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15:09
UGA AG Seminar: Eloise Hamilton?
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GIT: \(G\) a reductive group (trivial unipotent radical), \(G\curvearrowright X\) a projective variety, a lift of the action to an ample 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.
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Making GIT work more generally in non-reductive settings: adding a \({\mathbb{G}}_m\) grading seems to fix most issues!
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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 \(\lambda(k^{\times})\curvearrowright\operatorname{Lie}U\) (the Lie algebra) has strictly positive weights.
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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.
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\(\widehat{U}\) theorem can be used in situations addressed by classical GIT, e.g. curves, vector bundles or sheaves, Higgs bundles, quiver reps, 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
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Prismatic cohomology: a \(p{\hbox{-}}\)adic analog of crystalline cohomology
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Carries a Frobenius action.
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\(H^i_{\prism}(\mathfrak{X}_{/ { \mathfrak{S} }} )\) is finitely generated over $\mathfrak{S} = W { \left[\left[ {u} \right] \right] } $, some Witt ring?
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\(\phi_{\prism}\) is a semilinear operator.
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Any torsion must be \(p{\hbox{-}}\)power torsion, i.e. \(H^i_{\prism}({\mathfrak{X}}_{/ { {\mathfrak{S}}}} )_{{\operatorname{tors}}} = H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[p^{\infty}]\).
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The pathological bits in all integral \(p{\hbox{-}}\)adic Hodge theories come from \(H^i_{\prism}({\mathfrak{X}}_{/ {{\mathfrak{S}}}} )[u^{\infty}]\).
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To study finite flat \(p{\hbox{-}}\)power group schemes, study their Dieudonne modules
19:43
Idk I just like this:
