Algebraic cycles on quadric sections of cubics in \(\mathbb{P}^4\) under the action of symplectomorphisms (Q2806122)

Let \(\tau:\mathbb{CP}^4\to\mathbb{CP}^4\) be the involution which switches the sign of two of the homogeneous coordinates, and consider the intersection \(S\) of a general \(\tau\)-invariant cubic and quadric hypersurfaces in \(\mathbb{CP}^4\). The first main result shows that the induced action \(\tau^*:\mathrm{CH}^2(S)\to \mathrm{CH}^2(S)\) on the Chow group is the identity. Next, they consider a \(\tau\)-invariant general cubic hypersurface \(C\) in \(\mathbb{CP}^4\), and they determine the induced action \(\tau^*:\mathrm{CH}^2(C)\to \mathrm{CH}^2(C)\). The Chow group splits as \(\mathrm{CH}^2(C)=A^2(C)\oplus\mathbb{Z}\), where \(A^2(C)\) is generated by cycle algebraically equivalent to zero. Then \(\tau^*\) acts as the identity on the \(\mathbb{Z}\) factor, while \(A^2(C)\) is generated by two subgroups \(B_2,B_3\) of geometric origin, and \(\tau^*\) is the identity on \(B_3\) and minus the identity on \(B_2\).