---
date: 2021-04-28 17:16:50
title: Interesting Topological Spaces in Algebraic Geometry
---

Tags: #projects/my-talks #projects/my-talks   #AG 

# Intro/Motivation

## Space, but Which One?

- You run into a "space" in the wild. Which one is it? 

- How many possible spaces *could* it be? 

- How much information is needed to specify our space uniquely?



## Possible Applications: Physics






## Possible Applications: Data



Possible to fit data to a high-dimensional manifold, makes clustering/grouping easier (here, slice with a hyperplane).



Can extract information about an entire family of objects and how they vary.
Also useful for outlier detection!


## Where to Start

- Where to start the hunt: what structure does it have?
  What can you "measure", what does it look like locally?
  How might it vary in ways you can't measure?

- Important question before attempting to classify:

  1. **What does "space" mean**?
    Need to pick a category to work in.

  2. **What does "which space" mean**?
    Need an equivalence relation.

# Question 2: Which Space?

## The Greeks: Conics

- Early classification efforts: **conic sections**.
  
  - Apollonius, 190 BC, Ancient Greeks

- Key idea: realize as intersection loci in bigger space (projectivize $\RR^2$)

- Note the 6 coefficient parameters

- Each conic is a variety, and we can obtain *every* conic by "modulating* the 6 parameters.





## The Greeks: Conics

- We can imagine a *moduli space of conics* that parameterizes these:

- All of $\RR^6$ is too much information: scaling by a nonzero $\lambda \in \RR$ yields the same conic, so we can reduce the space
  \[  
  \thevector{A, B, C, D, E, F}\in \RR^6 \mapsto  \thevector{A: B: C: D: E: F} \in \RP^5
  .\]

- Important point: $\RP^5$ is a projective variety and a smooth manifold! Tools available:
  - Dimension (what does a generic point look like?)
  - Tangent and cotangent spaces, differential forms
  - Measures, metrics, volumes, integrals
  - Intersection theory (Bezout's Theorem!), subvarieties, curves
  - Linear algebra and Combinatorics (enumerative questions)





\newpage 

## Quadrics


\[  
\begin{array}{l}
A x^{2}+B y^{2}+C z^{2}+2 F y z+2 G z x+2 H x y+2 P x+2 Q y+2 R z +D=0 \\ \\
\text{Setting} \quad 
E\definedas \left[\begin{array}{llll}
A & H & Q & P \\
H & B & F & Q \\
G & F & C & R \\
P & Q & R & D
\end{array}\right] \qquad 
e\definedas \left[\begin{array}{lll}
A & H & G \\
H & B & F \\
G & F & C
\end{array}\right] \qquad 
\Delta \definedas \operatorname{det}(E)
\end{array}
\]

(discriminants), the equation becomes $\vector x^t E \vector x = 0$ and we have a classification:

![Classification of quadrics](attachments/17Quadrics.png)

> What is the moduli space? It sits inside $\RR^{16}$, possibly $\RP^{15}$ but not in the literature.


## Automorphisms 

- Problem: infinitely many points in these moduli spaces correspond to the same "class" of conic

- How to address: Klein's Erlangen program, understand the geometry of a space by understanding its structure-preserving automorphisms.

  - For affine space $\AA^n_\KK$: $\mathrm{Aff}(\AA^n_\KK) = \AA^n_\KK \semidirect_\psi \GL(\AA^n_\KK)$, i.e. "twist" a translation with an non-singular linear transformation.
  - For Euclidean space, want isomoetries.
    $\RR^n$: can "reduce structure groups" to get $\RR^n \semidirect_\psi O(n, \RR)$, i.e. a rotation and a translation.
    - Can restrict to orientation preserving: $\RR^n \semidirect_\psi SO(n, \RR)$.
  - For topological spaces: a Lie group.

