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  • Torelli: the map sending a curve to its Jacobian is an injection on points.

  • Intermediate Jacobian: introduce to prove irrationality of cubic threefolds. An abelian variety the parameterizes degree zero cycles in dimension 1, up to rational equivalence.

    • The pair \((J(X), \Theta)\) determines a cubic threefold, where \(\Theta\) is the theta divisor, which has a unique singular point.
  • Relationship between complex projective and geometry and symplectic topology: Kähler manifolds.

  • Abouzaid: interesting results about symplectic topology of Hamiltonian fibrations over the 2-sphere, and their consequences for smooth projective maps over the projective line.

  • The Grothendieck group of mixed Hodge modules, which enhances the Grothendieck group of \(G{\hbox{-}}\)modules.

  • A motivic semiorthogonal decomposition is the decomposition of the derived category of a quotient stack [X/G] into components related to the “fixed-point data”. They represent a categorical analog of the Atiyah-Bott localization formula in equivariant cohomology, and their existence is conjectured for finite G

  • Can define curvature and 2nd fundamental form for algebraic varieties?

  • Invariants like HOMFLY: invariants of quantum matrices

  • consider the stack of representations, its inertia stack and the nilpotent version of the inertia stack.

  • Hurwitz spaces H_{k,g}, parametrizing degree k, genus g covers of P^1

  • Kobayashi–Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite–Einstein metric if and only if the bundle is slope polystable

  • predicted that given two vector bundles V_1, V_2 whose first Chern classes both vanish and whose second Chern classes agree, the resulting line bundles Thom(V_1) and Thom(V_2) should agree in Pic(Ell_G(X)).

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