complex projective space

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###Context###
#### Topology
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[[!include topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Complex projective space $\mathbb{C}P^n$ is the [[projective space]] $\mathbb{A}P^n$ for $\mathbb{A} = \mathbb{C}$ being the [[complex numbers]] (and for $n \in \mathbb{N}$), a [[complex manifold]] of complex [[dimension]] $n$ (real dimension $2n$). Equivalently, this is the complex [[Grassmannian]] $Gr_1(\mathbb{C}^{n+1})$. For the special case $n = 1$ then $\mathbb{C}P^1 \simeq S^2$ is the [[Riemann sphere]].

As $n$ ranges, there are natural inclusions

$$
  \ast = \mathbb{C}P^0
    \hookrightarrow
  \mathbb{C}P^1
    \hookrightarrow
  \mathbb{C}P^2
    \hookrightarrow
  \mathbb{C}P^3
    \hookrightarrow
  \cdots
  \,.
$$

The [[sequential colimit]] over this sequence is the infinite complex projective space $\mathbb{C}P^\infty$. This is a model for the [[classifying space]] $B U(1)$ of [[circle principal bundles]]/[[complex line bundles]] (an [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$).

## Definition
 {#Definition}

+-- {: .num_defn #ComplexProjectiveSpace}
###### Definition

For $n \in \mathbb{N}$, then **complex $n$-dimensional complex projective space** is the [[complex manifold]] (often just regarded as its underlying [[topological space]]) defined as the [[quotient]]

$$
  \mathbb{C}P^n \coloneqq (\mathbb{C}^{n+1}-\{0\})/_\sim
$$

of the [[Cartesian product]] of $(n+1)$-copies of the [[complex plane]], with the origin removed, by the [[equivalence relation]]

$$
  (z \sim w) \Leftrightarrow (z = \kappa \cdot w)
$$

for some $\kappa \in \mathbb{C} - \{0\}$ and using the canonical multiplicative [[action]] of $\mathbb{C}$ on $\mathbb{C}^{n+1}$.

The canonical inclusions

$$
  \mathbb{C}^{n+1} \hookrightarrow \mathbb{C}^{n+2}
$$

induce canonical inclusions

$$
  \mathbb{C}P^n \hookrightarrow \mathbb{C}P^{n+1}
  \,.
$$

The [[sequential colimit]] over this sequence of inclusions is the _infinite complex projective space_

$$
  \mathbb{C}P^\infty
    \coloneqq
  \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n
  \,.
$$
=--

The following equivalent characterizations are immediate but useful:

+-- {: .num_prop #ComplexProjectiveSpaceAsGrassmannian}
###### Proposition

For $n \in \mathbb{N}$, we have that complex projective space, def. \ref{ComplexProjectiveSpace}, is equivalently the complex [[Grassmannian]]

$$
  \mathbb{C}P^n \simeq Gr_1(\mathbb{C}^{n+1})
  \,.
$$

=--

+-- {: .num_prop #ComplexProjectiveSpaceAsS1Quotient}
###### Proposition

For $n \in \mathbb{N}$ then complex projective space, def. \ref{ComplexProjectiveSpace}, is equivalently

1. the [[coset]]

   $$
     \mathbb{C}P^n \simeq U(n+1)/(U(n) \times U(1))
     \,,
   $$

1. the quotient of the [[n-sphere|(2n+1)-sphere]] by the [[circle group]] $S^1 \simeq \{ \kappa  \in \mathbb{C}| {\vert \kappa \vert} = 1\}$

  $$
    \mathbb{C}P^n \simeq S^{2n+1}/S^1
    \,.
  $$

=--

+-- {: .proof}
###### Proof

To see the second characterization from def. \ref{ComplexProjectiveSpace}:

