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For a vector bundle $F$ consider its slope $\mu(F)=deg(F)/rk(F)$.

Let the Harder-Narasimhan stratum of degree $(q_1, q_2, ..., q_n)$ with $q_1 \geq ... \geq q_n$ be the locus of vector bundles $E$ with Harder-Narasimhan filtration $0 \neq E_1 \subset E_2 \subset ... \subset E_m = E$ such that $(q_1, q_2, ..., q_n)$ is 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.

Suppose that a Harder-Narasimhan stratum of degree $(p_1, p_2, ..., p_n)$ intersects nontrivially the closure of a Harder-Narasimhan stratum of degree $(q_1, q_2, ..., q_n)$. Is it true that the sequence $(q_1, q_2, ..., q_n)$ dominates the sequence $(p_1, p_2, ..., p_n)$ in the sense that $q_n \geq p_n$, $q_n+q_{n-1} \geq p_n+p_{n-1}$, ..., $q_n + ... + q_2 \geq p_n + ... + q_2$, $q_n + ... + q_1 = p_n + ... + q_1$? Can this be deduced from Theorem 2.3.3 (part (f)?) in Simon Schieder's paper https://arxiv.org/abs/1212.6814 "The Harder-Narasimhan stratification of the moduli stack of G-bundles via Drinfeld's compactifications" ?

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  • $\begingroup$ It seems in your question the role of $p_i, q_i$ are almost symmetrical. If the claim holds, then you would have $p_i=q_i$ for most cases. I feel I must be confused with something really basic here. $\endgroup$
    – Bombyx mori
    Sep 3, 2019 at 5:04
  • $\begingroup$ I don't think their roles are symmetric because the condition in the question means that a bundle in the Harder-Narasimhan stratum of degree (p_1,...,p_n) is a limit of bundles in the Harder-Narasimhan stratum of degree (q_1,...,q_n). $\endgroup$
    – Yellow Pig
    Sep 3, 2019 at 5:49
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    $\begingroup$ This seems to amount to the semicontinuity of the Harder-Narasimhan polygon, which is proved in a paper by Shatz numdam.org/item/CM_1977__35_2_163_0 $\endgroup$
    – afh
    Jan 23, 2022 at 1:02

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