0
$\begingroup$

Suppose $(p_1, ..., p_n)$ and $(q_1, ..., q_n)$ are two (non-strictly) decreasing sequences of positive real numbers. We say that $(q_1, ..., q_n)$ dominates $(p_1, ..., p_n)$ if $q_n \geq p_n$, $q_n+q_{n-1} \geq p_n+p_{n-1}$, ..., $q_n + ... + q_2 \geq p_n + ... + q_2$, $q_n + ... + q_1 = p_n + ... + q_1$.

For a vector bundle $F$ consider its slope $\mu(F)=deg(F)/rk(F)$. For a vector bundle $E$ with Harder-Narasimhan filtration $0 \neq E_1 \subset E_2 \subset ... \subset E_m = E$ we consider the sequence $\mu(E_1),...,\mu(E_1), ..., \mu(E_m/E_{m-1}), ..., \mu(E_m/E_{m-1})$, where $\mu(E_i/E_{i-1})$ is repeated $rk(E_i/E_{i-1})$ times, and call this sequence the degree of the Harder-Narasimhan filtration of $E$.

Suppose that $A \subset B$ are two vector bundles of the same rank $n$ such that $deg B = deg A + 1$. Assume that the degree of the Harder-Narasimhan filtration of $B$ is dominated by a (non-strictly) decreasing integer sequence $(d_1, ..., d_n)$ such that $d_1>d_2$. Is it true that then the degree of the Harder-Narasimhan filtration of $A$ is dominated by $(d_1-1, d_2, ..., d_n)$?

$\endgroup$
0
$\begingroup$

Sorry, I remembered how to answer my own question. Consider the degree of a HN stratum as a Young diagram with rational lengths of rows. It is enough to show that the Young diagram of A is a subset of the Young diagram of B because we can just omit the last inequality in the definition of domination. Suppose the Young diagram of A is not a subset of the Young diagram of B. Then there is a subbundle C of A consisting of several members of HN filtration of highest slope whose Young diagram is not inside the Young diagram of B. The bundle C maps by 0 to the quotient E of B consisting of several members of the HN filtration of lowest slope whose slope is smaller than the slope of the parts of C. So the bundle C is a subbundle of the kernel of the surjection from B to E. But since the Young diagram of C is not inside the Young diagram of B, the kernel of the map from B to E has smaller rank than the rank of C. Contradiction.

Please see a related question which I still need Slopes of the Harder-Narasimhan filtration of a limit of vector bundles

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.