Loose ends of smooth manifolds

 Preliminary exam prep

Here we round up some theorems that have escaped previous roundings-up. Let $X,Y$ be smooth manifolds and $f:X\to Y$ a smooth map.

Theorem: (Inverse function theorem) If $df_p$ is invertible for some $p\in M$, then there exist $U\owns p$ and $V\owns f(p)$ connected such that $f|_U:U\to V$ is a diffeomorphism.

Corollary: (Stack of records theorem) If $\dim(X)=\dim(Y)$, then every regular value $y\in Y$ has a neighborhood $V\owns y$ such that $f^{-1}(Y)=U_1\sqcup \cdots \sqcup U_k$, where $f|_{U_i}:U_i\to V$ is a diffeomorphism.

Proof: Since $y\in Y$ is a regular value, $df_x$ is surjective for all $x\in f^{-1}(y)$. Since $\dim(X)=\dim(Y)$ and $df_x$ is linear, $df_x$ is an isomorphism, hence invertible. By the inverse function theorem, there exist $U\owns x$ and $V\owns y$ connected such that $f|_U:U\to V$ is a diffeomorphism. Before we actually apply this, we need to show that $f^{-1}(y)$ is a finite set.

First we note that by the preimage theorem, since $y$ is a regular value, $f^{-1}(y)$ is a submanifold of $X$ of dimension $\dim(X)-\dim(Y)=0$. Next, if $f^{-1}(y) = \{x_i\}$ were infinite, since $X$ is compact, there would be some limit point $p\in X$ of $\{x_i\}$. But then by continuity,
\[
y = \lim_{i\to \infty}\left[f(x_i)\right] = f\left(\lim_{i\to\infty}\left[x_i\right]\right) = f(p),\]so $p\in f^{-1}(y)$. But then either $p$ cannot be separated from other elements of $f^{-1}(y)$, meaning $f^{-1}(y)$ is not a manifold, or the sequence $\{x_i\}$ is finite in length. Hence $f^{-1}(y) = \{x_1,\dots,x_k\}$. Let $U_i\owns x_i$ and $V_i\owns y$ be the sets asserted to exist by the inverse function theorem (the $U_i$ may be assumed to be disjoint without loss of generality). Let $V = \bigcap_{i=1}^k V_i$ and $U_i' = f^{-1}(V)\cap U_i$, for which we still have $f|_{U_i'}:U_i'\to V$ a diffeomorphism. $\square$

Theorem: (Classification of manifolds) Up to diffeomorphism,

• the only 0-dimensional manifolds are collections of points,
• the only 1-dimensional manifolds are $S^1$ and $\R$, and
• the only 2-dimensional compact manifolds are $S^2\# (T^2)^{\#n}$ or $S^2\#(\R\P^2)^{\#n}$, for any $n\geqslant 0$.

Compact 2-manifolds are homeomorphic iff they are both (non)-orientable and have the same Euler characteristic. Note that
\[
\chi\left(S^2\# (T^2)^{\#n}\right) = 2-2n,
\hspace{1cm}
\chi\left(S^2\#(\R\P^2)^{\#n}\right) = 2-n.
\]
These surfaces are called orientable (on the left) and non-orientable (on the right) surfaces of genus $n$.

Theorem: (Stokes' theorem) For $X$ oriented and $\omega\in \Omega^{n-1}_X$, $\int_X d\omega = \int_{\dy X} \omega$.

Proposition: The tangent bundle $TX$ is always orientable.

