What is a scheme?

 Conference topic

This is from a problem session at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Tyler Lawson for explaining the material.

Definition: Affine schemes are the category $\Ring^{op}$. An object $R\in \Ring$ becomes an object $\Spec(R)$ in affine schemes, and a ring map $R\to S$ becomes a map $\Spec(S)\to \Spec(R)$, where $\Spec$ denotes the set of prime ideals.

We try to think of $Spec(R)$ as a geometrical object.

Example: Let $k$ be a field and consider the ring
\[
R = k[x_1,\dots,x_n] / (f_1(x_1,\dots,x_n),\dots,f_r(x_1,\dots,x_n)).
\]
$\Spec(R)$ is supposed to be a substitute for the set of solutions to a system of equations
\begin{align*}
f_1(x_1,\dots,x_n) & = 0,\\
\vdots \hspace{.7cm}\\
f_r(x_1,\dots,x_n) & = 0.
\end{align*}

The scheme $\Spec(R)$ has a more precise definition. It consists of a set, a topology, and a sheaf. 

1. Set: The underlying set of the scheme $\Spec(R)$ is the set of prime ideals of $R$. For example:

• if $R = \C[x]$, then the prime ideals are $(x-\alpha)$ and $(0)$;
• if $R = \C[x,y]$, then the prime ideals are $(x-\alpha,y-\beta)$, irreducible polynomials $(f(x,y))$, and $(0)$.

2. Topology: For every ideal $I\subset R$, the set $V(I) = \{P\subset R$ prime, $P\supset I\}$ is a closed set. Note that

\[
\bigcup_{n=1}^N V(I_n) = V\left(\bigcap_{n=1}^N I_n\right)
\hspace{1cm}\text{and}\hspace{1cm}
\bigcap_{\alpha\in I} V(I_\alpha) = V\left(\sum_{\alpha\in A} I_A\right).
\]
Geometrically, the closed sets are sets of points where one or more identities (like $f(x)=0$) can hold. For example, if $R=\C[x]$, then we have three different closed set types: $\Spec(C[x])$, $\emptyset$, or a finite union of $(x-\alpha_1,\dots, x-\alpha_n)$. Solutions to equations can be one of the following types below.

3. Sheaf: Let $X$ be a set with a topology. $\mathcal O_X$ is the sheaf for which:

• to each open set $U\subseteq X$ we get a ring $\mathcal O_X(U)$;
• to each containment $V\subseteq U\subseteq X$ of open sets, there exists a restriction map $\res_{UV}:\mathcal O_X(U)\to \mathcal O_X(V)$;
• the restriction maps are compatible, in the sense that $\res_{VW}\circ \res_{UV} = \res_{UW}$.

This is called the structure sheaf of $X$.

Say $R$ is our ring, $\Spec(R)$ our set of primes, and we have some open set $U\subseteq \Spec(R)$. We like to think of it in the following way:

• elements of $R$ are functions;
• elements of $\Spec(R)$ are points where we can evaluate a function $f\in P$ (or where the function vanishes);
• subsets $S\subset R$ are the sets $\{f\in R\ :\ f$ only vanishes at points outside $U\}$.

Note that $S$ is closed under multiplication. We localize $R$ at $S$ to get a set
\[
S^{-1}R = \left\{\left[\frac fs\right]\ :\ f\in R, s\in S\right\},
\]
for which $\mathcal O_X(U) = S^{-1}R$ (good enough for today's purposes). Now we have a triple $(\Spec(R),\tau,\mathcal O_X)$, for $\tau$ the Zariski topology, which we call a locally ringed space.

Definition: A scheme is a space $X$ with a topology and a sheaf of rings that is locally isomorphic to $\Spec(R)$.

Since the sheaf has the space $X$ and the topology (through the open sets) encoded in it, we may think of a scheme as a special type of sheaf. Also, isomorphism is meant in the category of locally ringed spaces.

Proposition: Morphisms of schemes $\Spec(R)\to \Spec(S)$ are the same as ring maps $S\to R$.

