--- title: Interesting Topological Spaces in Algebraic Geometry author: D. Zack Garza slideNumber: true width: 1200 height: 900 transition: fade --- Tags: #projects/my-talks #AG #personal/mood-board # 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? ![](ASpace.png) --- ## One Motivation: Physics --- ## Another Application: Data - Possible to fit data to a high-dimensional manifold - Makes clustering/grouping easier - *(Here, slice with a hyperplane)* --- - Extract info about an entire family of objects and how they vary. - Also useful for outlier detection! --- ## Where to Start - What structure does your space have? - What can you "measure" 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" mean**? - How to distinguish? Need an equivalence relation! --- # Q1: Types of Spaces - Plan: compare classification theorem in topology and algebraic geometry - Hopefully see some fun spaces along the way! - But first: address classification and the notion of "sameness" --- ## The Greeks: Conics - Early classification efforts: **conic sections**. - Apollonius, 190 BC, Ancient Greeks - Key idea: realize as intersection loci in bigger space - (Projectivize $\mathbb{R}^2$.) --- - A conic is specified by 6 parameters: \[ A x^{2}+B x y+C y^{2}+D x+E y+F=0 \\ \\ Q :=\left(\begin{array}{ccc} A & B / 2 & D / 2 \\ B / 2 & C & E / 2 \\ D / 2 & E / 2 & F \end{array}\right), \quad \mathbf{x} = [x, y, 1] \\ \implies \mathbf{x}^t Q \mathbf{x} = 0 \] | $\det(Q)$ | Conic | | --- | --- | | $<0$ | Hyperbola | | $=0$ | Parabola | | $>0$ | Ellipse | --- - Each conic is a variety - Can obtain *every* conic by "modulating* 6 parameters. - $\mathbb{R}^6$: too much information: scaling by a nonzero $\lambda \in \mathbb{R}$ yields the same conic, so reduce the space \[ [A, B, C, D, E, F]\in \mathbb{R}^6 \mapsto [A: B: C: D: E: F] \in \mathbb{RP}^5 .\] - Important point: $\mathbb{RP}^5$ 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) --- - We can imagine a *moduli space of conics* that parameterizes these: --- ## Quadrics Some Calc III review: --- General form: \[\begin{array}{l}\scriptsize A x^{2}+B y^{2}+C z^{2}+2 F y z+2 G z x+2 H x y+2 P x+2 Q y+2 R z +D=0 \\ \\ \text{Setting}\,\, E:= \left[\begin{array}{llll} A & H & Q & P \\ H & B & F & Q \\ G & F & C & R \\ P & Q & R & D \end{array}\right]\,\, e:= \left[\begin{array}{lll} A & H & G \\ H & B & F \\ G & F & C \end{array}\right] \\ \Delta := \operatorname{det}(E) \end{array} \] (discriminants), the equation becomes $\mathbf x^t E \mathbf x = 0$ and we have a classification: --- ![](attachments/17Quadrics.png){ height=600px } > What is the moduli space? It sits inside $\mathbb{R}^{10}$, possibly $\mathbb{RP}^{9}$ 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 topological spaces: a Lie group acting on the space. - Can then "mod out" by the appropriate morphisms to (hopefully) get finitely many equivalence classes --- # What Does "Space" Mean? --- ## Some Setup - **[Algebraic Variety](Algebraic%20Variety.md)**: Irreducible ,zero locus of some family $f\in \mathbb{k}[x_1, \cdots, x_n]$ in $\mathbb{A}^n/\mathbb{k}$. - Equivalently, a locally ringed space $(X, \mathcal{O}_X)$ where $\mathcal{O}_X$ is a sheaf of finite rational maps to $\mathbb{k}$. - **[Projective Variety](Projective%20Variety)**: Irreducible zero locus of some family $f_n \subset \mathbb{k}[x_0, \cdots, x_n]$ in $\mathbb{P}^n/\mathbb{k}$ - Admits an embedding into $\mathbb{P}^\infty/\mathbb{k}$ as a closed subvariety. - **Dimension** of a variety: the $n$ appearing above. --- - **[Topological Manifold](Topological%20Manifold)**: Hausdorff, 2nd Countable, topological space, locally homeomorphic to $\mathbb{R}^n$. - Equivalently, a locally ringed space where $\mathcal{O}_X$ is a sheaf of continuous maps to $\mathbb{R}^n$. - **[Smooth Manifold](Smooth%20Manifold)**: 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 $\mathbb{R}^n$. - **Algebraic Manifold**: A manifold that is also a variety, i.e. cut out by polynomial equations. Example: $S^n$. --- - 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 --- ## 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 --- ## Smooth Category Generally expect things to split into more classes. - Dimension 0: The point (terminal object) - Dimension 1: $\mathbb{S}^1, \mathbb{R}^1$ --- - Dimension 2: $\left\langle\mathbb{S}^2, \mathbb{T}^2, \mathbb{RP}^2 \mid \mathbb{S}^2 = 0,\,\,3\mathbb{RP}^2 = \mathbb{RP}^2 + \mathbb{T}^2 \right\rangle$. - Classified by $\pi_1$ (orientability and "genus"). Riemann, Poincaré, Klein. --- - Dimension 2: closed + orientable $\implies$ complex - **Uniformization**: Holomorphically equivalent to a quotient of one of three spaces/geometries: - $\mathbb{CP}^1$, positive curvature (spherical) - $\mathbb{C}$, zero curvature (flat, Euclidean) - $\mathbb{H}$ (equiv. $\mathbb{D}^\circ$), negative curvature (hyperbolic) - Stratified by genus - Genus 0: Only $\mathbb{CP}^1$ - Genus 1: All of the form $\mathbb{C}/\Lambda$, with a distinguished point $[0]$, i.e. an elliptic curve. - Has a topological group structure! - Genus $\geq 2$: Complicated? > Doesn't capture holomorphy type completely. --- ## 3-manifolds: Thurston's [Geometrization](Geometrization.md) - Geometric structure: a diffeo $M\cong \tilde M/\Gamma$ where $\Gamma$ is a discrete Lie group acting freely/transitively on $X$ (as in Erlangen program) - Decompose into pieces with one of 8 geometries: - Spherical $\sim S^3$ - Euclidean $\sim \mathbb{R}^3$ - Hyperbolic $\sim \mathbb{H}^3$ - $S^2\times \mathbb{R}$ - $\mathbb{H}^2\times \mathbb{R}$ - $\widetilde{\mathrm{SL}(2, \mathbb{R})}$ - "Nil" - "Sol" --- - Proved by Perelman 2003, [Ricci flow](Ricci%20flow.md) with surgery. --- --- - 4-manifolds: classified in the topological category by [surgery](surgery.md), but not in the smooth category - Hard! Will examine special cases of [Calabi-Yau](Calabi-Yau.md) - Open part of [Poincaré Conjecture](Poincaré%20Conjecture). - Dimension $\geq 5$: surgery theory, strong relation between diffeomorphic and [s-cobordism](s-cobordism.md). --- ## Toward Algebraic Manifolds: Berger's Classification - Every smooth manifold admits a [Riemannian metric](Riemannian%20metric), so consider Riemannian manifolds - Here $H\leq \mathrm{SO}(n)$ is the [holonomy group](holonomy%20group.md) : --- - [Berger's classification](Berger's%20classification) for smooth Riemannian manifolds, one of 7 possibilities. $$ \tiny \begin{array}{|c|c|c|c|c|} \hline n=\operatorname{dim} M & H & \text { Parallel tensors } & \text { Name } & \text { Curvature } \\ \hline n & \mathrm{SO}(n) & g, \mu & \text {orientable} & \\ \hline 2 m(m \geq 2) & \mathrm{U}(m) & g, \omega & \textbf{Kähler} & \\ \hline 2 m(m \geq 2) & \mathrm{SU}(m) & g, \omega, \Omega & \textbf{Calabi-Yau} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & \mathrm{Sp}(m) & g, \omega_{1}, \omega_{2}, \omega_{3}, J_{1}, J_{2}, J_{3} & \textbf{hyper-Kähler} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & (\mathrm{Sp}(m) \times \mathrm{Sp}(1)) / \mathbb{Z}_{2} & g, \Upsilon & \text {quaternionic-Kähler} & \text {Einstein} \\ \hline 7 & \mathrm{G}_{2} & g, \varphi, \psi & \mathrm{G}_{2} & \text {Ricci-flat} \\ \hline 8 & \operatorname{Spin}(7) & g, \Phi & \operatorname{Spin}(7) & \text {Ricci-flat} \\ \hline \end{array} $$ > Types in bold: amenable to Algebraic Geometry. > $G2$ shows up in Physics! - [Ricci-flat](Ricci-flat.md), i.e. Ricci curvature tensor vanishes - *(Measures deviation of volumes of "geodesic balls" from Euclidean balls of the same radius)* --- # Classification in Algebraic Geometry --- ## [Enriques-Kodaira Classification](Enriques-Kodaira%20Classification.md) > Work over $\mathbb{C}$ for simplicity, take all dimensions over $\mathbb{C}$. - [Minimal model program](Minimal%20model%20program.md) : classifying complex projective varieties. - Stratify the "[moduli space](moduli%20space.md)" of varieties by $\mathbb{k}-$dimension. - Dimension 1: - Smooth Algebraic curves = compact Riemann surfaces, classified by genus - Roughly known by Riemann: moduli space of smooth projective curves $\mathcal{M}_g$ is a connected open subset of a projective variety of dimension $3g-3$. > We'll come back to these! --- - Dimension 2: - Smooth Algebraic Surfaces: Hard. See Enriques