Vanishing results in Chow groups for the modified diagonal cycles (Q6589645)

Let \(C\) be a smooth projective curve of genus \(g\) over the complex numbers. Choosing a degree 1 divisor \(e\) on \(C\), let \(\iota_e(C)\subset{Jac} (C)\) denote the image of \(C\) in its Jacobian under the Abel-Jacobi map based at \(e\). The Ceresa cycle is defined as\N\[\N\kappa_e:= \iota_e(C) - [-1]^\ast \iota_e(C)\ \in \mathrm{CH}^{g-1}(\mathrm{Jac}(C))_{\mathbb{Q}}\N\]\Nwhere \(CH()_{\mathbb{Q}}\) denotes the Chow group with \(\mathbb{Q}\)-coefficients. While the Ceresa cycle is always homologically trivial, it is non-zero in \(\mathrm{CH}^{g-1}(\mathrm{Jac}(C))_{\mathbb{Q}}\) for \(C\) very general of genus \(g\ge 3\) [\textit{G. Ceresa}, Ann. Math. (2) 117, 285--291 (1983; Zbl 0538.14024)]. On the other hand, when \(C\) is a hyperelliptic curve and \(e\in C\) is a Weierstraß point, it is known that \(\kappa_e=0\) (in \(\mathrm{CH}^{g-1}(\mathrm{Jac}(C))_{\mathbb{Q}}\)). For a long time, not a single example was known of a non-hyperelliptic curve with vanishing Ceresa cycle; indeed, some people even believed that such examples could not exist.\N\NThe vanishing of the Ceresa cycle is related to the vanishing of the modified diagonal (first defined by \textit{B. H. Gross} and \textit{C. Schoen} [Ann. Inst. Fourier 45, No. 3, 649--679 (1995; Zbl 0822.14015)])\N\[\N\begin{split} \Delta_e:= \Delta^{sm}_C &- p_1^\ast(e)\cdot p_{23}^\ast(\Delta_C) - p_2^\ast(e)\cdot p_{13}^\ast(\Delta_C)- p_3^\ast(e)\cdot p_{12}^\ast(\Delta_C)\\\N&+ p_1^\ast(e)\cdot p_2^\ast(e) + p_1^\ast(e)\cdot p_3^\ast(e) + p_2^\ast(e)\cdot p_3^\ast(e)\ \in\ \mathrm{CH}^2(C\times C\times C)_{\mathbb{Q}}\ ,\\\N\end{split}\N\]\Nwhere \(\Delta_C^{sm}\) denotes the class of the small diagonal, and \(p_i,p_{ij}\) are the various projections. The modified diagonal is always homologically trivial, in general it is non-zero in \(\mathrm{CH}^2(C\times C\times C)_{\mathbb{Q}}\); indeed, the vanishing of \(\Delta_e\) is known to be equivalent to the vanishing of \(\kappa_e\) by \textit{S.-W. Zhang} [Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031). (It is also equivalent to the fact that the divisor \(e\) defines an MCK decomposition, in the sense of \textit{M. Shen} and \textit{C. Vial} [The Fourier transform for certain hyperkähler fourfolds. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1386.14025)].)\N\NThe paper under review contains a nice cohomological criterion for the vanishing of \(\Delta_e\) (and hence of \(\kappa_e\)). This criterion is useful when the group \(\Aut(C)\) of automorphisms of the curve is large:\N\N{Theorem:} Let \(C\) be a curve of genus \(g\ge 2\) such that\N\[\N\bigl( H^1(C,\mathbb{C})^{\otimes 3}\bigr){}^{\Aut(C)}=0\ .\N\]\NThen the modified diagonal \(\Delta_\xi\) is \(0\) in \(\mathrm{CH}^2(C\times C\times C)_{\mathbb{Q}}\), where \(\xi:= \frac{1}{2g-2}K_C\). (Hence, also \(\kappa_\xi=0\) in \(\mathrm{CH}^{g-1}(\mathrm{Jac}(C))_{\mathbb{Q}}\).)\N\N(This is Theorem 1.2.1 in the paper, which is not exactly stated this way, but the formulation is equivalent thanks to Theorem 1.5.5 in [\textit{S.-W. Zhang}, Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031)]).\N\NIn the paper under review, this criterion is successfully applied to find many non-hyperelliptic curves with vanishing Ceresa cycle: among the examples presented are the Fricke-Macbeath curve (the unique curve of genus \(7\) with maximal order automorphism group), the Bring curve (genus 4 with largest possible automorphism group), and some more examples in genus 4 and 5.\N\NThis is currently a very active area of research. There is pioneering work of \textit{D. Bisogno} et al. [Épijournal de Géom. Algébr., EPIGA 7, Article 8, 19 p. (2023; Zbl 1527.11054)] and of \textit{A. Beauville} [C. R., Math., Acad. Sci. Paris 359, No. 7, 871--872 (2021; Zbl 1470.14057)], containing non-hyperelliptic examples where \(\kappa_e\) vanishes modulo Abel-Jacobi equivalence. This was followed by work of \textit{A. Beauville} and \textit{C. Schoen} [Int. Math. Res. Not. 2023, No. 5, 3671--3675 (2023; Zbl 1518.14045)] giving examples where \(\kappa_e\) vanishes modulo algebraic equivalence. In more recent work by Laga-Shnidman, a generalization of the criterion of the paper under review is obtained, allowing for many more examples of non-hyperelliptic curves with vanishing Ceresa cycle [\textit{J. Laga} and \textit{A. Shnidman}, ``Vanishing criteria for Ceresa cycles'', Preprint, \url{arXiv:2406.03891}].\N\NA complete understanding of the locus inside \({\mathcal M}_g\) where the Ceresa cycle vanishes still remains elusive; the paper under review contains some conjectures about this locus.