Example: In the Zariski topology, take $U\subseteq \Spec(k[x,y])$. Locally $U$ looks like it is covered by rings, though that may not be the case globally. Indeed:

Example: Consider projective space $\P^2$, where $[x:y:z] = [\lambda x: \lambda y:\lambda z]$. We may write
\[
\begin{array}{r c c c c c c}
\P^2 & = & U_0 & \cup & U_1 & \cup & U_2. \\
& & [1:y:z] & & [x:1:z] & & [x:y:1] \\
& & \Spec(k[y,z]) & & \Spec(k[x,z]) & & \Spec(k[x,y])
\end{array}
\]
How can we express $U_0\cap U_1$? This is left as an exercise.

Some facts about formal group laws

 Conference topic

Here we solve some problems from the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Piotr Pstragowski and Carolyn Yarnall for the solutions. First we recall some definitions.

Definition: Let $R$ be a commutative ring with unit. A formal group law $F$ over $R$ is an element $F\in R[[x,y]]$ satisfying

1. $F(x,y) = F(y,x)$ (symmetry), 
2. $F(x,0) =x$ and $F(0,y)=y$ (uniticity),
3. $F(F(x,y),z) = F(x,F(y,z))$ (associativity).

It follows from these three properties that $F(x,y)=x+y+($higher order terms$)$ for all $F$.

Proposition: For any formal group law $F(x,y)$ over $R$, $x$ has a formal inverse. That is, there exists an element $i(x)\in R[[x]]$ such that $F(x, i(x)) = 0$.

Proof: Consider $F(x,y+z)$, with $|z| = n$. Note that
\begin{align*}
F(x,y+z) & = x+y +z +\sum_{i,j\geqslant 1} a_{ij} x^i(y+z)^j \\
& = x+y +z+\sum_{i,j\geqslant 1} a_{ij} x^i \sum_{k=0}^j \binom jk y^k z^{j-k} \\
& = x+y+z+\sum_{i,j\geqslant 1} a_{ij} x^i \left(y^j + \sum_{k=0}^{j-1} \binom jk y^k z^{j-k} \right)\\
& = x+y+z+\sum_{i,j\>1} a_{ij} x^i y^j  + \underbrace{\sum_{i,j\geqslant 1} a_{ij} x^i}_{\text{deg }\geqslant \ 1}\underbrace{\sum_{k=0}^{j-1} \binom jk y^k z^{j-k}}_{\text{deg }=\ k+n(j-k)\geqslant n}\\
& = F(x,y) + z + (\text{terms of deg }\geqslant\ n+1).
\end{align*}
First choose $z_1$ to be the negative of all the degree-1 terms of $F(x,0)$, so that $F(x,z_1)$ has terms of degree 2 and higher. Now choose $z_2$ to be the negative of all the degree-2 terms of $F(x,z_1)$, so $F(x,z_1+z_2)$ has terms of degree 3 and higher. Continue in this manner ad infinitum to get a formal inverse $\sum_i z_i$ (this will be a power series) of $x$. $\square$

Recall that we call $f_a(x,y) = x+y$ the additive formal group law and $F_m(x,y) = x+y+xy$ the multiplicative formal group law. Via the universal Lazard ring of formal group laws, these turn out to be the formal group laws of ordinary singular cohomology theory (additive) and complex $K$-theory $KU$ (multiplicative). Recall also nested notation: for $F$ a formal group law, we write
\begin{align*}
[1]_F(x) & = x, \\
[2]_F(x) & = F(x,x), \\
[3]_F(x) & = F(F(x,x),x), \\
[4]_F(x) & = F(F(F(x,x),x),x),
\end{align*}
and so on.

Definition: Let $F$ be a formal group law over $R$. A morphism of formal group laws is an element $\varphi\in R[[u]]$, giving a formal group law $\varphi F\in R[[x,y]]$ by $\varphi F(x,y):= F(\varphi(x),\varphi(y))$.

An isomorphism of formal group laws is a morphism where the formal power series $\varphi$ is an isomorphism.

Proposition: The additive formal group law and the multiplicative formal group law are not isomorphic over $F_p$.

