# The Harder-Narasimhan filtration with inverse slopes.

Let $C$ be a complex curve. Recall that the slope of a coherent sheaf $\mathcal{E}$ is defined by $$\mu(\mathcal{E})=\mathrm{Arg}(-\mathrm{deg}(\mathcal{E})+i\mathrm{rank}(\mathcal{E}))\in(0,\pi].$$ We say that $\mathcal{E}$ is semistable if any subsheaf $\mathcal{F}\subset \mathcal{E}$ satisfies $\mu(\mathcal{F})\le \mu(\mathcal{E})$. The Harder-Narashimhan filtration says that for any coherent sheaf $\mathcal{F}$ there is a unique filtration $$0=\mathcal{F}_{0}\subset \mathcal{F}_{1}\subset \dots \subset \mathcal{F}_{n}=\mathcal{F}$$ such that the filtration quotient $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ are semistable of slope with $\mu_{i}$ with $\mu_{1}>\mu_{2}>\dots>\mu_{n}$.

Is it possible to obtain a filtration with modified condition $\mu_{1}<\mu_{2}<\dots<\mu_{n}$?

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