--- date: 2022-04-04 21:10 modification date: Monday 4th April 2022 21:10:50 title: "Reading notes - Singular Points on Complex Hypersurfaces" aliases: [Reading notes - Singular Points on Complex Hypersurfaces] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links --- # Reading notes - Singular Points on Complex Hypersurfaces - Idea: exotic spheres are higher dimensional knot theory - Singular points on a complex curve can be associated to knots in $S^3$ - $f: \CC^{n+1} \to \CC$, consider the hypersurface $V(f) \subseteq \CC^{n+1}$. Fix $\vector z_0 \in \CC^{n+1}$, intersect with a sphere to obtain $K = V(f) \intersect S^{n+1}_\eps$. - $\vector z_0$ regular implies $V$ is a smooth manifold of dimension $2n$, $K$ is a smooth $2n-1$ manifold with $K \iso_{\Diffeo} S^{2n-1}$ and $K\embeds S^{2n-1}_\eps$ is unknotted. - Result of Brauner: for $f(z_1, z_2) = z_1^p + z_2^q$ and $V = V(p, q) = f\inv(0)$, $K = T(p, q)$ is a torus knot in $S^3$. To form: take $p$ braids and $q$ boxes. Every strand descends one level, except the bottom which goes under all others to the top: ![](figures/2022-04-04_21-35-53.png) - Note $0$ is isolated since $\grad f = \tv{pz_1^{p-1}, qz_2^{q-1}} = 0 \iff \tv{z_1,z_2} = \tv{0, 0}$. - Verify: ![](attachments/Pasted%20image%2020220404220456.png) - More generally set $V(n_1, \cdots, n_k) = \sum_{i\leq k} z_i^{n_i}$. Take $V(3, 2, 2,\cdots, 2) \subseteq \CC^{n+1}$, then $K \embeds S^{2n-1}_\eps$ is knotted -- these are Brieskorn spheres. - For $n$ odd, $K \cong S^{2n-1}$, while for $n$ odd (eg $n=5$) this provvably produces an exotic sphere. - Main theorems: - ![](attachments/Pasted%20image%2020220404221109.png) - Improvement when $z_0$ is an isolated critical point: ![](attachments/Pasted%20image%2020220404221229.png) ![](attachments/Pasted%20image%2020220404221409.png) - Note that $H_*(\bigvee_i S^n) \cong \oplus_i H_*(S^n)$ by Mayer-Vietoris. Here the fibers are $2n\dash$dimensional manifolds but have homotopy types of $\bigvee_{1\leq i\leq m} S^n$ which has Poincare polynomial $1 + mx^n + 0x^{2n}$, so thhe middle Betti number is in dimension $2n/2 = n$ and records that number of spheres. - The common boundary of the $F_\theta$ is $K$ ![](attachments/Pasted%20image%2020220404224128.png) ![](attachments/Pasted%20image%2020220404234107.png) # Definitions - Algebraic variety: a subset $X \subseteq \AA^n\slice k$ of the form $V(f_1,\cdots, f_n)$. Ideal of defining functions: $I(V) \normal \kxn$. - For $X$ algebraic, by Hilbert's basis theorem $X$ is cut out by finitely many polynomials. Pick some, $F \da \tv{f_1, \cdots f_n}$, and form the Jacobian $$\mathbf{J}=\left[\begin{array}{ccc} \frac{\partial \mathbf{f}}{\partial x_{1}} & \cdots & \frac{\partial \mathbf{f}}{\partial x_{n}} \end{array}\right]=\left[\begin{array}{cc} \nabla^{\mathrm{T}} f_{1} \\ \vdots \\ \nabla^{\mathrm{T}} f_{m} \end{array}\right]=\left[\begin{array}{ccc} \frac{\partial f_{1}}{\partial x_{1}} & \cdots & \frac{\partial f_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_{m}}{\partial x_{1}} & \cdots & \frac{\partial f_{m}}{\partial x_{n}} \end{array}\right]$$ - Jacobian criterion: $X$ is singular if $J$ drops below its maximal rank $\rho$ anywhere. - The singular locus $\Sing(X) \subseteq X$ is an algebraic subvariety cut out by the collection of all $\rho\times \rho$ minors of $J$ vanishing. - Cones: ![](attachments/Pasted%20image%2020220404232131.png) - Link of a singularity: ![](attachments/Pasted%20image%2020220404232727.png) ![](attachments/Pasted%20image%2020220404233503.png) - Transversality: ![](attachments/Pasted%20image%2020220404233113.png) - Critical point terminology: ![](attachments/Pasted%20image%2020220404233144.png) - Monodromy ![](attachments/Pasted%20image%2020220404234032.png) - Singular set: ![](attachments/Pasted%20image%2020220405001112.png) - Complex gradient: ![](attachments/Pasted%20image%2020220405001946.png) - Use: ![](attachments/Pasted%20image%2020220405002024.png) - Alexander duality: ![](attachments/Pasted%20image%2020220405003641.png) - Brieskorn spheres: ? # Results (Section 2) - Theorem (Whitney): over $k= \CC$, $X\sm \Sing(X)$ is a smooth complex-analytic manfiold of codimension $\rho$ - ![](attachments/Pasted%20image%2020220404231915.png) - ![](attachments/Pasted%20image%2020220404232006.png) - ![](attachments/Pasted%20image%2020220404232022.png) - ![](attachments/Pasted%20image%2020220404232314.png)![](attachments/Pasted%20image%2020220404232117.png) - This has a long proof! (2-3 pages) - Curve selection lemma: ![](attachments/Pasted%20image%2020220404233859.png) - ![](attachments/Pasted%20image%2020220404235942.png) - ![](attachments/Pasted%20image%2020220405000033.png) # Results (Section 3) - ![](attachments/Pasted%20image%2020220405000210.png) - Application: ![](attachments/Pasted%20image%2020220405000400.png) - Proof ## Proof of curve selection lemma - ![](attachments/Pasted%20image%2020220405000733.png) - Shrink $V_1$ is necessary to... - Assume $V$ irreducible - Assume $U \intersect \Sing(X)$ is empty in a small enough neighborhood $D_\eta$ of zero - ![](attachments/Pasted%20image%2020220405001128.png) - Proof - ![](attachments/Pasted%20image%2020220405001155.png) - ![](attachments/Pasted%20image%2020220405001212.png) - Proof: - ![](attachments/Pasted%20image%2020220405001343.png) - Proof: proved for real 1-dimensional varieties. - ![](attachments/Pasted%20image%2020220405001503.png) - Assembling these into the full proof: - ![](attachments/Pasted%20image%2020220405001656.png) - ![](attachments/Pasted%20image%2020220405001708.png) # Results (Section 4) - ![](attachments/Pasted%20image%2020220405002107.png) - Part of proof: ![](attachments/Pasted%20image%2020220405002154.png) - ![](attachments/Pasted%20image%2020220405002207.png) - ![](attachments/Pasted%20image%2020220405002239.png) - ![](attachments/Pasted%20image%2020220405002253.png) - ![](attachments/Pasted%20image%2020220405002311.png) - ![](attachments/Pasted%20image%2020220405002343.png) - ![](attachments/Pasted%20image%2020220405002410.png) - ![](attachments/Pasted%20image%2020220405002451.png) ![](attachments/Pasted%20image%2020220405002504.png) - Theorem: ![](attachments/Pasted%20image%2020220405002520.png) # Random notes - Holomorphic Morse theory: ![](attachments/Pasted%20image%2020220405004140.png) ![](attachments/Pasted%20image%2020220405004346.png) - When a $\ZHS^n$ is a topological sphere: ![](attachments/Pasted%20image%2020220405003953.png) ![](attachments/Pasted%20image%2020220405005524.png) ![](attachments/Pasted%20image%2020220405005604.png) ![](attachments/Pasted%20image%2020220405005717.png) ![](attachments/Pasted%20image%2020220405005750.png) ![](attachments/Pasted%20image%2020220405005813.png)