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)