# 2021-11-09

Tags: #web/quick-notes

Refs: ?

## 15:51

• 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)).

#web/quick-notes