Proof: Let $U,V\subset X$ with $\vp:U\to \R^n$ and $\psi:V\to \R^n$ trivializing maps, and $\psi\circ \vp^{-1}:\R^n\to \R^n$ the transition function. To show that $TX$ is always orientable, we need to show the Jacobian of the induced transition function (determinant of the derivative) on $TX$ is always non-negative. On $TU$ and $TV$, we have trivializing maps $(\varphi,d\varphi)$ and $(\psi,d\psi)$, giving a transition function
\[
(\psi\circ \varphi^{-1}, d\psi \circ d\varphi^{-1}) = (\psi\circ \varphi^{-1}, d(\psi \circ \varphi^{-1})). 
\]
The Jacobian of this is
\[
\det(d(\psi\circ \varphi^{-1}, d(\psi \circ \varphi^{-1}))) = \det(d(\psi\circ \varphi^{-1}), d(\psi \circ \varphi^{-1}))) = \det(d(\psi\circ \varphi^{-1}))\cdot \det(d(\psi \circ \varphi^{-1})) \>0,
\]
and since $d(\psi\circ \varphi^{-1})\neq 0$ (as $\psi \circ \varphi^{-1}$ is a diffeomorphism, its derivative is an isomorphism), the result is always positive. $\square$

References: Lee (Introduction to smooth manifolds, Chapter 4), Guillemin and Pollack (Differential topology, Chapter 1

The tangent space and differentials

 Preliminary exam prep

Let $M,N$ be smooth $n$-manifolds. Here we discuss different definitions of the tangent space and differentials, or pushforwards, of smooth maps $f:M\to N$.

Derivations (Lee)

Definition: A derivation of $M$ at $p\in M$ is a linear map $v:C^\infty(M)\to \R$ such that for all $f,g\in C^\infty(M)$,
\[
v(fg) = f(p)v(g) + g(p)v(f).
\]
The tangent space $T_pM$ to $M$ at $p$ is the set of all derivations of $M$ at $p$.

Given a smooth map $F:M\to N$ and $p\in M$, define the differential $dF_p:T_pM\to T_{f(p)}N$, which, for $v\in T_pM$ and $f\in C^\infty(N)$ acts as
\[
dF_p(v)(f) = v(f\circ F)\in \R.
\]

Dual of cotangent (Hitchin)

Definition: Let $Z_p\subset C^\infty(M)$ be the functions whose derivative vanishes at $p\in M$. The cotangent space $T_p^*M$ to $M$ at $P$ is the quotient space $C^\infty(M)/Z_p$. The tangent space to $M$ at $P$ is the dual of the cotangent space $T_pM = (T_p^*M)^* = \Hom(T_p^*M,\R)$.

Given a smooth map $F:M\to N$ and $p\in M$, define the differential
\[
\begin{array}{r c l}
dF_p\ :\ T_pM & \to & T_{F(p)}N, \\
\left(f:C^\infty(M)/Z_p \to \R\right) & \mapsto & \left(\begin{array}{r c l}
g\ :\ C^\infty(N)/Z_{F(p)} & \to & \R, \\ h & \mapsto & f(h\circ F).
\end{array}\right)
\end{array}
\]
This definition makes clear the relation to the first approach. Since $h\not\in Z_{F(p)}$, the derivative of $h$ does not vanish at $F(p)$. Hence the derivative of $h\circ F$ at $p$, which is the derivative of $h$ at $F(p)$ multiplied by the derivative of $F$ at $p$, does not a priori vanish at $p$.

Derivative of chart map (Guillemin and Pollack)

Definition: Let $f:\R^n\to \R^m$ be a smooth map. Then the derivative of $f$ at $x\in \R^n$ in the direction $y\in \R^n$ is defined as
\[
df_x(y) = \lim_{h\to 0}\left[\frac{f(x+yh)-f(x)}h\right].
\]
Given $x\in M$ and charts $\varphi:\R^n\to M\subset \R^m$, the tangent space to $M$ at $p$ is the image $T_pM = d\varphi_0(\R^n)$, where we assume $\varphi(0)=p$.

Given a smooth map $F:M\to N$ and charts $\varphi:\R^n\to M$, $\psi:\R^n\to N$, with $\varphi(0)=p$ and $\psi(0)=F(p)$, define the differential $dF_p:T_pM\to T_{F(p)}N$ via the diagrams below.