Proof: We compare $[p]_{F_m}(x)$ and $[p]_{F_a}(x)$ and show they are not the same. If there were an isomorphism $\varphi$ between $F_a$ and $F_m$, we should have that
\[
F_m(x,x) = F_a(\varphi(x),\varphi(x)) = \varphi(F_a(x,x))
\ \ \implies\ \
[p]_{F_m}(x) = \varphi([p]_{F_a}(x)),
\]
since $\varphi$ is a homomorphism. However, we first see that
\[
[1]_{F_a}(x) = x
,\hspace{1cm}
[2]_{F_a}(x) = F_a(x,x) = 2x
,\hspace{1cm}
[3]_{F_a}(x) = F_a(F_a(x,x),x) = 3x,
\]
and so continuing this pattern we get that $[p]_{F_a}(x) = px = 0$ in $F_p$. Next, for the multiplicative formal group law we find that
\[
[1]_{F_m}(x) = x,
,\hspace{1cm}
[2]_{F_m}(x) = F_m(x,x) = 2x + x^2
,\hspace{1cm}
[3]_{F_m}(x) = F_m(2x+x^2,x) = 3x + 3x^2 + x^3.
\]
Here the pattern  is not immediate, but continuing these small examples we find that $[p]_{F_m}(x) = (x+1)^p-1 = 1+x^p-1 = x^p$ in $F_p$. An isomorphism sends only 0 to 0, but in this case $\varphi$ should send $x^p\neq 0$ to $0$, a contradiction. Hence no such isomorphism exists over $F_p$. $\square$

(Co)fibrations, suspensions, and loop spaces

 Seminar topic

Recall the exponential object $Z^Y$, which, in the category of topological spaces, is the set of all continuous functions $Y\to Z$. In general, the definition involves a commuting diagram and gives an isomorphism $\Hom(X\times Y,Z)\cong \Hom(X,Z^Y)$. The subspace $F(Y,Z)$ of $Z^Y$ consists of based functions $Y\to Z$.

Definition: Let $F,E,B,X$ be topological spaces. A map $i:F\to E$ is a cofibration if for every map $f:E\to X$ and every homotopy $h:F\times I\to X$, there exists a homotopy $\tilde h:E\times I\to X$ (extending $h$) making either of the equivalent diagrams below commute.

The horizontal maps on the left are the natural inclusion maps $x\mapsto (x,0)$ and the map on the right is the natural evaluation map $\varphi \mapsto \varphi(0)$. Similarly, a map $p:E\to B$ is a fibration if for every map $g:X\to E$ and every homotopy $h:X\times I\to B$, there exists a homotopy $\tilde h:X\times I\to E$ (lifting $h$) making either of the equivalent diagrams below commute.

The horizontal maps on the right are the natural evaluation maps and the map on the right is the natural inclusion map.

Instead of this terminology, often we say the pair $(F,E)$ has the homotopy extension property and the pair $(E,B)$ has the homotopy lifting property. Now, let let $(X,x)$ be a pointed topological space.

Definition: The (reduced) suspension $\Sigma X$ of $X$ is
\[
\Sigma X := X\times I/X\times \{0\} \cup X\times \{1\} \cup \{x\}\times I.
\] 
The unreduced suspension $SX$ of $X$ is
\[
S X := X\times I/X\times \{0\} \cup X\times \{1\}.
\]
The loop space $\Omega X$ of $X$ is
\[
\Omega X := F(S^1,X).
\]
Remark: If $X$ is well-pointed (the inclusion $i:\{x\}\hookrightarrow X$ is a cofibration), then the natural quotient map $SX\to \Sigma X$ is a homotopy equivalence. Moreover, there is an adjunction $F(\Sigma X,Y)\cong F(X,\Omega Y)$. In the fundamental group this gives the adjunction
\[
[\Sigma X,Y]\cong [X,\Omega Y],
\]
where $[A,B]$ is the set of based homotopy classes of maps $A\to B$.

References: May (A concise course in algebraic toplogy, Chapters 6, 7, 8), Aguilar, Gitler, and Prieto (Algebraic topology from a homotopical viewpoint, Chapter 2.10)

Connections, curvature, and Higgs bundles

Recall (from a previous post) that a Kähler manifold $M$ is a complex manifold (with natural complex structure $J$) with a Hermitian metic $g$ whose fundamental form $\omega$ is closed. In this context $M$ is Kähler. Previously we used upper-case letters $V,W$ to denote vector fields on $M$, but here we use lower-case letters $s,u,v$ and call them sections (to consider vector bundles more generally as sheaves).

