"theta characteristics" on general motives?

Question

Yesterday I read in some texts on theta characteristics of algebraic curves, which are structures on their Jacobians... Do you know if someone has thought if such, but more general, things can exist on motives?

Answer 1

The existence of theta characteristics on a curve is a refinement of the splitting of its motive. The reference to this is Oliver Ronding's theta characteristics and stable homotopy type paper. The point is that the existence of a theta characteristic on a curve (with a rational point) implies a splitting in the motivic stable homotopy category $SH$: $X \simeq S^{0,0} \vee J(X) \vee S^{2,1}$

This realizes to the splitting in category of motives, $M(X) \simeq \mathbb{Z} \oplus Jac(X) \oplus \mathbb{Z}(1)[2]$ (which always happens since the motive of an algebraic variety of dimension $d$ with a rational point splits into $\mathbb{Z} \oplus Pic(X) \oplus ?? \oplus Alb(X) \oplus \mathbb{Z}(d)[2d]$). But the splitting in the $SH$ is much finer - for example it implies splitting as a module over the motivic Steenrod algebra.

My guess is that a theta characteristic on a motive should be a structure on its stable motivic homotopy type, as opposed to its motive and it should lead to a splitting of the top cell. Hints of this is given in the Rondings paper: he shows that all you need for this refined splitting is that the Spanier-Whitehead dual of $X$ is of a certain shape (Corollary 5.2).