Here $h = \psi^{-1}\circ F\circ \varphi$, so $dh_0$ is well-defined. Hence $dF_p = d\psi_0\circ dh_0\circ d\varphi_0^{-1}$ is also well-defined.

Sometimes the differential is referred to as the pushforward, in which case it is denoted by $(F_*)_p$.

References: Lee (Introduction to Smooth Manifolds, Chapter 3), Hitchin (Differentiable manifolds, Chapter 3.2), Guillemin and Pollack (Differential topology, Chapter 1.2)

The Grassmannian is a complex manifold

 Lecture topic

Let $Gr(k,\C^n)$ be the space of $k$-dimensional complex subspaces of $\C^n$, also known as the complex Grassmannian. We will show that it is a complex manifold of dimension $k(n-k)$. Thanks to Jinhua Xu and professor Mihai Păun for explaining the details.

To begin, take $P\in Gr(k,\C^n)$ and an $n-k$ subspace $Q$ of $\C^n$, such that $P\cap Q = \{0\}$. Then $P\oplus Q = \C^n$, so we have natural projections

A neighborhood of $P$, depending on $Q$ may be described as $U_Q = \{S\in Gr(k,\C^n)\ :\ S\cap Q = \{0\}\}$. We claim that $U_Q \cong \Hom(P,Q)$. The isomorphism is described by
\[
\begin{array}{r c c c l}
\Hom(P,Q) & \to & U_Q & \to & \Hom(P,Q), \\
A & \mapsto & \{v+Av\ :\ v\in P\}, \\
& & S & \mapsto & \left(\pi_Q|_S\right) \circ \left(\pi_P|_S\right)^{-1}.
\end{array}
\]
The map on the right, call it $\varphi_Q$, is also the chart for the manifold structure. The idea of decomposing $\C^n$ into $P$ and $Q$ and constructing a homomorphism from $P$ to $Q$ may be visualized in the following diagram.

Then $\Hom(P,Q) \cong \Hom(\C^k,\C^{n-k})\cong \C^{k(n-k)}$, so $Gr(k,\C^n)$ is locally of complex dimension $k(n-k)$. To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let $P,P'\in Gr(k,\C^n)$ and $Q,Q'\in Gr(n-k,\C^n)$ such that $P\cap Q = P'\cap Q' = \{0\}$. Let $X\in \Hom(P,Q)$ such that $X\in \varphi_Q(U_Q\cap U_{Q'})$, with $\varphi_Q(S)=X$ and $\varphi_{Q'}(S)=X'$ for some $S\in U_Q\cap U_{Q'}$. Define $I_X(v) = v+Xv$, and note the transition map takes $X$ to
\begin{align*}
X' & = \varphi_{Q'}\circ \varphi_Q^{-1}(X) & (\text{definition}) \\
& = \varphi_{Q'}(S) & (\text{assumption})\\
& = \left(\pi_{Q'}|_S\right)\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{definition}) \\
& = \left(\pi_{Q'}|_S\right)\circ I_X\circ I_X^{-1}\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{creative identity}) \\
& = \left(\pi_{Q'}|_S\circ I_X\right)\circ \left(\pi_{P'}|_S\circ I_X\right)^{-1} & (\text{redistribution}) \\
& = \left(\pi_{Q'}|_P +\pi_{Q'}|_Q\circ X\right) \circ \left(\pi_{P'}|_P +\pi_{P'}|_Q\circ X\right). & (\text{definition})
\end{align*}
At this last step we have compositions and sums of homomorphisms and linear maps, which are all holomorphic. Hence the transition functions of $Gr(k,\C^n)$ are holomorphic, so it is a complex manifold.

Equations on Riemann surfaces

Recall that a Riemann surface is a complex 1-manifold $M$ with a complex structure $\Sigma$ (a class of analytically equivalent atlases on $X$). Here we consider equations that relate connections and Higgs fields with solutions on Riemann surfaces. Let $G=SU(2)$ (complex 2-matrices with determinant 1) or $SO(3)$ (real orthogonal 3-matrices with determinant 1), $\theta$ a Higgs field over $M$, and  $P$ a principal $G$-bundle over $M$.

Definition: The curvature of a principal $G$-bundle $P$ is the map
\[
\begin{array}{r c l}
F_\nabla\ :\ \mathcal A^0_M(P) & \to & \mathcal A^2_M(P), \\
\omega s & \mapsto & (d_\nabla \circ \nabla)(\omega s),
\end{array}
\]
where the extension $d_\nabla:\mathcal A^k_M(P)\to \mathcal A^{k+1}_M(P)$ is defined by the Leibniz rule, that is $d_\nabla (\omega\otimes s) = (d\omega)\otimes s +(-1)^k\omega \wedge \nabla s$, for $\omega$ a $k$-form and $s$ a smooth section of $P$.

Since we may write $\mathcal A^1 = \mathcal A^{1,0}\oplus\mathcal  A^{0,1}$ as the sum of its holomorphic and anti-holomorphic parts, respectively (see post "Smooth projective varieties as Kähler manifolds," 2016-06-16), we may consider the restriction of $d_\nabla$ to either of these summands.

Definition: For a vector space $V$, define the Hodge star $*$ by
\[
\begin{array}{r c l}
*\ :\ \bigwedge^k(V^*) & \to & \bigwedge^{n-k}(V^*), \\
e^{i_1}\wedge \cdots \wedge e^{i_k} & \mapsto & e^{j_1}\wedge\cdots \wedge e^{j_{n-k}},
\end{array}
\]
so that $e^{i_1}\wedge\cdots \wedge e^{i_k}\wedge e^{j_1}\wedge\cdots \wedge e^{j_{n-k}} = e^1\wedge \cdots \wedge e^n$. Extend by linearity from the chosen basis.

The dual of the generalized connection $d_\nabla$ is written $d_\nabla^* = (-1)^{m+mk+1}*d_\nabla *$, where $\dim(M)=m$ and the argument of $d_\nabla^*$ is in $\mathcal A^k_M$ (this holds for manifolds $M$ that are not necessarily Riemann surfaces as well).

Now we may understand some equations on Riemann surfaces. They all deal with the connection $\nabla$, its generalization $d_\nabla$, its curvature $F_\nabla$, and the Higgs field $\theta$. Below we indicate their names and where they are mentioned (and described in further detail).
\begin{align*}
\text{Hitchin equations} && \left.d_\nabla\right|_{A^{0,1}}\theta & = 0 && [2],\ \text{Introduction}\\
&& F_\nabla + [\theta,\theta^*] & = 0\\[10pt]
\text{Yang-Mills equations} && d^*_\nabla d_\nabla \theta + *[*F_\nabla,\theta] & = 0 && [1],\ \text{Section 4} \\
&& d_\nabla^*\theta & = 0 \\[10pt]
\text{self-dual Yang-Mills equation} && F_\nabla  - *F_\nabla & = 0 && [2],\ \text{Section 1}\\[10pt]
\text{Yang-Mills-Higgs equations} && d_\nabla *F_\nabla + [\theta,d_\nabla \theta] & = 0 && [4],\ \text{equation (1)} \\
&& d_\nabla * d_\nabla \theta & = 0
\end{align*}