With ${\vert -\vert} \colon \mathbb{C}^{n} \longrightarrow \mathbb{R}$ the standard [[norm]], we have that every element $\vec z \in \mathbb{C}^{n+1}$ is identified under the defining equivalence relation with

$$
  \frac{1}{\vert \vec z\vert}\vec z \in S^{2n+1} \hookrightarrow \mathbb{C}^{n+1}
$$

lying on the unit $(2n+1)$-sphere. This fixes the action of $\mathbb{C}-0$ up to a remaining action of complex numbers of unit [[absolute value]]. These form the [[circle group]] $S^1$. This shows that we have a [[commuting diagram]] of functions of underlying sets of the form

$$
  \array{
    S^{2n+1} &\hookrightarrow& \mathbb{C}^{n+1} \setminus \{0\}
    \\
    {}^{\mathllap{q_{S^{2n+1}}}}\downarrow &\searrow^{\mathrlap{f}}& \downarrow^{\mathrlap{q_{\mathbb{C}^{n+1}}}}
    \\
    S^{2n+1}/S^1 &\longrightarrow& \mathbb{C}P^n
  }
$$

where the top horizontal and the two vertical functions are [[continuous function|continuous]], and where the bottom function is a [[bijection]].  Since the diagonal [[composition|composite]] is also continuous, the nature of the [[quotient space topology]]
implies that the bottom function is also continuous. To see that it is a [[homeomorphism]] it hence remains to
see that it is an [[open map]] (by [this prop.](homeomorphism#HomeoContinuousOpenBijection)).

So let $U \subset S^{2n+1}/S^1$ be an open set, which means that $q_{S^{2n+1}}^{-1}(U) \subset S^{2n+1}$ is an open set.
We need to see that $f(q_{S^{2n+1}}^{-1}(U)) \subset \mathbb{C}P^{n}$ is open, hence that
$q_{\mathbb{C}^{n+1}}^{-1}(f(q_{S^{2n+1}}^{-1}(U))) \subset \mathbb{C}^{n+1}$ is open.
Now by the nature of the [[Euclidean space|Euclidean]] [[metric topology]], the open subset $q_{S^{2n+1}}^{-1}(U)$
is a union of open balls $B^\circ_x(\epsilon)$ in $\mathbb{C}^{n+1}$ intersected with $S^{2n+1}$. But then $q_{\mathbb{C}^{n+1}}^{-1}(f(B^\circ_x(\epsilon)\vert_{S^{2n+1}}))$
is their [[orbit]] under the multiplicative action by $\mathbb{C} \setminus \{0\}$,
hence is a [[cylinder]] $B^\circ_x(\epsilon)\vert_{S^{2n+1}} \times (\mathbb{C} \setminus \{0\})$. This is clearly open.

The first characterization follows via prop. \ref{ComplexProjectiveSpaceAsGrassmannian} from the general discusion at _[[Grassmannian]]_. With this the second characterization follows also with the [[coset]] identification of the $(2n+1)$-sphere: $S^{2n+1} \simeq U(n+1)/U(n)$ ([exmpl.](unitary+group#nSphereAsUnitaryCosetSpace)).

=--

## Examples

* $\mathbb{C}P^1$ is also known as the [[complex projective curve]] or [[Riemann sphere]]

* $\mathbb{C}P^2$ is also known as the [[complex projective plane]]

* $\mathbb{C}P^3$ is also known as [[complex projective 3-space]]

## Properties

### General

\begin{proposition}
Every continuous map $\mathbb{C}P^n\rightarrow\mathbb{C}P^n$ for $n$ even has a [[fixed point]]. This does not hold for $n$ odd as in this case the continuous map $\mathbb{C}P^n\rightarrow\mathbb{C}P^n, [x_0:x_1:\ldots:x_{n-1}:x_n]\mapsto[x_1:-x_0:\ldots:x_n:-x_{n-1}]$ does not have a fixed point.
\end{proposition}

([Hatcher 02, page 180](#Hatcher02))

\begin{proposition}
One has
$$
\operatorname{TC}(\mathbb{C}P^n)
=2n+1
$$
for the [[topological complexity]].
\end{proposition}

([Farber & Tabachnikov & Yuzvinsky 02,  Corollary 2](#FarberTabachnikovYuzvinsky02))

### Cell structure

+-- {: .num_prop #CellComplexStructureOnComplexProjectiveSpace}
###### Proposition
**([[cell structure of projective spaces]])**

There is a [[CW-complex]] structure on complex projective space $\mathbb{C}P^n$ (def. \ref{ComplexProjectiveSpace}) for $n \in \mathbb{N}$, given by [[induction]], where $\mathbb{C}P^{n+1}$ arises from $\mathbb{C}P^n$ by [[attaching space|attaching]] a single cell of dimension $2(n+1)$ with [[attaching map]] the [[projection]] $S^{2n+1} \longrightarrow \mathbb{C}P^n$ from prop. \ref{ComplexProjectiveSpaceAsS1Quotient}:

$$
  \array{
     S^{2n+1} &\longrightarrow& S^{2n+1}/S^1 \simeq \mathbb{C}P^n
     \\
     {}^{\mathllap{\iota_{2n+2}}}\downarrow^{\mathrlap{i_n}}
       &(po)& \downarrow
     \\
     D^{2n+2} &\underset{q}{\longrightarrow}& \mathbb{C}P^{n+1}
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

