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# 2021-11-03

## 15:09

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

• GIT: $$G$$ a reductive group (trivial unipotent radical), $$G\curvearrowright X$$ a projective variety, a lift of the action to an very ample divisor line bundle $${\mathcal{L}}\to X$$ so that $$G$$ “acts on functions on $$X$$”.

• Define GIT quotient as $$X{ \mathbin{/\mkern-6mu/}}G\coloneqq\mathop{\mathrm{Proj}}\bigoplus _{i\geq 0} H^*(X; {\mathcal{L}}{ {}^{ \scriptscriptstyle\otimes_{k}^{i} } } )^G$$, where by Hilbert if $$G$$ is reductive then the invariants are finitely generated.
• Making GIT work more generally in non-reductive settings: adding a $${\mathbb{G}}_m$$ grading seems to fix most issues!

• Definition: $$H = U\rtimes R\in {\mathsf{Alg}}{\mathsf{Grp}}$$ linear with $$U$$ unipotent and $$R$$ reductive is internally graded if there is a 1-parameter subgroup $$\lambda: k^{\times}\to Z(R)$$ such that the adjoint action of $$\lambda(k^{\times})\curvearrowright\mathsf{Lie}U$$ (the Lie algebra) has strictly positive weights.

• Of interest: the hyperbolicity conjecture. Call a projective variety over $${\mathbb{C}}$$ Brody hyperbolic if any entire holomorphic map $${\mathbb{C}}\to X$$ is constant.

• Kobayashi #conjecture (1970): any generic hypersurface $$X \subseteq {\mathbb{P}}^{n+1}$$ of degree $$d_n \gg 1$$ is Brody hyperbolic.
• Griffiths-Lang #conjecture (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.
• $$\widehat{U}$$ 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 $$p{\hbox{-}}$$adic analog of crystalline cohomology

• Carries a Frobenius action.

• $$H^i_{\prism}(\mathfrak{X}_{/ { \mathfrak{S} }} )$$ is finitely generated over $\mathfrak{S} = W {\left[\left[ u \right]\right] }$, some Witt ring?

• $$\phi_{\prism}$$ is a semilinear operator.

• Any torsion must be $$p{\hbox{-}}$$power torsion, i.e. $$H^i_{\prism}({\mathfrak{X}}_{/ { {\mathfrak{S}}}} )_{{\operatorname{tors}}} = H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[p^{\infty}]$$.

• The pathological bits in all integral $$p{\hbox{-}}$$adic Hodge theories come from $$H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[u^{\infty}]$$.

• To study finite flat $$p{\hbox{-}}$$power group schemes, study their Dieudonne modules

## 19:43

Idk I just like this: