The Fubini-Study metric and length in projective space

In this post we inspect how the Fubini-Study metric works and compute an example. Professor Mihai Paun for helpful discussions. Recall that from projective space $\P^n$ there are natural maps

\[
[x_0:x_1:\cdots:x_n]\tov{\vp_i}\left(\frac{x_0}{x_i},\dots,\widehat{\frac{x_i}{x_i}},\dots,\frac{x_n}{x_i}\right)
\]
for $i=0,\dots,n$. The maps land in $\C^n$ with coordinates $(z_1,z_2,\dots,z_n)$. We use $\vp_0$ as the main map, and conflate notation for objects in $\P^n$ and in $\C^n$ under $\vp_0$. Most of this post deals with the $n=2$ case.

The metric

The metric used on $\P^n$ is the Fubini-Study metric. Directly from Section 3.1 of Huybrechts, for $n=2$ the associated differential 2-form and its image in $\C^2$ are
\begin{align*}
\omega & = \frac i{2\pi}\partial \bar\partial \log\left(1+\left|\frac{x_1}{x_0}\right|^2+\left|\frac{x_2}{x_0}\right|^2\right), \\
\vp_0(\omega) & = \frac i{2\pi}\partial \bar\partial \log\left(1+\left|z_1\right|^2+\left|z_2\right|^2\right) \\
& =  \underbrace{\frac{i}{2\pi (1+|z_1|^2+|z_2|^2)^2}}_{\lambda_2}\sum_{k,\ell=1}^2\underbrace{(1+|z_1|^2+|z_2|^2)\delta_{k\ell} -\overline{z_k}z_\ell}_{\chi_{k\ell}}dz_k\wedge d\overline{z_\ell}. \hspace{1cm} (1)
\end{align*}
A Hermitian metric on a complex manifold $X$ may be described as a 2-tensor $h=g-i\omega$, where $g$ is a Riemannian metric (also a 2-tensor) on the underlying real manifold and $\omega$ is a Kahler form, a 2-form. As in Lemma 3.3 of Voisin, the relationship between $g$ and $\omega$ is given by
\[
g(u,v)=\omega(u,Iv)=\omega(Iu,v), \hspace{1cm} (2)
\]
where $I:T_xX\to T_xX$ is a tangent space endomorphism defined by
\[
\begin{array}{r c l}
I|_{T^{1,0}_xX} & = & i\cdot \id, \\
\frac{\dy}{\dy z_i} & \mapsto & i\frac{\dy}{\dy z_i},
\end{array}
\hspace{1cm}
\begin{array}{r c l}
I|_{T^{0,1}_xX} & = & -i\cdot \id, \\
\frac{\dy}{\dy \overline{z_i}} & \mapsto & -i\frac{\dy}{\dy \overline{z_i}},
\end{array}
\]
as in Proposition 1.3.1 of Huybrechts.

An application

Let $\gamma:[0,1]\to \C^2$ be a path, described as $\gamma(t)=(\gamma_1(t),\gamma_2(t))$. The derivative of $\gamma$ with respect to $t$, in the basis $\frac{\dy}{\dy z_1}$, $\frac{\dy}{\dy \overline{z_1}}$, $\frac{\dy}{\dy z_2}$, $\frac{\dy}{\dy \overline{z_2}}$ is given by
\[
\frac{d\gamma_1}{dt} = \frac{du_1}{dt}\frac\dy{\dy x_1} + i\frac{dv_1}{dt}\frac\dy{\dy y_1} = \frac{du_1}{dt}\left(\frac\dy{\dy \overline{z_1}}+\frac\dy{\dy z_1}\right) + i\frac{dv_1}{dt}\left(\frac\dy{\dy \overline{z_1}} -\frac{\dy}{\dy z_1}\right) = \underbrace{\left(\frac{du_1}{dt} + i\frac{dv_1}{dt}\right)}_{\gamma_1'}\frac\dy{\dy \overline {z_1}} + \underbrace{\left(\frac{du_1}{dt}-i\frac{dv_1}{dt}\right)}_{\overline \gamma_1'}\frac\dy{\dy z_1},\]
and analogously for $\gamma_2$. Hence
\[
\frac{d\gamma}{dt} =
\overline \gamma_1'\frac\dy{\dy z_1} + \gamma_1'\frac\dy{\dy \overline{z_1}} + \overline \gamma_2' \frac\dy{\dy z_2} + \gamma_2' \frac{\dy}{\dy \overline{z_2}}. \hspace{1cm} (3)
\]
The length of $\gamma$ is
\[
\int_0^1\sqrt{g\left(\frac{d\gamma}{dt},\frac{d\gamma}{dt}\right)}\ dt = \int_0^1\sqrt{\omega\left(\frac{d\gamma}{dt},I\frac{d\gamma}{dt}\right)}\ dt,
\]

using equation (2). Recall that the pairing of vectors with covectors is given by\[
\left(d\alpha_1\wedge \cdots \wedge d\alpha_n\right)\left(\frac\dy{\dy \beta_1},\dots,\frac\dy{\dy \beta_n}\right) = \det\begin{bmatrix}
d\alpha_1\frac\dy{\dy \beta_1} & d\alpha_1\frac{\dy}{\dy \beta_2} & \cdots & d\alpha_1\frac\dy{\dy \beta_n} \\
d\alpha_2\frac\dy{\dy \beta_1} & d\alpha_2\frac{\dy}{\dy \beta_2} & \cdots & d\alpha_2\frac\dy{\dy \beta_n} \\
\vdots & \vdots & \ddots & \vdots \\
d\alpha_n\frac\dy{\dy \beta_1} & d\alpha_n\frac{\dy}{\dy \beta_2} & \cdots & d\alpha_n\frac\dy{\dy \beta_n}
\end{bmatrix}
 \ \ = \ \
\det\left(d\alpha_i\frac\dy{\dy \beta_j}\right),
\]
for $\alpha_i,\beta_j$ a basis of the underlying real manifold (as in the previous post "Vector fields," 2016-10-10). The components of the vector (3) may be viewed as given in directions $z_1,\overline{z_1}, z_2,\overline{z_2}$, respectively, which also indicates how the coefficient functions $\chi_{k\ell}$ act on (3). Apply the definition of $\omega$ from equation (1), and note that we are always at the tangent space to the point $\gamma(t)=(\gamma_1(t),\gamma_2(t))$, to get that
\begin{align*}
& \omega\left(\frac{d \gamma}{dt},I\frac{d\gamma}{dt}\right) \\
& = \lambda_2(\gamma(t)) \sum_{k,\ell=1}^2 \chi_{k\ell}(\gamma(t)) dz_k\wedge d\overline{z_\ell}\left(\overline \gamma_1'\frac\dy{\dy z_1} + \gamma_1'\frac\dy{\dy \overline{z_1}} + \overline \gamma_2' \frac\dy{\dy z_2} + \gamma_2' \frac{\dy}{\dy \overline{z_2}}, i\overline \gamma_1'\frac\dy{\dy z_1} - i\gamma_1'\frac\dy{\dy \overline{z_1}} + i\overline \gamma_2' \frac\dy{\dy z_2} -i\gamma_2' \frac{\dy}{\dy \overline{z_2}}\right) \\
& = \lambda_2(\gamma(t)) \sum_{k,\ell=1}^2 \chi_{k\ell}(\gamma(t))\det
\begin{bmatrix}
\overline \gamma_k'(t) & i\overline \gamma_k'(t) \\[5pt] \gamma_\ell'(t) & -i\gamma_\ell'(t)
\end{bmatrix} \\
& = \frac{(1+|\gamma_2(t)|^2)|\gamma_1'(t)|^2 - \overline\gamma_1(t)\gamma_2(t)\overline\gamma_1'(t)\gamma_2'(t) - \overline\gamma_2(t)\gamma_1(t)\overline \gamma_2'(t)\gamma_1'(t) + (1+|\gamma_1(t)|^2) |\gamma_2'(t)|^2}{\pi\left(1+\left|\gamma_1(t)\right|^2+\left|\gamma_2(t)\right|^2\right)^2}.\end{align*}
Unfortunately this expression does not simplify too much. In $\P^n$, with $\gamma = (\gamma_1,\dots,\gamma_n):[0,1]\to \C^n$, we have that
\[
g\left(\frac{d \gamma}{dt},\frac{d\gamma}{dt}\right) = \lambda_n(\gamma(t)) \sum_{k,\ell=1}^n \chi_{k\ell}(\gamma(t))\det
\begin{bmatrix}
\overline \gamma_k'(t) & i\overline \gamma_k'(t) \\[5pt] \gamma_\ell'(t) & -i\gamma_\ell'(t)
\end{bmatrix}.
\]

An example

Here we compute the distance between two points in $\P^2$. Let $\gamma$ be the straight line segment connecting $p=[p_0:p_1:p_2]$ and $q=[q_0:q_1:q_2]$. The word "straight" is used loosely, and means the segment may be parametrized as
\[
\gamma(t) = [(1-t)p_0+tq_0:(1-t)p_1+tq_1:(1-t)p_2+tq_2],
\]
so $\gamma(0)=p$ and $\gamma(1)=q$. The image of $\gamma$ under $\vp_0$ and its derivative are given by
\[
\vp_0(\gamma(t)) = \left(\frac{(1-t)p_1+tq_1}{(1-t)p_0+tq_0}, \frac{(1-t)p_2+tq_2}{(1-t)p_0+tq_0}\right) = (\gamma_1,\gamma_2),
\hspace{2cm}
\gamma_i' = \frac{q_ip_0-q_0p_i}{((1-t)p_0+tq_0)^2}.
\]
If, for example, $p=[1:1:0]$ and $q=[1:0:1]$, then
\[
\text{length}(\gamma) = \frac{3}{4\pi}\int_0^1\frac1{(t^2-t+1)^2}\ dt = \frac{9+2\pi\sqrt 3}{18\pi}.
\]

A further goal is to consider the path $\gamma$ as lying on a projective variety, beginning with a complete intersection. This would allow some of the $dz_i$ to be expressed in terms of other $dz_j$.

References: Huybrechts (Complex geometry, Section 3.1), Voisin (Hodge theory and complex algebraic geometry 1, Chapter 3.1), Wells (Differential analysis on complex manifolds, Chapter V.4)

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)

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)