Definition: A connection on $M$ is a $\C$-linear homomorphism $\nabla: A^0_M\to A^1_M$ satisfying the Leibniz rule $\nabla(fs) = (df)\wedge s + f\nabla (s)$, for $s$ a section of $TM$ and $f\in C^\infty(M)$.

For ease of notation, we often write $\nabla_us$ for $\nabla(s)(u)$, where $s,u$ are sections of $TM$. On Kähler manifolds there is a special connection that we will consider.

Proposition: On $M$ there is a unique connection $\nabla$ that is (for any $u,v\in A^0_M$)

1. Hermitian (satisfies $dg(u,v) = g(\nabla (u),v) + g(u,\nabla (v))$),
2. torsion-free (satisfies $\nabla_uv - \nabla_vu-[u,v] = 0$), and
3. compatible with the complex structure $J$ (satisfies $\nabla_uv = \nabla_{Ju}(Jv)$).

If $\nabla$ satisfies the first two conditions, it is called the Levi-Civita connection, and if it satisfies the first and third conditions, it is called the Chern connection. If $g$ is not necessarily Hermitian, $\nabla$ is called metric if it satisfies the first condition. From here on out $\nabla$ denotes the unique tensor described in the proposition above.

Definition: The curvature tensor of $M$ is defined by
\[
R(u,v) = \nabla_u\nabla_v - \nabla_v\nabla_u-\nabla_{[u,v]}.
\]
It may be viewed as a map $A^2 \to A^1$, or $A^3\to A^0$, or $A^0\to A^0$. The Ricci tensor of $M$ is defined by
\[
r(u,v) = \trace(w\mapsto R(u,v)w) = \sum_i g(R(a_i,u)v,a_i),
\]
for the $a_i$ a local orthonormal basis of $A^0 = TM$. This is a map $A^2\to A^0$. The Ricci curvature of $M$ is defined by
\[
\Ric(u,v) = r(Ju,v).
\]
This is a map $A^2\to A^0$.

Definition: An Einstein manifold is a pair $(M,g)$ that is Riemannian and for which the Ricci curvature is directly proportional to the Riemannian metric. That is, there exists a constant $\lambda\in \R$ such that $\Ric(u,v) = \lambda g(u,v)$ for any $u,v\in A^1$.

Recall that a holomorphic vector bundle $\pi:E\to M$ has complex fibers and holomorphic projection map $\pi$. Here we consider two special vector bundles (as sheaves), defined on open sets $U\subset M$ by
\begin{align*}
\End(E)(U) & = \{f:\pi^{-1}(U)\to \pi^{-1}(U)\ :\ f|_{\pi^{-1}(x)}\text{\ is a homomorphism}\}, \\
\Omega_M(U) & = \left\{\sum_{i=0}^n f_idz_1\wedge\cdots \wedge dz_i\ :\ f_i\in C^\infty(U)\right\},
\end{align*}
where $z_1,\dots,z_n$ are local coordinates on $U$. The first is the endomorphism sheaf of $E$ and the second is the sheaf of differential forms of $M$, or the holomorphic cotangent sheaf. The cotangent sheaf as defined is a presheaf, so we sheafify to get $\Omega_M$.

Definition: A Higgs vector bundle over a complex manifold $M$ is a pair $(E,\theta)$, where $\pi:E\to M$ is a holomorphic vector bundle and $\theta$ is a holomorphic section of $\text{End}(E)\otimes \Omega_M$ with $\theta\wedge\theta = 0$, called the Higgs field.

References: Huybrechts (Complex Geometry, Chapters 4.2, 4.A), Kobayashi and Nomizu (Foundations of Differential Geometry, Volume 1, Chapter 6.5)

On the separation of nearest neighbors

We work through Lemma 3 (called the "$A-B$ Lemma" or the "cleaning procedure") of [2], adopting a cleaner and more thorough approach.

Necessary tools

Definition: The inverse of the complex-valued function $f(z) = ze^z$ is called the Lambert $W$-function and denoted by $W = f^{-1}$. When restricted to the real numbers, it is multi-valued on part of its domain, so it is split up into two branches $W_0$ (for positive values) and $W_{-1}$ (for negative values).

Hoeffding's inequality gives an upper bound on how much we should expect a sum of random variables to deviate from their combined mean. The authors of [2] use a similar inequality called the Chernoff bound, but Hoeffding gives a tighter bound on the desired event.

