---
date: 2021-11-09
tags: [ web/quick-notes ]
---

# 2021-11-09

Tags:
#web/quick-notes

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## 00:11


![](figures/2021-11-09_00-11-41.png)

## 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 $G\dash$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)).