Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $L$ be an ample line bundle on $X$. Let $F$ be a $\mu_L$ semistable rank 2 vector bundle on $X$ (semistability in the sense of Mumford-Takemoto).

We have the projections $p_i:X\times X\longrightarrow X$ for $i=1,2$. Then $L'=p_1^*L\otimes p_2^*L$ is an ample line bundle on $X\times X$. I want to know if \ (a) $p_1^*F\oplus p_2^*F$ and (b) $p_1^*F\otimes p_2^*F$ are $\mu_{L'}$-semistable on $X\times X$.

The first doubt I have is as follows.

1) We have the morphism $f:X\times X\longrightarrow X\times X$, $(x,y)\mapsto (y,x)$. This is an isomorphism. So under this isomorphism $f^*p_2^*F=p_1^* F$ right? So doesn't this mean that the chern classes of $p_i^*F$ are the same?

If (1) is correct, in order to check (a) it is enough to check that the pull back $p_i^*F$ is $\mu_{L'}$ semistable (since the direct sum of semistable sheaves with the same slopes is semistable).

The toy case I tried is $\mathbb{P}^1\times\mathbb{P}^1\longrightarrow\mathbb{P}^1$. But in this case it looks to be trivial. Because any $F$ on $\mathbb{P}^1$ is a direct sum of line bundles of the same degree. So the pull back will also be a direct sum of line bundles with same slope, and hence will be semistable. So it does not seem to be a very illuminating example.

(2) In case when $X$ is a curve or a surface, can we say that (a) and (b) are $\mu_{L'}$ semistable?

Thanks in advance!


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