Proposition: (Hoeffding - Theorem 2 and Equation (1.4) of [1])
Let $X_1,\dots,X_n$ be independent random variables, with $X_i$ bounded on the interval $[a_i,b_i]$. Then
\[
P\left(\left| \frac1n\sum_{i=1}^n X_i - \frac 1n\sum_{i=1}^n E[X_i]\right|\geqslant t\right) \leqslant 2\exp\left(\frac{-2t^2n^2}{\sum_{i=1}^n(b_i-a_i)^2}\right).
\]

The union bound (or Boole's inequality) says that the probability of one of a collection of events happening is no larger than the sum of the probabilities of each of the events happening.

Proposition: Let $A_1,A_2,\dots$ be a countable collection of events. Then $P(\bigcup_i A_i) \leqslant \sum_i P(A_i)$.

The setup

Let $P$ be a probability distribution $P$ on $\R^n$ and $X=\{x_1,\dots,x_k\}\subseteq \R^n$ a finite set of points drawn according to $P$. These points may be considered as random variables $X_1,X_2,\dots,X_k$ on the sample space $\R^n$, with $X_i$ evaluating to 1 only on $x_i$, and 0 otherwise. Choose $s>0$ and construct the nearest neighbor graph $G$ on $X$, with parameter $s$. Write $X=A\cup B$ and set
\[
\eta := \inf_{a\in A,b\in B}\left\{|| a-b||\right\}
\hspace{1cm},\hspace{1cm}
\alpha_s := \inf_{a\in A}\left\{P(B^n(s,a))\right\}
\hspace{1cm},\hspace{1cm}
\beta_s \:= \sup_{b\in B}\left\{P(B^n(s,b))\right\},
\]
with $h = (\alpha_s-\beta_s)/2$. We assume that

• $\eta>0$, so $A$ and $B$ are disjoint;
• $s<\eta/2$, so $A$ and $B$ are in separate components of $G$; and
• $\alpha_s >\beta_s$, so any point in $A$ is more likely to be chosen than every point in $B$.

Proposition: Choose $\delta\in (0,1)$. If $|X| >-W_{-1}(-\delta h^2e^{-2h^2})/(2h^2)$, then for all $a\in A$ and $b\in B$, with probability $1-\delta$,
\[
\frac{\deg_G(a)}{k - 1} > \frac{\alpha_s+\beta_s}2
\hspace{1cm}\text{and}\hspace{1cm}
\frac{\deg_G(b)}{k - 1} < \frac{\alpha_s+\beta_s}2.
\]
The statement holds also for $\alpha,\beta$ instead of $\alpha_s,\beta_s$, such that $\alpha_s\> \alpha >\beta \> \beta_s$, which may be useful to bound the degree of vertices in $G$.

The proof

For each $i=1,\dots,k$, define new random variables $Y_{ij}$ on the sample space $X$, with $Y_{ij}$ evaluating to 1 on $x_j$ iff $x_j\in B^n(s,x_i)$, and evaluating to 0 otherwise. The mean of $Y_{ij}$ is $P(B^n(s,x_i))$. Since the $Y_{ij}$ are independent with the same mean, Hoeffding's inequality gives that
\[
\left(\begin{array}{c}
\text{the probability that the sampled $x_j$}\\
\text{have clustered around a point more than} \\
\text{a distance $h$ away from $B^n(s,x_i)$}
\end{array}\right)
=
P\Bigg(\underbrace{\left|\frac{1}{k-1}\sum_{j\neq i} Y_{ij} - P(B^n(s,x_i))\right| \> h}_{\text{event}\ A_i}\Bigg)  \leqslant 2e^{-2h^2(k-1)}.
\]
The union bound gives that
\[
\left(\begin{array}{c}
\text{the probability that at}\\
\text{least one $A_i$ occurs}
\end{array}\right)
=
P\left(\bigcup_{i=1}^k A_i\right) < \sum_{i=1}^k P(A_i) \leqslant 2ke^{-2h^2(k-1)}.
\]
Note that $\sum_{j\neq i} Y_{ij} = \deg_G(x_i)$ for every $i$, so whenever $\delta>2ke^{-2h^2(k-1)}$, with probability $1-\delta$
\[
\left| \frac{\deg_G(x_i)}{k-1} - P(B^n(s,x_i))\right| < h
\hspace{1cm}\text{or}\hspace{1cm}
P(B^n(s,x_i)) -h < \frac{\deg_G(x_i)}{k-1} < P(B^n(s,x_i)) + h.
\]
When $x_i\in A$ ($x_i\in B$) we have a lower (upper) bound of $\alpha_s$ ($\beta_s$) on $P(B^n(s,x_i))$. Indeed:
\[
\frac{\deg_G(a)}{k-1} > \alpha_s- h =  \frac{\alpha_s+\beta_s}2
\hspace{1cm}\text{and}\hspace{1cm}
\frac{\deg_G(b)}{k-1} < \beta_s + h = \frac{\alpha_s+\beta_s}2.
\]
To find how many points we need to sample, we solve for $k$ in the inequality $ \delta > 2ke^{-2h^2(k-1)}$. With the aid of a computer algebra system, we find that
\[
k > \frac{-1}{2h^2}W_{-1}\left(-\delta h^2e^{-2h^2}\right),
\]
completing the proof.