Recall the definitions of $\theta$ and $\theta*$ from a previous post ("Higgs fields of principal bundles," 2016-08-24). Now we look at these equations in more detail. The first of the Hitchin equations says that $\theta$ has no anti-holomorphic component, or in other words, that $\theta$ is holomorphic. In the second equation, the Lie bracket $[\cdot,\cdot]$ of the two 1-forms is
\begin{align*}
[\theta,\theta^*] & = \left[\textstyle\frac12f(dz+i\ dy), \frac12\bar f(dz - i\ dy) \right] \\
& = \textstyle -\frac i4f\bar f\ dx\wedge dy + \frac i4 f\bar f\ dy \wedge dx -\frac i4 f\bar f\ dx\wedge dy +\frac i4 f\bar f\ dy\wedge dx \\
& = -i|f|^2\ dx\wedge dy.
\end{align*}
In the Yang-Mills and Yang-Mills-Higgs equations, we can simplify some parts by noting that, for a section $s$ of the complexification of $P\times_\ad \mathfrak g$,
\begin{align*}
d_\nabla (\theta\otimes s) & = \textstyle \frac12d_\nabla (fdx\otimes s) + \frac i2 d_\nabla (fdy \otimes s) \\
& = \textstyle \frac12 (df\wedge dx \otimes s - fdx \wedge \nabla s) +\frac i2 (df\wedge dy - fdy \wedge \nabla s) \\
& = \left(\frac i2\frac{\dy f}{\dy x} - \frac 12 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s - \underbrace{\textstyle \frac12f(dx+idy)}_{\theta}\wedge \nabla s.
\end{align*}
The Hodge star of $\theta$ is $*\theta = \frac 12f(dy -idx)$, so
\begin{align*}
d_\nabla *(\theta\otimes s) & = \textstyle \frac12d_\nabla (fdy\otimes s) - \frac i2 d_\nabla (fdx \otimes s) \\
& = \textstyle \frac12 (df\wedge dy \otimes s - fdy \wedge \nabla s) -\frac i2 (df\wedge dx - fdx \wedge \nabla s) \\
& = \left(\frac 12\frac{\dy f}{\dy x} + \frac i2 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s + \underbrace{\textstyle \frac12f(idx-dy)}_{i\theta}\wedge \nabla s.
\end{align*}
We could express $\nabla s = (s_1dx + s_2dy)\otimes s^1$, but that would not be too enlightening. Next, note the self-dual Yang-Mills equation only makes sense over a (real) 4-dimensional space, since the degrees of the forms have to match up. In that case, with a basis $dz_1=dx_1+idy_1, dz_2 = dx_2+idy_2$ of $\mathcal A^1$, we have
\begin{align*}
F_\nabla & = F_{12} dx_1\wedge dy_1 + F_{13} dx_1 \wedge dx_2 + F_{14} dx_1\wedge dy_2 + F_{23} dy_1\wedge dx_2 + F_{24} dy_1\wedge dy_2 + F_{34} dx_2\wedge dy_2, \\
*F_\nabla & = F_{12} dx_2\wedge dy_2 - F_{13} dy_1 \wedge dy_2 + F_{14} dy_1\wedge dx_2 + F_{23} dx_1\wedge dy_2 - F_{24} dx_1\wedge dx_2 + F_{34} dx_1\wedge dy_1.
\end{align*}
Then the self-dual equation simply claims that
\[
F_{12} = F_{34}
\hspace{1cm},\hspace{1cm}
F_{13} = -F_{24}
\hspace{1cm},\hspace{1cm}
F_{14} = F_{23}.
\]

Remark: This title of this post promises to talk about equations on Riemann surfaces, yet all the differential forms are valued in a principal $G$-bundle over $\R^2$ (or $\R^4$). However, since the given equations are conformally invariant (this is not obvious), and as a Riemann surface locally looks like $\R^2$, we may consider the solutions to the equations as living on a Riemann surface.

References:
[1] Atiyah and Bott (The Yang-Mills equations over Riemann surfaces)
[2] Hitchin (Self-duality equations on a Riemann surface)
[3] Huybrechts (Complex Geometry, Chapter 4.3)
[4] Taubes (On the Yang-Mills-Higgs equations)

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)

Reconstructing a manifold from sample data, with noise

We follow the article [3] and add more background and clarifications. Some assumptions are made that are not explicitly mentioned in the article, to make calculations easier.

Background in probability, measure theory, topology

Let $X$ be a random variable over a space $A$. Recall that the expression $P(X)$ is a number in $[0,1]$ describing the probability of the event $X$ happening. This is called a probability distribution. Here we will consider continuous random variables, so $P(X=x)=0$ for any single element $x\in A$.

Definition: The probability density function of $X$ is the function $f:A\to \R$ satisfying

• $f(x)\geqslant 0$ for all $x\in A$, and
• $\int_B f(x)\ dx = P(X\in B)$ for any $B\subseteq A$.

The second condition implies $\int_A f(x)\ dx=1$.

Often authors use just $P$ instead of $f$, and write $P(x)$ instead of $P(X=x)$.

Definition: Let $Y=g(X)$ be another random variable. The expected value of $Y$ is
\[
E[Y] = E[g(X)] = \int_Ag(x)f(x)\ dx.
\]
The mean of $X$ is $\mu= E[X]$, and the variance of $X$ is $\sigma^2 = E[(X-\mu)^2]$. If $\vec X=(X_1\ \cdots\ X_n)^T$ is a multivariate random variable, then $\vec \mu=E[\vec X]$ is an $n$-vector, and the variance is an $(n\times n)$-matrix given as
\[
\Sigma = E[(\vec X-E[\vec X])(\vec X-E[\vec X])^T]
\hspace{1cm}
\text{or}
\hspace{1cm}
\Sigma_{ij} = E[(X_i-E[X_i])(X_j-E[X_j])].
\]
The covariance of $X$ and $Y$ is $E[(X-E[X])(Y-E[Y])]$. Note that the covariance of $X$ with itself is just the usual variance of $X$.

Example: One example of a probability distribution is the normal (or Gaussian) distribution, and we say a random variable with the normal distribution is normally distributed. If a random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, then the probability density function of $X$ is
\[
f(x) = \frac{\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)}{\sigma\sqrt{2\pi}}.
\]
If $\vec X=(X_1\ \cdots\ X_n)^T$ is a normally distributed multivariate random variable, then $\vec \mu = (E[X_1]\ \cdots\ E[X_n])^T$ and the probability density function of $\vec X$ is
\[
f(\vec x) = \frac{\exp\left(-\frac 12 (\vec x-\vec \mu)^T\Sigma^{-1}(\vec x-\vec \mu)\right)}{\sqrt{(2\pi)^n\det(\Sigma)}}.
\]

Definition: A measure on $\R^D$ is a function $m:\{$subsets of $\R^D\}\to [0,\infty]$ such that $m(\emptyset) = 0$ and $m(\bigcup_{i\in I} E_i) = \sum_{i\in I} m(E_i)$ for $\{E_i\}_{i\in I}$ a countable sequence of disjoint subsets of $\R^D$. A probability measure on $\R^D$ is a measure $m$ on $\R^D$ with the added condition that $m(\R^D)=1$.

A probability distribution is an example of a probability measure.

Definition: Let $U= \{U_i\}_{i\in I}$ be a covering of a topological space $M$. The nerve of the covering $U$ is a set $N$ of subsets of $I$ given by
\[
N = \left\{J\subset I\ :\ \bigcap_{j\in J} U_j \neq\emptyset\right\}.
\]
Note that this makes $N$ into an abstract simplicial complex, as $J\in N$ implies $J'\in N$ for all $J'\subseteq J$.

Let $M$ be a smooth compact submanifold of $\R^d$. By the tubular neighborhood theorem (see Theorem 2.11.4 in [3]), every smooth compact submanifold $M$ of $\R^d$ has a tubular neighborhood for some $\epsilon>0$.

Definition: For a particular embedding of $M$, let the condition number of $M$ be $\tau=\sup\{\epsilon\ :$ $M$ has an $\epsilon-$tubular neighborhood$\}$.