- Can then "mod out" by the appropriate morphisms to (hopefully) get finitely many equivalence classes




# What Does "Space" Mean?

## Some Setup

- **Algebraic Variety**: 
  Irreducible ,zero locus of some family $f\in \kk[x_1, \cdots, x_n]$ in $\AA^n/\kk$.
  - Equivalently, a locally ringed space $(X, \OO_X)$ where $\OO_X$ is a sheaf of finite rational maps to $\kk$.
- **Projective Variety**: 
  Irreducible zero locus of some family $f_n \subset \kk[x_0, \cdots, x_n]$ in $\PP^n/\kk$
  - Admits an embedding into $\PP^n/\kk$ as a closed subvariety.
- **Dimension** of a variety: the $n$ appearing above.
- **Topological Manifold**: Hausdorff, 2nd Countable, topological space, locally homeomorphic to $\RR^n$. 
  - Equivalently, a locally ringed space where $\OO_X$ is a sheaf of continuous maps to $\RR^n$.
- **Smooth Manifold**: Topological manifold with a smooth structure (maximal smooth atlas) with $C^\infty$ transition functions.
  - Equivalently, a locally ringed space where $\OO_X$ is a sheaf of smooth maps to $\RR^n$.
- **Algebraic Manifold**: A manifold that is also a variety, i.e. cut out by polynomial equations. 
  Example: $S^n$.

## Impossible Goal

- Notation: most dimensions will be over $\RR$, manifolds will be compact and without boundary, varieties are (probably) smooth, separated, of finite type.

- **Impossible Goal**: pick a category, understand *all* of the objects (identifying a moduli "space") and *all* of the maps.
  - Understand all topological spaces up to ???
    - Homeomorphism?
    - Diffeomorphism?
    - Homotopy-Equivalence?
    - Cobordism?
  - Understand all algebraic and/or projective varieties up to 
    - Biregular maps? 
    - Birational maps?
    - Locally ringed morphisms?

# Classification in Topology

- Two main categories with a forgetful functor: $\mathbf{Diff} \to \mathbf{Top}$.

  - What's in the "image" of this functor? 
    - Manifolds that admit a differentiable structure.

  - What is the "fiber" above a given topological manifold? 
    - Distinct differentiable structures.

- Classifying manifolds: considered open in a few directions, current work in classifying morphisms (mapping class groups, Torelli groups), knot theory, embeddings/immersions/submersions/isometries

- General slogan: classified by geometric data in low dimensions ($\leq 4$), algebraic data in high dimensions

## Topological Category

Classifying manifolds up to homeomorphism: stratify "moduli space" of topological manifolds by dimension.

- Dimensions 0,1,2,3:
   - Smooth = Top. See smooth classification.
- Dimension 4: 
  - *Topologically* classified by surgery, but barely, and not smoothly.
- Dimension $n\geq 5$: 
  - Uniformly "classified" by surgery, s-cobordism, with a caveat:
  - $\pi_1$ can be any finitely presented group -- word problem
  - Instead, breaks homotopy type of a fixed manifold up into homeomorphism classes









# Classification in Algebraic Geometry

## [../Enriques-Kodaira Classification](../Enriques-Kodaira%20Classification.md)



## Interesting Space: Elliptic Curves

- Equivalently, Riemann surfaces with one marked point.
- Equivalently, $\CC/\Lambda$ a lattice, where homothetic lattices (multiplication by  $\lambda \in \CC\smz$) are equivalent.
- Generalize to $\CC^n/\Lambda$ to obtain *abelian varieties*.






## Interesting Space: Moduli of Elliptic Curves
- $\mcm_g$: the moduli space of compact Riemann surfaces (curves) of genus $g$, i.e. elliptic curves.

- Parameterized by a moduli space:
  - For $X = \CC/\Lambda$ choose a positively oriented basis 
   $\Lambda = z\ZZ \oplus w\ZZ$.
    - Note: push into meridians on a torus, generators of $H_1(X)$, and require that their intersection is $+1$.
  - Replace $\thevector{z, w}$ with $\thevector{1, \tau}$ where $\tau = {w\over z}$; the orientation condition forces $\Im(\tau) > 0$ so this yields a point $\tau \in \HH$.
  - Account for automorphisms: roughly $\SL(2, \ZZ)$.