References:
[1] Hoeffding (Probability inequalities for sums of bounded random variables)
[2] Niyogi, Smale, and Weinberger (A topological view of unsupervised learning from noisy data)

The conditioning number of a projective curve

Let $C$ be a smooth algebraic curve in $\P^2$. That is, for some homogeneous $f\in \C[x_0,x_1,x_2]$ we let $C = \{x\in \P^2\ :\ f(x)=0\}$. Describe $C$ as a manifold via the usual open sets $U_i = \{x\in \P^2\ :\ x_i\neq 0\}$ and charts
\[
\begin{array}{r c l}
\varphi_0\ :\ U_0 & \to & \C^2, \\\
[x_0:x_1:x_2] & \mapsto & (\frac{x_1}{x_0},\frac{x_2}{x_0}),
\end{array}
\hspace{1cm}
\begin{array}{r c l}
\varphi_1\ :\ U_1 & \to & \C^2, \\\
[x_0:x_1:x_2] & \mapsto & (\frac{x_0}{x_1},\frac{x_2}{x_1}),
\end{array}
\hspace{1cm}
\begin{array}{r c l}
\varphi_2\ :\ U_2 & \to & \C^2, \\\
[x_0:x_1:x_2] & \mapsto & (\frac{x_0}{x_2},\frac{x_1}{x_2}).
\end{array}
\]
Let $w=[w_0:w_1:w_2]\in \P^2$ for which $f(w)=0$. The Jacobian of $C$ at $w$ is then
\[
J_w = \left[
\left.\frac{\dy f}{\dy x_0}\right|_w \ :\  \left.\frac{\dy f}{\dy x_1}\right|_w \ :\  \left.\frac{\dy f}{\dy x_2}\right|_w
\right] \in \P^2.
\]
Assume that $\left.\frac{\dy f}{\dy x_0}\right|_w\neq 0$ and pass to $\varphi_0(U_0)$ to get the Jacobian to be
\[
J_w^0 = \left(
\frac{\dy f/\dy x_1|_w}{\dy f/\dy x_0|_w}\ ,\ \frac{\dy f/\dy x_2|_w}{\dy f/\dy x_0|_w}\right)  \in \C^2.
\]
Assume that $w_0\neq 0$, so the tangent line to $\varphi_0(C)\subset \C^2$ at $\varphi_0(w)=(w_1/w_0,w_2/w_0)$ is
\[
T_{\varphi_0(w)}= \{\varphi_0(w)+tJ_w^0\ :\ t\in \C\}\subset \C^2.
\]
A vector orthogonal to the Jacobian $J_w^0$ is
\[
\bar J_w^0 = \left(-\frac{\dy f/\dy x_2|_w}{\dy f/\dy x_0|_w}\ ,\ \frac{\dy f/\dy x_1|_w}{\dy f/\dy x_0|_w}\right) \in \C^2,
\]
so the space space normal to $T_{\varphi_0(w)}$ is given by
\[
T_{\varphi_0(w)}^\perp = \{\varphi_0(w)+t\bar J_w^0\ :\ t\in \C\}\subset \C^2.
\]

Example: Let $C\subset \P^2$ be the zero locus of $f(x_0,x_1,x_2) = x_0^2+x_1x_2-x_1x_0$. The Jacobian is $J = [2x_0-x_1:x_2-x_0:x_1]$, and as $J=0$ implies $x_0=x_1=x_2=0$, but $0\not\in\P^2$, the curve $C$ is smooth. Consider two points $w=[1:1:0],z=[2:1:-2]\in C$, at which the Jacobian is
\[
J_w = [1:-1:1]
\hspace{1cm},\hspace{1cm}
J_z = [3:-4:1].
\]
Both $w_0$ and $z_0$ are non-zero, with $\varphi_0(w)=(1,0)$ and $\varphi_0(z)=(1/2,-1)$, giving the tangent and normal spaces to be
\begin{align*}
T_{(1,0)} & = \{(1,0)+t(-1,1)\ :\ t\in \C\}, & T_{(1/2,-1)} & = \{(1/2,-1)+s(-4/3,1/3)\ :\ s\in \C\}, \\
T^\perp_{(1,0)} & = \{(1,0)+t(-1,-1)\ :\ t\in \C\}, & T_{(1/2,-1)}^\perp & = \{(1/2,-1)+s(-1/3,-4/3)\ :\ s\in \C\}.
\end{align*}
The two normal spaces intersect at $(t,s)=(1/3,-1/2)$ at distances of $1/3\cdot ||(-1,-1)|| = \sqrt 2/3\approx 0.471$ and $1/2\cdot||(-1/3,-4/3)|| = \sqrt{17}/3\approx 1.374$ from the points $\varphi_0(w),\varphi_0(z)$, respectively. Hence the conditioning number of $C$ is at most $\sqrt 2/3$.

Given a smooth projective curve and a finite set of points, this Sage code will calculate the conditioning number from that collection of points.

Smooth projective varieties as Kähler manifolds

Definition: Let $k$ be a field and $\P^n$ projective $n$-space over $k$. An algebraic variety $X\subset \P^n$ is the zero locus of a collection of homogeneous polynomials $f_i\in k[x_0,\dots,x_n]$.

Here we let $k=\C$, the complex numbers. Complex projective space $\C\P^n$ may be described as a complex manifold, with open sets $U_i = \{(x_0:\cdots:x_n)\ :\ x_i\neq 0\}$ and maps
\[
\begin{array}{r c l}
\varphi_i\ :\ U_i & \to & \C^n, \\
(x_0:\cdots:x_n) & \mapsto & \left(\frac{x_0}{x_i},\dots,\widehat{\frac{x_i}{x_i}},\dots,\frac{x_n}{x_i}\right),
\end{array}
\]
which can be quickly checked to agree on overlaps. In this context we assume all varieties are smooth, so they are submanifolds of $\C\P^n$.

Definition: An almost complex manifold is a real manifold $M$ together with a vector bundle endomorphism $J:TM\to TM$ (called a complex structure) with $J^2=-\id$.

Note that every complex manifold admits an almost complex structure on its underlying real manifold. Indeed, given standard coordinates $z_i=x_i+y_i$ for $i=1,\dots,n$ on $\C^n$, we get a basis $\partial/\partial x_1, \dots, \partial /\partial x_n$, $\partial/\partial y_1, \dots, \partial/\partial y_n$ on the underlying real tangent space $T_pU$, for $p\in M$ and $U\owns p$ a neighborhood. Then $J$ is defined by
\[
J\left(\frac\partial{\partial x_i}\right) = \frac\partial{\partial y_i}
\hspace{1cm},\hspace{1cm}
J\left(\frac\partial{\partial y_i}\right) = -\frac\partial{\partial x_i}.
\]
Write $T_\C M=TM\otimes_\R\C$ for the complexification of the tangent bundle, which admits a canonical decomposition $T_\C M = T^{1,0}M\oplus T^{0,1}M$, where $J|_{T^{1,0}}=i\cdot \id$ and $J|_{T^{0,1}}=(-i)\cdot \id$. We call $T^{1,0}M$ the holomorphic tangent bundle of $M$ and $T^{0,1}M$ the antiholomorphic tangent bundle of $M$, even though it is extraneous to consider any related map here as holomorphic. Define vector bundles (or sheaves, to consider sections on open sets)
\[
A^k_M = \textstyle \bigwedge^k(T_\C M)^*,
\hspace{1cm}
A^{p,q}_M = \textstyle \bigwedge^p(T^{1,0}M)^* \otimes_\C \bigwedge^q(T^{0,1}M)^*,
\]
where we drop the subscript $M$ when the context makes it clear. There is a canonical decomposition $A^k = \bigoplus_{p+q=k} A^{p,q}$, which yields projection maps $\pi^{p,q}:A^k \to A^{p,q}$. The exterior differential $d$ on $T^*M$ may be extended $\C$-linearly to $(T_\C M)^*$, and hence also to $A^k$. Define two new maps
\begin{align*}
\partial = \pi^{p+1,q}\circ d|_{A^{p,q}}\ :\ &\ A^{p,q} \to A^{p+1,q}, \\
\bar\partial = \pi^{p,q+1}\circ d|_{A^{p,q}}\ :\ &\ A^{p,q} \to A^{p,q+1}.
\end{align*}
These satisfy the Leibniz rule and (under mild assumptions) $\partial^2 = \bar\partial^2 = 0$ and $\partial \bar \partial = -\bar \partial \partial$.

