Do we know anything about Harder-Narasimhan filtrations of tensor products of vector bundles?

Question

I am interested in vector bundles over a nonsingular complete algebraic curve $C$ over $\mathbb C$. For a vector bundle $E$, its Harder-Narasimhan filtration is a filtration of subbundles $$0=E_0\subset E_1\subset\cdots\subset E_n=E$$ such that each $E_i/E_{i-1}$ is semitable and $\frac{\deg(E_1/E_0)}{\mathrm{rank}(E_1/E_0)}>\cdots>\frac{\deg(E_n/E_{n-1})}{\mathrm{rank}(E_n/E_{n-1})}$.

Now if we have two vector bundles $E,F$ and we know their HN filtation $\{E_i\},\{F_j\}$, can we get any information about the HN filtration of $E\otimes F$?

One information I am interested in is the upper bound of $\frac{\deg}{\mathrm{rank}}$ of subbundles of $E\otimes F$.

Thanks.

Answer 1

It is a result of Narasimhan and Seshadri that if $V$ is semistable and $W$ is semistable then $V \otimes W$ is semistable.

If $E$ has a filtration with associated graded $E_i/ E_{i-1}$, and $F$ has a filtration with associated graded $F_j/ F_{j-1}$, then $E \otimes F$ has a filtration with associated graded $(E_i/ E_{i-1}) \otimes (F_j/ F_{j-1})$. By the previous claim, the associated graded pieces of this filtration are semistable, and we can choose the filtration so that these pieces are in order of increasing slope. Hence it is the Harder-Narasimhan filtration.

Thus the slopes of $E \otimes F$ are the slopes of $E$ plus the slope of $F$, and in particular the maximal slope of $E \otimes F$ is the sum of the maximal slope of $E$ and the maximal slope of $F$.