expectation of something about K-stability or slope stability

Question

I'm studying K-stability of $(X,L)$ where $X$ is an $L$-polarized variety and $L$ is an ample line bundle from paper of Y. Odaka. I expect following things, but I can't get an answer.

1) polarized varieties $(X_1,L_1)$,$(X_2,L_2)$ are K-(semi)stable, then $(X_1\times X_2,p_1^*L_1\otimes p_2^*L_2)$ is K-(semi)stable, where $p_i$ i-th projection.

2) polarized varieties $(X,L_1)$,$(X,L_2)$ are K-(semi)stable, then $(X,L_1\otimes L_2)$ is K-(semi)stable.

I know that K-(semi)stable $\Longrightarrow$ slope-(semi)stable and computing slope-stability is much easier than computing K-stability. So I tried slope-stable case first.

By calculation, $\mu(X_1\times X_2,p_1^*L_1\otimes p_2^*L_2)=\mu(X_1,L_1)+\mu(X_2,L_2)$ and $\mu_c(\mathcal{O}_{Z\times X_2},p_1^*L_1\otimes p_2^*L_2)=\mu_c(\mathcal{O}_Z,L_1)+\mu(X_2,L_2)$ for subscheme $Z \subset X_1$ where $(\mu(X,L)$=the slope of $(X,L)$ and $\mu_c(\mathcal{O}_Z,L)$=the quotient slope of $Z)$. But slope stability must be checked with respect to all subschemes of $X_1\times X_2$. I can't make progress.

In case of curve, a smooth curve $(C,L)$ is slope semistable. And from a paper of J.Ross, I know that for $g\geq 5$, there exist smooth curve $C$ of genus $g$ s.t. $X=C\times C$ is not slope semisable with respect to certain polarizations. But these polarizations are not of the form $p_1^*L_1\otimes p_2^*L_2$.

Do you know the answer of 1),2)(for K-stability or slope stability)? Can you give helpful information or reference?

Answer 1

Yomi,

I have thought about (1) for a while. I am sure it is true but am unable to prove it algebraically. The problem for K-stability is really the same as that for slope stability -- it is easy to rule out test configurations that "split" but I see no a priori reason to rule out more complicated test configurations. It is a curiosity since the corresponding statement for cscK metrics (or balanced metrics in fact) is clearly correct. I understand some work has been done, or is being done, in this direction for the toric case.

I suspect the answer for (2) is no. In terms of metrics you are asking that if $L_i$ has a cscK metric $\omega_i$ for $i=1,2$ then $L_1\otimes L_2$ has a cscK metric. But the scalar curvature of $\omega_1 + \omega_2$ need not be constant. I imagine coming up with a specific example is not too hard (perhaps this can be done even for slope stability on the product of curves that you already mention).