\newpage

## Dimension 2: Algebraic Surfaces



- Definition: **Kodaira Dimension** 
  - Given a projective variety $X$ of dimension complex dimension 2..
  - Use the canonical bundle to *try* to get a rational map $f: \Sigma \to \CP^\infty$
  So define $\kappa(\Sigma) = \dim_\CC(f(\Sigma))$
    - *(really, take a maximum dimension over a linear system)*
  - If this doesn't work, set dimension to $-\infty$.

- Fact:
\[
\kappa(X) \in \theset{-\infty, 0, 1, \cdots, \dim_\CC(X)}
.\]

- Alternative definition: 

  - $X$ has some canonical sheaf $\omega_X$, you can take some sheaf cohomology and get a sequence of integers (*plurigenera*)
\[  
P_{\mathbf{n}} (X) &\definedas h^0(X, \omega_X^{\tensor \mathbf{n}}) \quad n\in \ZZ^{\geq 0} \\ \\
\implies \kappa(X) &\definedas \limsup_{\mathbf n \to \infty} {\log P_{\mathbf n}(X) \over \log(\mathbf{n})  }
.\]



## Dimension 2: Algebraic Surfaces

Every such surface has a minimal model of one of 10 types:

$\kappa = -\infty$ (2 main types)

1. Rational: $\cong \CP^2$
2. Ruled: $\cong X$ for $\CP^1 \to X \to C$ a bundle over a curve.
  Called "ruled" because every point is on some $\CP^1$.
  3. Type VII (non-algebraic)



$\kappa = 0$ (Elliptic-ish, 4 types)

4. Enriques (all (quasi)-elliptic fibrations)
5. Hyperelliptic
  - Taking Albanese embedding (generalizes Jacobian for curves) yields an elliptic fibration 
    - *(i.e. a surface bundle, potentially with singular fibers)*
6. $K3$ (Kummer-Kahler-Kodaira) surfaces
7. Toric and Abelian Surfaces:
  - 2 dimensional abelian varieties (projective algebraic variety + algebraic group structure).
  - Compare to 1 dimensional case: all 1d complex torii are algebraic varieties, 
    - Riemann discovered that most 2d torii are *not*.
8. Kodaira Surfaces 

$\kappa = 1$: Other elliptic surfaces

9. Properly quasi-elliptic.
  Elliptic fibration, but almost all fibers have a node.

$\kappa = 2$ (Max possible, "everything else")
10. General type


# Interesting Space: Toric Varieties

- Flavor: spaces modeled on convex polyhedra
- Examples: bundles over $\CP^n$.
- Why study:
 - Model spaces by rigid geometry, generalize things like Bezier curves
  - Some are determined by rigid combinatorial data ("fan", or polytopes)
  - Combinatorial data for constructions in mirror symmetry, e.g. Calabi-Yaus (1/2 of one billion threefolds!)
- Definition:
  - Define a *complex torus* as $(C\units)^n \subseteq \CC^n$
  - Can be written as the zero set of some $f\in \CC[x_0, \cdots, x_n]$ in $\CC^{n+1}$.
    
    > Generalizes to algebraic groups over a field: $(\GG_m)^n$ (analogy: maximal torus/Cartan subalgebra in Lie theory)
  
  - **Toric variety**: $X$ contains a dense Zariski-open torus $\TT$, where the action of $\TT$ on itself as a group extends to $X$.



## Interesting Space: Kahler Manifolds/Varieties


- As complex manifolds:
  - A symplectic manifold $(X, \omega)$ with an integrable almost-complex structure $J$ compatible with $\omega$.
  - Yields an inner product on tangent vectors: $g(u, v) \definedas \omega(u, Jv)$ (i.e. a metric)

- Includes smooth projective varieties, but not all complex manifolds (exception: Stein manifolds)
- Specialize to Calabi-Yaus: compact, Ricci-flat, trivial canonical
  - Calabi's Conjecture and Yau's field medal: existence of Ricci-flat Kahlers (Calabi-Yaus) 

> Trivial canonical class = vanishing chern class = exists a nowhere vanishing top form = top wedge of $T\dual X$ is the trivial line bundle

\newpage 

## Calabi-Yaus