---
date: 2021-11-03
modification date: Sunday 15th May 2022 16:18:05
title: "2021-11-03-quick_note"
aliases: [2021-11-03-quick_note]
---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #web/quick-notes 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# 2021-11-03

## 15:09

> UGA AG Seminar: Eloise Hamilton #projects/notes/seminars 

- GIT: $G$ a [reductive](Unsorted/Borel.md) group (trivial [unipotent radical](Unsorted/Borel.md)), $G\actson X$ a projective variety, a lift of the action to an [[ample]] line bundle $\mcl \to X$ so that $G$ "acts on functions on $X$".
  - Define [[GIT quotient]] as $X\gitquot G\da \Proj \bigoplus _{i\geq 0} H^*(X; \mcl\tensorpowerk{i} )^G$, where by Hilbert if $G$ is reductive then the invariants are finitely generated.

- Making GIT work more generally in non-reductive settings: adding a $\GG_m$ grading seems to fix most issues!

- Definition: $H = U\semidirect R\in \Alg\Grp$ linear with $U$ unipotent and $R$ reductive is **internally graded** if there is a 1-parameter subgroup $\lambda: k\units \to Z(R)$ such that the adjoint action of $\lambda(k\units)\actson \Lie U$ (the Lie algebra) has strictly positive weights.

- Of interest: the [hyperbolicity conjecture](hyperbolicity%20conjecture).
  Call a projective variety over $\CC$ **Brody hyperbolic** if any entire holomorphic map $\CC\to X$ is constant. 
  - Kobayashi #open/conjectures (1970): any generic hypersurface $X \subseteq \PP^{n+1}$ of degree $d_n \gg 1$ is Brody hyperbolic.
  - Griffiths-Lang #open/conjectures (1979): any projective variety $X$ of general type is weakly Brody hyperbolic.
  - **Theorem**, Riedl-Young 2018: if for all $n$ there exists a $d_n$ such that GL holds for generic hypersurfaces of degree $d\geq d_n$, then the Kobayashi conjecture is true for them.

- $\hat U$ theorem can be used in situations addressed by classical GIT, e.g. curves, vector bundles or sheaves, [[Higgs bundles]], [[quiver representations]], etc. 
- There is a notion of [[semistability]] in classical situations, and this allows defining moduli for unstable things.  
	- Really gives a moduli space parameterizing "stable" objects of a fixed instability type.
	- Gives a stratification by instability types.



## 16:23


- [[Prismatic cohomology]]: a \(p\dash \)adic analog of [[crystalline cohomology]]
- Carries a Frobenius action.
- $H^i_{\prism}(\mathfrak{X}\slice{ \mathfrak{S} })$ is finitely generated over $\mathfrak{S} = W\formalpowerseries{u}$, some Witt ring?
- $\phi_{\prism}$ is a semilinear operator.
- Any torsion must be $p\dash$power torsion, i.e. $H^i_{\prism}(\mfX \slice{ \mfS })_{\tors} = H^i_{\prism}(\mfX\slice \mfS)[p^{\infty}]$.
- The pathological bits in all integral \(p\dash \)adic Hodge theories come from $H^i_{\prism}(\mfX\slice{\mfS})[u^{\infty}]$.

- To study finite flat $p\dash$power group schemes, study their Dieudonne modules

## 19:43

Idk I just like this:

![](figures/2021-11-03_19-44-08.png)