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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 \({\mathbf{G}}_m\) 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 \({\mathbf{C}}\) Brody hyperbolic if any entire holomorphic map \({\mathbf{C}}\to X\) is constant.

    • Kobayashi #open/conjectures (1970): any generic hypersurface \(X \subseteq {\mathbf{P}}^{n+1}\) 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.
  • \(\widehat{U}\) 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.


  • 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


Idk I just like this:

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