Given [[homogeneous coordinates]] $(z_1 ,  \cdots ,  z_n ,  z_{n+1} ,  z_{n+2}) \in \mathbb{C}^{n+2}$ for $\mathbb{C}P^{n+1}$, let

$$
  \phi \coloneqq -arg(z_{n+2})
$$

be the [[phase]] of $z_{n+2}$. Then under the equivalence relation defining $\mathbb{C}P^{n+1}$ these coordinates represent the same element as

$$
  \frac{1}{\vert \vec z\vert}(e^{i \phi} z_1, e^{i \phi}z_2,\cdots, e^{i \phi}z_{n+1}, r)
  \,,
$$

where

$$
  r = {\vert z_{n+2}\vert}\in [0,1] \subset \mathbb{C}
$$

is the [[absolute value]] of $z_{n+2}$. Representatives $\vec z'$ of this form (${\vert \vec z' \vert = 1}$ and $z'_{n+2} \in [0,1]$) parameterize the [[n-disk|2n+2-disk]] $D^{2n+2}$,  with [[boundary]] being the $(2n+1)$-sphere at $r = 0$.

The resulting function $q \colon D^{2n+2} \to \mathbb{C}P^{n+1}$ is continuous: It may be factored as

$$
  \array{
    q_{D^{2n+2}} \colon & D^{2n+2} &\overset{\phantom{AAA}}{\hookrightarrow}& \mathbb{C}^{n+2} \setminus \{0\} &\overset{q_{\mathbb{C}^{n+2}}}{\longrightarrow}&  \mathbb{C}P^{n+1}
    \\
    & (Re(z_1), Im(z_1),  \cdots, Re(z_{n+1}), Im(z_{n+1}), r) &\mapsto& (z_1, \cdots, z_{n+1}, r)  &\mapsto& [ z_1 : \cdots : z_{n+1} : r ]
  }
  \,.
$$

Here the first map is the [[embedding of topological spaces|embedding]] of the disk $D^{2n+2}$ as a [[hemisphere]] in $\mathbb{R}^{2n+1} \hookrightarrow \mathbb{R}^{2n+2} \simeq \mathbb{C}^{2n+2}$, while the second is the defining quotient space projection.
Both of these mare are evidently continuous, and hence so is their composite.

The only remaining part of the action of $\mathbb{C}-\{0\}$ which fixes the conditions ${\vert z'\vert} = 0$ and $z'_{n+2}$ is
$S^1 \subset \mathbb{C} \setminus \{0\}$ acting on the elements with $ r = \{z'_{n+2}\} = 0$ by phase shifts on the $z_0, \cdots, z_{n+1}$.
The quotient of this remaining action on $D^{2(n+1)}$ identifies its boundary $S^{2n+1}$-sphere with $\mathbb{C}P^{n}$, by prop. \ref{ComplexProjectiveSpaceAsS1Quotient}.

This shows that the above square is a [[pushout]] diagram of underlying sets.

By the nature of [[colimits]] in [[Top]] ([this prop.](Top#DescriptionOfLimitsAndColimitsInTop))
it remains to see that the
[[topological space|topology]] on $\mathbb{C}P^{n+1}$ is the [[final topology]] induced by the functions
$D^{2n+2} \to \mathbb{C}P^{n+1}$ and $\mathbb{C}P^n \to \mathbb{C}P^{n+1}$, hence that a subset of
$\mathbb{C}P^{n+1}$ is open precisely if its pre-images under these two functions are open.

We saw above that $q_{D^{2n+2}}$ is continuous.
Moreover, also the function
$i_n \colon \mathbb{C}P^n \to \mathbb{C}P^{n+1}$ is continuous (by [this lemma](projective+space#CanonicalInclusionOfProjectiveSpaces)).

This shows that if a subset of $\mathbb{C}P^{n+1}$ is open, then its pre-images under these functions are open.
It remains to see that if $S \subset \mathbb{C}P^{n+1}$ is a subset with $q_{S^{2n+2}}^{-1}(S) \subset D^{2n+2}$ open and
$i_n^{-1}(S) \subset \mathbb{C}P^n$ open, then $S \subset \mathbb{C}P^{n+1}$ is open.

