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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:

  • What does “space” mean? Need to pick a category to work in.

  • 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 \({\mathbf{R}}^2\))

• 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 \({\mathbf{R}}^6\) is too much information: scaling by a nonzero \(\lambda \in {\mathbf{R}}\) yields the same conic, so we can reduce the space \begin{align*} {\left[ {A, B, C, D, E, F} \right]}\in {\mathbf{R}}^6 \mapsto {\left[ {A: B: C: D: E: F} \right]} \in {\mathbf{RP}}^5 .\end{align*}

• Important point: \({\mathbf{RP}}^5\) 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)

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Quadrics

\begin{align*} \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\coloneqq\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\coloneqq\left[\begin{array}{lll} A & H & G \\ H & B & F \\ G & F & C \end{array}\right] \qquad \Delta \coloneqq\operatorname{det}(E) \end{array} \end{align*}

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

What is the moduli space? It sits inside \({\mathbf{R}}^{16}\), possibly \({\mathbf{RP}}^{15}\) 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 \({\mathbf{A}}^n_{\mathbb{K}}\): \(\mathrm{Aff}({\mathbf{A}}^n_{\mathbb{K}}) = {\mathbf{A}}^n_{\mathbb{K}}\rtimes_\psi \operatorname{GL}({\mathbf{A}}^n_{\mathbb{K}})\), i.e. “twist” a translation with an non-singular linear transformation.
  • For Euclidean space, want isomoetries. \({\mathbf{R}}^n\): can “reduce structure groups” to get \({\mathbf{R}}^n \rtimes_\psi O(n, {\mathbf{R}})\), i.e. a rotation and a translation.
    • Can restrict to orientation preserving: \({\mathbf{R}}^n \rtimes_\psi SO(n, {\mathbf{R}})\).
  • 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 \(f\in {\mathbf{k}}[x_1, \cdots, x_n]\) in \({\mathbf{A}}^n/{\mathbf{k}}\).
  • Equivalently, a locally ringed space \((X, {\mathcal{O}}_X)\) where \({\mathcal{O}}_X\) is a sheaf of finite rational maps to \({\mathbf{k}}\).
• Projective Variety: Irreducible zero locus of some family \(f_n \subset {\mathbf{k}}[x_0, \cdots, x_n]\) in \({\mathbf{P}}^n/{\mathbf{k}}\)
  • Admits an embedding into \({\mathbf{P}}^n/{\mathbf{k}}\) as a closed subvariety.
• Dimension of a variety: the \(n\) appearing above.
• Topological Manifold: Hausdorff, 2nd Countable, topological space, locally homeomorphic to \({\mathbf{R}}^n\).
  • Equivalently, a locally ringed space where \({\mathcal{O}}_X\) is a sheaf of continuous maps to \({\mathbf{R}}^n\).
• Smooth Manifold: Topological manifold with a smooth structure (maximal smooth atlas) with \(C^\infty\) transition functions.
  • Equivalently, a locally ringed space where \({\mathcal{O}}_X\) is a sheaf of smooth maps to \({\mathbf{R}}^n\).
• 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 \({\mathbf{R}}\), 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

Interesting Space: Elliptic Curves

• Equivalently, Riemann surfaces with one marked point.
• Equivalently, \({\mathbf{C}}/\Lambda\) a lattice, where homothetic lattices (multiplication by \(\lambda \in {\mathbf{C}}\setminus\left\{{0}\right\}\)) are equivalent.
• Generalize to \({\mathbf{C}}^n/\Lambda\) to obtain abelian varieties.

Interesting Space: Moduli of Elliptic Curves

• \({\mathcal{M}}_g\): the moduli space of compact Riemann surfaces (curves) of genus \(g\), i.e. elliptic curves.

• Parameterized by a moduli space:

  • For \(X = {\mathbf{C}}/\Lambda\) choose a positively oriented basis \(\Lambda = z{\mathbf{Z}}\oplus w{\mathbf{Z}}\).
    • Note: push into meridians on a torus, generators of \(H_1(X)\), and require that their intersection is \(+1\).
  • Replace \({\left[ {z, w} \right]}\) with \({\left[ {1, \tau} \right]}\) where \(\tau = {w\over z}\); the orientation condition forces \(\Im(\tau) > 0\) so this yields a point \(\tau \in {\mathbb{H}}\).
  • Account for automorphisms: roughly \({\operatorname{SL}}(2, {\mathbf{Z}})\).

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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 \(f: \Sigma \to {\mathbf{CP}}^\infty\) So define \(\kappa(\Sigma) = \dim_{\mathbf{C}}(f(\Sigma))\)
    • (really, take a maximum dimension over a linear system)
  • If this doesn’t work, set dimension to \(-\infty\).

• Fact: \begin{align*} \kappa(X) \in \left\{{-\infty, 0, 1, \cdots, \dim_{\mathbf{C}}(X)}\right\} .\end{align*}

• Alternative definition:

  • \(X\) has some canonical sheaf \(\omega_X\), you can take some sheaf cohomology and get a sequence of integers (plurigenera) \begin{align*} P_{\mathbf{n}} (X) &\coloneqq h^0(X, \omega_X^{\otimes\mathbf{n}}) \quad n\in {\mathbf{Z}}^{\geq 0} \\ \\ \implies \kappa(X) &\coloneqq\limsup_{\mathbf n \to \infty} {\log P_{\mathbf n}(X) \over \log(\mathbf{n}) } .\end{align*}

Dimension 2: Algebraic Surfaces

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

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

• Rational: \(\cong {\mathbf{CP}}^2\)
• Ruled: \(\cong X\) for \({\mathbf{CP}}^1 \to X \to C\) a bundle over a curve. Called “ruled” because every point is on some \({\mathbf{CP}}^1\).
• Type VII (non-algebraic)

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

• Enriques (all (quasi)-elliptic fibrations)
• Hyperelliptic

• Taking Albanese embedding (generalizes Jacobian for curves) yields an elliptic fibration
  • (i.e. a surface bundle, potentially with singular fibers)

• \(K3\) (Kummer-Kahler-Kodaira) surfaces
• 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.

• Kodaira Surfaces

\(\kappa = 1\): Other elliptic surfaces

• 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 \({\mathbf{CP}}^n\).
• 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 \((C^{\times})^n \subseteq {\mathbf{C}}^n\)

  • Can be written as the zero set of some \(f\in {\mathbf{C}}[x_0, \cdots, x_n]\) in \({\mathbf{C}}^{n+1}\).

    Generalizes to algebraic groups over a field: \(({\mathbf{G}}_m)^n\) (analogy: maximal torus/Cartan subalgebra in Lie theory)

  • Toric variety: \(X\) contains a dense Zariski-open torus \({\mathbb{T}}\), where the action of \({\mathbb{T}}\) 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: \(g(u, v) \coloneqq\omega(u, Jv)\) (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 \(T {}^{ \vee }X\) is the trivial line bundle

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Calabi-Yaus