{ind}}^{(2)}(X)^{\circ} \to J(H_{\text{tr}}^2(X, \mathbb{Z}(2))) = \frac{(H^{2,0}(X) \oplus H_{\text{tr}}^{1,1}(X))^{\vee}}{H_2(X, \mathbb{Z})}, $$

given by (for $\omega \in H^{2,0}(X) \oplus H_{\text{tr}}^{1,1}(X))$:

$$ \{\xi\} \to \Phi(\xi)(\omega) := \frac{1}{2\pi i} \left( \sum_{j=1}^{N} \int_{D_j} \log(f_j)\omega - 2\pi i \int_{\zeta} \omega \right), $$

where log has the principal branch. It turns out that we can say something about this map. For this we recall the (limit) Betti cycle class map

$$ K_2(\mathbb{C}(X)) \xrightarrow{\mathrm{dlog}_2} \Gamma H^2(\mathbb{C}(X), \mathbb{Z}(2)), \{f, g\} \mapsto \mathrm{dlog}\,f \wedge \mathrm{dlog}\,g, $$

where

$$ \Gamma H^2(\mathbb{C}(X), \mathbb{Z}(2)) = \lim_{\vec{U}} \Gamma H^2(U, \mathbb{Z}(2)), U \subset X \text{ Zariski open.} $$

**Theorem 1.** $\Phi$ is injective iff $\mathrm{dlog}_2$ is surjective.

**Proof.** See [2, Corollary 6.5], where it should also be pointed out that $H^2(\mathbb{C}(X), \mathbb{Z}(2))$ is torsion free. $\square$

The following conjecture seems to have survived critical examination (see [2] and the references cited there).

**Conjecture 2 (Beilinson–Milnor–Hodge conjecture).** $\mathrm{dlog}_2$ is surjective.

**Remark 3.** As argued in [2], this conjecture is equivalent to the corresponding conjecture with $\mathbb{Q}$-coefficients.

Let $X = \mathcal{X}_0 := \rho^{-1}(0)$ be a very general<fnref n="1" /> member of a family of surfaces $\rho : \mathcal{X} \to S$, $0 \in S$. Here $\rho$ is a smooth and proper morphism of smooth quasi-projective varieties. We recall the Kodaira–Spencer map:

$$ \kappa : T_0(S) \to H^1(X, \Theta_X), $$

and put $H_{\text{alg}}^1(X, \Theta_X) := \kappa(T_0(S))$. We prove the following:

<footnote n="1">¹Very general in this context means outside of a countable union of proper analytic subsets.</footnote>
**Theorem 4.** Assume that $H_{\text{alg}}^{1}(X, \Theta_X) \otimes H^{2,0}(X) \xrightarrow{\circ} H_{\text{tr}}^{1,1}(X)$ is surjective. Then the correspondence $\Phi(\xi) \mapsto \Phi(\xi)$ is injective.

**Proof.** This proof takes inspiration from [1]. First of all, the assumptions in the theorem do not change if we shrink $S$ and/or replace $S$ by a finite cover $S' \to S$, which will be unramified over $0 \in S$ by Sard's lemma, together with 0 a very general point of $S$. Indeed a very general point of $S'$ will map to a very general point of $S$, and we can just as easily work over $S'$. However we can assume $0 \in S$ corresponds to a very general point of $S'$ and instead work over a polydisk neighbourhood of $0 \in S$. So for simplicity, we will replace $S$ by a polydisk, and we will assume this. Thus a cycle $\xi \in K_{1,\text{ind}}^{(2)}(X)^\circ$ lifts to a relative spread cycle $\tilde{\xi} \in K_{1}^{(2)}(\mathcal{X}/S)^\circ$, and corresponding normal function $v_{\tilde{\xi}}$, where $v_{\tilde{\xi}}(0) = \Phi(\tilde{\xi})$. Let $\nabla = \partial \otimes 1$ be the Gauss–Manin connection associated to the Hodge bundle $\mathcal{H} := \mathcal{O}_S \otimes R^2 \rho_* \mathbb{C}$, $\mu \in H^0(S, \Theta_S)$ a linear differential operator, which we identify with the corresponding operator $\nabla_\mu$ on $\mathcal{H}$. If $\omega \in H^{2,0}(X)$, there is a variational $\tilde{\omega} \in \mathcal{K}(\mathcal{X}/S)$ (relative canonical sheaf), with $\tilde{\omega}_0 = \omega$. Now suppose that $\Phi(\xi) \neq 0$, and yet $\langle v_{\tilde{\xi}}, \tilde{\omega} \rangle = 0$ for all $\tilde{\omega} \in \mathcal{K}(\mathcal{X}/S)$. This translates to saying that $\langle v_{\tilde{\xi}}, \tilde{\omega} \rangle = \langle \gamma, \tilde{\omega} \rangle$, for some period $\gamma \in H^0(S, R^2 \rho_* \mathbb{Z}(2))$. Now for all $\mu \in H^0(S, \Theta_S)$, we arrive at:

$$ \langle \gamma, \nabla_{\mu} \tilde{\omega} \rangle = \mu \langle \gamma, \tilde{\omega} \rangle = \mu \langle v_{\tilde{\xi}}, \tilde{\omega} \rangle = \langle \nabla_{\mu} v_{\tilde{\xi}}, \tilde{\omega} \rangle + \langle v_{\tilde{\xi}}, \nabla_{\mu} \tilde{\omega} \rangle = \langle v_{\tilde{\xi}}, \nabla_{\mu} \tilde{\omega} \rangle, $$

where we use the well-known fact that $v_{\tilde{\xi}}$ is quasi-horizontal, implying that $\langle \nabla_{\mu} v_{\tilde{\xi}}, \tilde{\omega} \rangle = 0$ for Hodge type reasons. By our assumption on the Kodaira–Spencer map, we have $v_{\tilde{\xi}} \equiv 0$, a fortiori $\Phi(\xi) = 0$, a contradiction. This tells us that $\Phi(\tilde{\xi}_t) \neq 0$ for very general $t \in S$. Since $0 \in \Delta$ already corresponds to a very general $X = \mathcal{X}_0$, we can assume that $t = 0$, and hence $\Phi(\xi) \neq 0 \Rightarrow \Phi(\xi) \neq 0$. $\square$

**Corollary 5.** Same assumptions as given in the above theorem. Further, let us assume Conjecture 2. Then the correspondence $\Phi \mapsto \Phi$ is injective. In particular, $\Phi$ is injective.

**Proof.** Beilinson rigidity and Conjecture 2, imply that $K_{1,\text{ind}}^{(2)}(X)^\circ$ is countable (Voisin's conjecture). Let $\xi \in K_{1,\text{ind}}^{(2)}(X)^\circ$. A spread of $\xi$ will involve a finite cover $S'_\xi \to S$ and the very general points of $S'_\xi$ map to a dense subset $S_\xi$ of $S$, being a countable intersection of open dense subsets (Baire). But again by Baire, the countable intersection

$$ \bigcap_{\xi \in K_{1,\text{ind}}^{(2)}(X)^\circ} S_\xi $$

is likewise dense in $S$; indeed and more explicitly, it amounts to the complement of a countable union of proper analytic subsets of $S$. Since $0 \in S$ is already very general, it belongs to that intersection. Therefore if $\Phi \mapsto \Phi$ were not injective, then it would fail to be injective for some $\xi$ as well. Now apply the above theorem. $\square$

## References

[1] X. Chen, C. Doran, M. Kerr, J. D. Lewis, “Normal Functions, Picard-Fuchs Equations, and Elliptic Fibrations on $K3$ Surfaces”, J. Reine Angew. Math. 721 (2016), p. 43-80.

[2] R. de Jeu, J. D. Lewis, “Beilinson's Hodge conjecture for smooth varieties, with an appendix by Masanori Asakura”, J. K-Theory 11 (2013), no. 2, p. 243-282.

[3] M. Kerr, J. D. Lewis, S. Müller-Stach, “The Abel-Jacobi map for higher Chow groups”, Compos. Math. 142 (2006), no. 2, p. 374-396.