# Slopes of the Harder-Narasimhan filtration of a limit of vector bundles

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" ?

• 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. Commented Sep 3, 2019 at 5:04
• 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). Commented Sep 3, 2019 at 5:49
• 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
– afh
Commented Jan 23, 2022 at 1:02