From now on, the manifold $M$ will be complex with the natural complex structure described above.

Definition: A Riemannian metric on $M$ is a function $g:TM\times TM \to C^\infty(M)$ such that for all $V,W\in TM$,

• $g(V,W)=g(W,V)$, and
• $g_p(V_p,V_p)\geqslant 0$ for all $p\in M$, with equality iff $V=0$.

A Riemannian manifold is a pair $(M,g)$ where $g$ is Riemannian.

Locally we write $g_p:T_pM\times T_pM \to \R$, defined as $g_p(V_p,W_p)=g(V,W)(p)$. If $x_1,\dots,x_n$ are local coordinates on some open set $U\subset M$, then $g=\sum_{i,j}g_{ij}dx_i\wedge dx_j\in A^2(M)$, for $g_{ij} = g(\frac\partial{\partial x_i},\frac \partial{\partial x_j})\in C^\infty(U)$. Writing $V = \sum_if_i\frac\partial{\partial x_i}$ and $W=\sum_jg_j\frac\partial{\partial x_j}$, we get the local expression
\[
g_p(V_p,W_p) = \sum_{i,j}g_{ij}(p)f_i(p)g_j(p).
\]

Definition: A Hermitian metric on a complex manifold $M$ is a Riemannian metric $g$ such that $g(JV,JW)=g(V,W)$ for all $V,W\in TM$. A Hermitian manifold is a pair $(M,g)$ where $g$ is Hermitian.

There is an induced form $\omega:TM \times TM\to C^\infty(M)$ given by $\omega (V,W)=g(JV,W)$, called the fundamental form. From $g$ being Hermitian it follows that $\omega\in A^{1,1}(M)\subset A^2(M)$. Note also that any two of the structures $J,g,\omega$ determine the remaining one.

Definition: A Kähler metric on a complex manifold $M$ is a Hermitian metric whose fundamental form is closed (that is, $d\omega = 0$). A Kähler manifold is a pair $(M,g)$ where $g$ is Kähler.

Example: Recall the atlas given to $\C\P^n$ above. There is a metric (canonical in some sense) on each $U_j$ given by
\[
\omega_j = \frac i{2\pi} (\partial \circ \bar\partial) \left(\log\left(\sum_{\ell=0}^n \left|\frac{x_\ell}{x_j}\right|^2 \right)\right),
\]
called the Fubini--Study metric. Each $\omega_j$ is a section of $A^{1,1}(U_j)$, and as a quick calculation shows that $\omega_j|_{U_j\cap U_k} = \omega_k|_{U_j\cap U_k}$, there is a global metric $\omega_{FS}\in A^{1,1}(\C\P^n)$ such that $\omega_{FS}|_{U_j} = \omega_j$ for all $j$.

Hence $\C\P^n$ is a Kähler manifold. If we have a smooth projective variety $X\subset \C\P^n$, then it is a submanifold of $\C\P^n$, so by restricting $\omega_{FS}$ to $X$, we get that $X$ is also a Kähler manifold. Therefore all smooth projective varieties are Kähler.

References: Huybrechts (Complex Geometry, Chapters 1.3, 2.6, 3.1), Lee (Riemannian manifolds, Chapter 3)