classification. - Setting of classical theorem: always 27 lines on a cubic surface! - Example Clebsch surface, satisfies the system \[ \begin{array}{l} x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0 \\ \\ x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0 \end{array} \] --- --- ## Interesting Space: [Projects/2022 Advanced Qual Projects/Elliptic Curves/Elliptic Curves](Projects/2022%20Advanced%20Qual%20Projects/Elliptic%20Curves/Elliptic%20Curves.md) - Equivalently, Riemann surfaces with one marked point. - Equivalently, $\mathbb{C}/\Lambda$ a lattice, where homothetic [lattices](lattices) (multiplication by $\lambda \in \mathbb{C}- \{0\}$) are equivalent. - Generalize to $\mathbb{C}^n/\Lambda$ to obtain [abelian varieties](abelian%20varieties). --- --- ## Interesting Space: [Moduli of Elliptic Curves](Moduli%20of%20Elliptic%20Curves) - $\mathcal{M}_g$: the moduli space of compact Riemann surfaces (curves) of genus $g$, i.e. elliptic curves. --- ## Dimension 2: Algebraic Surfaces --- --- - Definition: **[Kodaira Dimension](Kodaira%20Dimension.md)** - $X$ has some canonical sheaf $\omega_X$, you can take some [sheaf cohomology](sheaf%20cohomology.md) and get a sequence of integers ([plurigenera](plurigenera)) \[ P_{\mathbf{n}} (X) &:= h^0(X, \omega_X^{\otimes \mathbf{n}}) \quad n\in \mathbb{Z}^{\geq 0} \\ \\ \implies \kappa(X) &:= \limsup_{\mathbf n \to \infty} {\log P_{\mathbf n}(X) \over \log(\mathbf{n}) } \] --- ## Dimension 2: Algebraic Surfaces Every such surface has a minimal model of one of 10 types: $\kappa = -\infty$ (2 main types) - 1. Rational: $\cong \mathbb{CP}^2$ - 2. Ruled: $\cong X$ for $\mathbb{CP}^1 \to X \to C$ a bundle over a curve. - Called "ruled" because every point is on some $\mathbb{CP}^1$. - 3. Type VII --- --- $\kappa = 0$ (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: [Unsorted/toric](Unsorted/toric.md) - Definitions: - Define a *complex torus* as $\mathbb{T} = (\mathbb{C}^{\times})^n \subseteq \mathbb{C}^n$ - Can be written as the zero set of some $f\in \mathbb{C}[x_0, \cdots, x_n]$ in $\mathbb{C}^{n+1}$. > Generalizes to algebraic groups over a field: $(\mathbb{G}_m)^n$ (analogy: maximal torus/Cartan subalgebra in Lie theory) - [Toric variety](Toric%20variety) : $X$ contains a dense Zariski-open torus $\mathbb{T}$, where the action of $\mathbb{T}$ on itself as a group extends to $X$. --- --- - Flavor: spaces modeled on convex polyhedra - Examples: bundles over $\mathbb{CP}^n$. - 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!) --- ## 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: $g(u, v) := \omega(u, Jv)$ (i.e. a metric) --- - Examples: *all* smooth complex 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 $\implies$ exists a nowhere vanishing top form = top wedge of $T^* X$ is the trivial line bundle --- ## Calabi-Yau - Another from Berger's classification, special case of Kahler --- - Applications: Physicists want to study $G_2$ manifolds (an exceptional Lie group, automorphisms of octonions) - Part of $M$-theory uniting several superstring theories, but no smooth or complex structures. - Indirect approach: compactify an 11-dimension space, one small $S^1$ dimension $\to$ 10 dimensions - 4 spacetime and 6 "small" Calabi-Yau - Superstring theory: a bundle over spacetime with fibers equal to Calabi-Yaus. > *Roughly*, genera of fibers will correspond to families of observed particles. --- ## Calabi-Yaus - As manifolds: - Ricci-flat: vacuum solutions to (analogs of) Einstein's equations with zero cosmological constant - Setting for mirror symmetry: the symplectic geometry of a Calabi-Yau is "the same" as the complex geometry of its mirror. - Yau, Fields Medal 1982: There are Ricci flat but non-flat (nontrivial holonomy) projective complex manifolds of dimensions $\geq 2$. --- - As varieties: the canonical bundle $\Lambda^n T^* V$ is trivial - Compact classification for $\mathbb{C}-$dimension: - Dimension 1: 1 type, all elliptic curves (up to homeomorphism) - Dimension 2: 1 type, $K3$ surfaces --- - Dimension 3: (threefolds) conjectured to be a bounded number, but unknown. - At least 473,800,776!