A motivic study of generalized Burniat surfaces (Q1713449)

The article under review is concerned with the Chow motives of the so-called generalised Burniat surfaces as defined by \textit{I. Bauer} et al. [J. Math. Sci., Tokyo 22, No. 1, 55--111 (2015; Zbl 1375.14128)]. The main result (Theorem 4.2) is an application of a more general theorem due to \textit{R. Laterveer} et al. [Q. J. Math. 70, No. 1, 71--104 (2019; Zbl 1444.14017)]. It reproves Bloch's conjecture for those generalised Burniat surfaces satisfying \(p_g=0\) (a result independently due to \textit{C. Pedrini} and \textit{C. Weibel} [Lond. Math. Soc. Lect. Note Ser. 427, 308--329 (2016; Zbl 1388.14114)] for the classical and \textit{I. Bauer} and \textit{D. Frapporti} [Rend. Circ. Mat. Palermo (2) 64, No. 1, 27--42 (2015; Zbl 1330.14009)] for the generalised Burniat surfaces), namely, that the cycle class map \(\mathrm{CH}_0\to \mathrm{H}_0\) has trivial kernel. Moreover, it proves a suitable variant of Bloch's conjecture for the remaining cases, as well as the finite-dimensionality of their motives (resp. slight variants thereof) after Kimura. It should be recalled that Kimura, who introduced the concept of finite-dimensionality of Chow motives, conjectured that the motives of all varieties are finite-dimensional [\textit{S.-I. Kimura}, Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)]. Smooth generalised Burniat surfaces are certain minimal complex surfaces of general type satisfying \(K^2=6\) and \(q=p_g=0,1,2,3\). They arise as quotients \(Y=X/G\) where \(X\subset A\) is a suitable hypersurface in a product of three elliptic curves, \(A\), by a group \(G\cong (\mathbb Z/2\mathbb Z)^3\) of involutions of \(A\). There are \(16\) families \(\mathcal S_1,\dots,\mathcal S_{16}\), corresponding to different choices of the action. The surfaces of the first four families \(\mathcal S_1,\dots,\mathcal S_4\) are those satisfying \(p_g=0\). (The classical Burniat surface is in the family parametrised by \(\mathcal S_2\).) For the remaining cases, \(p_g>0\). In the above setup of a smooth generalised Burniat surface \(Y=X/G\), let \(\chi_A\) be the character corresponding to the one-dimensional \(G\)-representation \(\mathrm H^0(\Omega_A^3)\) and let \(\pi_{\chi_A}=|G|^{-1}\sum_{g\in G}\chi(g)\Gamma_g\in\mathrm{Corr}^0(X)\) be the canonically associated idempotent self-correspondence. The main theorem is concerned with the motive \((X, \pi_{\chi_A})\). For example, if the character is trivial, this is just the Chow motive of \(Y\). The Chow groups of \((X, \pi_{\chi_A})\) shall be denoted by \(\mathrm{CH}(X)^{\chi_A}\). For the surfaces parametrised by \(\mathcal S_3,\ldots,\mathcal S_{16}\), the main theorem of the article under review states that the motive \((X, \pi_{\chi_A})\) is finite-dimensional and that the kernel of the map \(i_*\colon \mathrm{CH}_0(X)^{\chi_A}\to \mathrm{CH}_0(A)^{\chi_A}\) is trivial, where \(i\colon X\to A\) is the inclusion map. By inspection of the actions involved, this implies the finite-dimensionality of the motive of \(Y\) for the families \(\mathcal S_3\), \(\mathcal S_4\), \(\mathcal S_{11}\), \(\mathcal S_{12}\) and \(\mathcal S_{16}\). Moreover, the kernel vanishing implies Bloch's conjecture for the cases \(\mathcal S_3\) and \(\mathcal S_4\). The same technique also proves the finite-dimensionality of the motive of \(X\) and Bloch's conjecture in the case of the families \(\mathcal S_1\) and \(\mathcal S_2\), as was shown by the author and their coauthors in [Q. J. Math. 70, No. 1, 71--104 (2019; Zbl 1444.14017)]. In a concluding remark, the author notes that the same technique applies to the so-called Sicilian surfaces, which are a common generalisation of the surfaces in the families \(\mathcal S_{10}\) and \(\mathcal S_{11}\); [\textit{I. Bauer} et al., J. Math. Sci., Tokyo 22, No. 1, 55--111 (2015; Zbl 1375.14128)].