Reading notes - Singular Points on Complex Hypersurfaces

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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 $S^3$
$f: {\mathbf{C}}^{n+1} \to {\mathbf{C}}$, consider the hypersurface $V(f) \subseteq {\mathbf{C}}^{n+1}$. Fix $\mathbf{z}_0 \in {\mathbf{C}}^{n+1}$, intersect with a sphere to obtain $K = V(f) \cap S^{n+1}_{\varepsilon}$.
$\mathbf{z}_0$ regular implies $V$ is a smooth manifold of dimension $2n$, $K$ is a smooth $2n-1$ manifold with $K { \, \xrightarrow{\sim}\, }_{{\operatorname{Diffeo}}} S^{2n-1}$ and $K\hookrightarrow S^{2n-1}_{\varepsilon}$ is unknotted.
Result of Brauner: for $f(z_1, z_2) = z_1^p + z_2^q$ and $V = V(p, q) = f^{-1}(0)$, $K = T(p, q)$ 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:
- Note $0$ is isolated since $\operatorname{grad}f = {\left[ {pz_1^{p-1}, qz_2^{q-1}} \right]} = 0 \iff {\left[ {z_1,z_2} \right]} = {\left[ {0, 0} \right]}$.
- Verify:
More generally set $V(n_1, \cdots, n_k) = \sum_{i\leq k} z_i^{n_i}$. Take $V(3, 2, 2,\cdots, 2) \subseteq {\mathbf{C}}^{n+1}$, then $K \hookrightarrow S^{2n-1}_{\varepsilon}$ 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:
- Improvement when $z_0$ is an isolated critical point:
Note that $H_*(\bigvee_i S^n) \cong \oplus_i H_*(S^n)$ by Mayer-Vietoris. Here the fibers are $2n{\hbox{-}}$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 $K$

Definitions

Algebraic variety: a subset $X \subseteq {\mathbf{A}}^n_{/ {k}} $ of the form $V(f_1,\cdots, f_n)$. Ideal of defining functions: $I(V) {~\trianglelefteq~}k[x_1, \cdots, x_{n}]$.
For $X$ algebraic, by Hilbert’s basis theorem $X$ is cut out by finitely many polynomials. Pick some, $F \coloneqq{\left[ {f_1, \cdots f_n} \right]}$, and form the Jacobian \begin{align*}\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]\end{align*}
Jacobian criterion: $X$ is singular if $J$ drops below its maximal rank $\rho$ anywhere.
The singular locus ${\operatorname{Sing}}(X) \subseteq X$ is an algebraic subvariety cut out by the collection of all $\rho\times \rho$ minors of $J$ vanishing.
Cones:
Link of a singularity:
- Transversality:
Critical point terminology:
Monodromy
Singular set:
Complex gradient:
- Use:
Alexander duality:
Brieskorn spheres: ?

Results (Section 2)

Theorem (Whitney): over $k= {\mathbf{C}}$, $X\setminus{\operatorname{Sing}}(X)$ is a smooth complex-analytic manfiold of codimension $\rho$
- This has a long proof! (2-3 pages)
Curve selection lemma:

Results (Section 3)

Application:
- Proof

Proof of curve selection lemma

Shrink $V_1$ is necessary to…
- Assume $V$ irreducible
- Assume $U \cap{\operatorname{Sing}}(X)$ is empty in a small enough neighborhood $D_\eta$ of zero
- Proof
- Proof:
- Proof: proved for real 1-dimensional varieties.
Assembling these into the full proof:

Results (Section 4)

- Part of proof:
Theorem:

Random notes

Holomorphic Morse theory:
When a ${\mathbf{Z}}\operatorname{HS}^n$ is a topological sphere: