Author: Eiko

Tags: uniformity, isocrystal, harder-narasimhan polygon

Time: 2024-11-12 21:58:14 - 2024-11-12 21:58:14 (UTC)

Reference: Etale And Crystalline Companions, II

Harder-Narasimhan Polygons

Let $C$ be a proper curve of genus $g$ over an arbitrary field $L$ (of arbitrary characteristic).

• The degree $\mathrm{deg}(V)$ of a nonzero bundle $V$ on $C$ is defined as

  $$\mathrm{deg}V=\mathrm{deg}({\wedge }^{\text{top}}V).$$

• The slope $\mu (V)$ of a nonzero bundle $V$ on $C$ is defined as

  $$\mu (V)=\frac{\mathrm{deg}V}{\text{rank}(V)}.$$

• $E$ is semistable if $\mu (E)\ge \mu (E^{\prime })$ for all nonzero subbundles $E^{\prime }$ of $E$, stable if $\mu (E)>\mu (E^{\prime })$ for all nonzero proper subbundles $E^{\prime }$ of $E$.

Basic Properties

• If $F$ is a subbundle of $E$ of the same rank, then $\mathrm{deg}(F)\le \mathrm{deg}(E)$.

  Side note: How can there be a subbundle of the same rank? I’m not sure what this means. Because $\mathcal{O}(-1)$ is not a subbundle of $\mathcal{O}$, since the quotient ${\mathcal{O}}_P$ is not locally free and hence not a bundle.

• If $0\to F\to E\to G\to 0$ is a short exact sequence of bundles, then

  $$\mathrm{deg}(E)=\mathrm{deg}(F)+\mathrm{deg}(G),\mathrm{rank}(E)=\mathrm{rank}(F)+\mathrm{rank}(G).$$

  you can think of $\mu (E)$ as an intermediate slope between $\mu (F)$ and $\mu (G)$. As a result of this viewpoint, we have

  • $\mu (F)<\mu (E)\iff \mu (G)>\mu (E)$

  • $\mu (F)>\mu (E)\iff \mu (G)<\mu (E)$

  • $\mu (F)=\mu (E)\iff \mu (G)=\mu (E)\iff \mu (F)=\mu (G)$

• $\mathrm{deg}(V^{\ast })=-\mathrm{deg}(V)$ and $\mu (V^{\ast })=-\mu (V)$.

The Polygon

The HN polygon of $E$ is the convex hull of the set of points $(\mathrm{rank}(E^{\prime }),\mathrm{deg}(E^{\prime }))$ where $E^{\prime }$ goes over all subbundles of $E$.

What does this mean? By the formalism of stability in abelian categories, we can see that semi-stability implies that for any two term filtration $0\to E^{\prime }\to E\to E^{″}\to 0$ we have $\mu (E^{\prime })\le \mu (E)\le \mu (E^{″})$. Think $E^{\prime }$ and $E^{″}$ be part of the slopes in the diagram, with $E^{\prime }$ connecting the origin and on the left of $E^{″}$, then this is similar to the convex polygon we are familiar with in Newton polygons. This clearly generalises to the case of many-term filtrations.

Theorems

• Using Riemann-Roch, we can prove certain existence. If $E$ is a line bundle, a classical example of Riemann-Roch is that $h^0(E)\ge \mathrm{deg}E+1-g$, so $\mathrm{deg}E\ge g\Rightarrow H^0(E)\ne 0$.

• Let $E$ be a bundle on $C$, if all slopes of $\mathrm{HN}(E)>2g-1$, then $E$ is generated by global sections (that means there exists a surjection ${\mathcal{O}}^I\to E$ for some index set $I$).