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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 2sphere, 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 “fixedpoint data”. They represent a categorical analog of the AtiyahBott 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)).