Distributions on a manifold

Let $M$ be a $d$-dimensional manifold embedded in $\R^D$, with $D>d$. Recall that every element in $NM\subseteq \R^D$, the normal bundle of $M$, may be represented as a pair $(\vec x,\vec y)$, where $\vec x\in M$ and $\vec y\in T^\perp$ (since $M$ is a manifold, all the normal spaces are isomorphic). Hence we may consider a probability distribution $P$ on $NM$, with $\vec X$ the $d$-multivariate random variable representing points on $M$ and $\vec Y$ the $(D-d)$-multivariate random variable representing points on the space normal to $M$ at a point on $M$. We make the assumption that $\vec X$ and $\vec Y$ are independent, or that
\[
P(\vec X, \vec Y) = P_M(\vec X)P_{T^\perp}(\vec Y).
\]
That is, $P_{T^\perp}$ is a probability distribution that is the same at any point on the manifold.

Definition: Let $P$ be a probability distribution on $NM$ and $f_M$ the probability density function of $P_M$. In the context described above, $P$ satisfies the strong variance condition if

• there exist $a,b>0$ such that $f_M(\vec x)\in [a,b]$ for all $\vec x\in M$, and
• $P_{T^\perp}(\vec Y)$ is normally distributed with $\vec \mu = 0$ and $\Sigma = \sigma^2I$.

The second condition implies that the covariance of $Y_i$ with $Y_j$ is trivial iff $i\neq j$, and that the vairance of all the $Y_i$s is the same. From the normally distributed multivariate example above, this also tells us that the probability density function $f^\perp$ of $\vec Y$ is
\[
f^\perp(\vec y) = \frac{\exp\left(\displaystyle-\frac{\sigma^2}{2}\sum_{i=1}^{D-d}y_i^2\right)}{\sigma^{D-d}\sqrt{(2\pi)^{D-d}}}.
\]

Theorem: In the context described above, let $P$ be a probability distribution on $NM$ satisfying the strong variance condition, and let $\delta>0$. If there is $c>1$ such that
\[
\sigma <\frac{c\tau(\sqrt9-\sqrt 8)}{9\sqrt{8(D-d)}},
\]
then there is an algorithm that computes the homology of $M$ from a random sample of $n$ points, with probability $1-\delta$. The number $n$ depends on $\tau,\delta,c,d,D$, and the diameter of $M$.

The homology computing algorithm

Below is a broad view of the algorithm described in sections 3, 4, and 5 of [1]. Let $M$ be a $d$-manifold embedded in $\R^D$, and $P$ a probability measure on $NM$ satisfying the strong variance condition.

1. Calculate the following numbers:
\begin{align*}
\tau & = \text{condition number of $M$}\\
\text{vol}(M) & = \text{volume of $M$}\\
\sigma^2 & = \text{variance of $P$}
\end{align*}
2. Define (or choose) the following numbers:

\begin{align*}
\delta & \in (0,1) \\
r & \in \left(2\sqrt{2(D-d)}\sigma,\textstyle\frac\tau9 (3-2\sqrt 2)\right) \\
n & > \text{function}(a,r,\tau,d,\delta,\text{vol}(M)) & (\max(A,B)\  \text{in Proposition 9 of [1])} \\
s & = 4r \\
deg & > \textstyle \frac{3a}4 \left(1-\left(\frac r{2\tau}\right)^2\right)^{d/2}\text{vol}\left(B^d(r,0)\right)\\
R & = (9r+\tau)/2
\end{align*}
3. Choose $n$ points randomly from $NM$ according to $P$.
4. From these $n$ points, construct the nearest neighbor graph $G$ with distance $s$.
5. Remove from $G$ all the vertices of degree $<deg$ to get a refined graph $G'$.
6. Set $U=\bigcup_{\vec x\in V(G')}B^D(R,\vec x)$ and construct the simplicial complex $K$ of its nerve.
7. Compute the homology of $K$, which is the homology of $M$, with probability $1-\delta$.

References:
[1] Niyogi, Smale, and Weinberger (A topological view of unsupervised learning from noisy data)
[2] Folland (Real analysis, Chapter 10.1)
[3] Bredon (Topology and Geometry, Chapter 2.11)