Notice that $q_{\mathbb{C}^{n+2}}^{-1}(S)$ contains with every point also its [[orbit]] under the [[action]]
of $\mathbb{C} \setminus \{0\}$, and that every open subset of $D^{2n+2}$ is a unions of open balls.
By the above factorization of $q_{D^{2n+2}}$ this means that if $q_{D^{2n+2}}^{-1}(S)$ is open, then
$q_{\mathbb{C}^{n+2}}^{-1}(S)$ is a union of open cyclinders, hence is open. By the nature of the
[[quotient topology]], this means that $S \subset \mathbb{C}P^n$ is open.

=--

### Homotopy groups
  {#HomotopyGroups}

\begin{prop}
  For $n \geq 1$, the [[homotopy groups]] of [[complex projective space]] $\mathbb{C}P^n$ are the [[integers]] in degree 2, the [[homotopy groups of spheres|homotopy groups of]] the [[n-sphere|2n+1-sphere]] in degrees $\geq 2n+1$ and [[trivial group|trivial]] otherwise:
\[
  \label{ListOfHomotopyGroupsOfComplexProjectiveSpace}
    \pi_k
    \big(
      \mathbb{C}P^n
    \big)
    \;=\;
    \left\{
    \array{
      \ast &\vert& k = 0
      \\
      1 &\vert& k = 1
      \\
      \mathbb{Z} &\vert& k = 2
      \\
      \mathbb{Z} &\vert& k = 2n + 1
      \\
      \pi_k
      \big(
        S^{2n+1}
      \big)
      &\vert&
      k \geq 2n+1
      \\
      0 &\vert& otherwise
    }
    \right.
\]
\end{prop}

\begin{proof}
  Essentially by definition, $\mathbb{C}P^n$ is the [[quotient space]] of the [[circle group]]-[[action]] on the [[unit sphere]]  $S^{2n+1} \simeq S\big(\mathbb{C}^{2n+2}\big)$ (e.g. [Bott & Tu 1982, Exp. 14.22](#BottTu82)).

First of all, this implies that $\mathbb{C}P^n$ is [[connected topological space|connected]], since $S^{2n+1}$ is, hence $\pi_0\big( \mathbb{C}P^n\big) = \ast$.

Moreover, since this is a [[free action]], we have a [[circle principal bundle]] and hence a [[fiber sequence]] of the form

\begin{xymatrix@C=20pt@R=20pt}
  S^1
  \ar[r]
  &
  S^{2n+1}
  \ar[d]
  \\
  &
  \mathbb{C}P^n
  \,.
\end{xymatrix}

Therefore the homotopy groups of $\mathbb{C}P^n$ sit in the corresponding [[long exact sequence of homotopy groups]], hence in a [[long exact sequence]] of [[abelian groups]] of this form:

\begin{xymatrix@C=20pt@R=20pt}
  \cdots
  \ar[r]
  &
  \pi_{k+1}
  \big(
    \mathbb{C}P^k
  \big)
  \ar[r]
  &
  \pi_k
  \big(
    S^1
  \big)
  \ar[r]
  &
  \pi_k
  \big(
    S^{2n+1}
  \big)
  \ar[r]
  &
  \pi_k
  \big(
    \mathbb{C}P^n
  \big)
  \ar[r]
  &
  \pi_{k-1}
  \big(
    S^1
  \big)
  \ar[r]
  &
  \cdots
\end{xymatrix}

By the following two basic facts about the [[homotopy groups of spheres]]

$$
  \pi_{k \geq 2}
  \big(
    S^1
  \big)
  \;=\;
  0
  \,,
  \phantom{AAAA}
  \pi_{k \leq 2n}
  \big(
    S^{2n+1}
  \big)
  \;=\;
  0
$$

this long exact sequence contains the parts (using here the assumption that $n \geq 1$):

\begin{xymatrix@C=20pt@R=20pt}
  \pi_1
  \big(
    S^{2n+1}
  \big)
  \ar@{=}[d]
  \ar[r]
  &
  \pi_1
  \big(
    \mathbb{C}P^n
  \big)
  \ar@{=}[d]
  \ar[r]
  &
  \pi_0
  \big(
    S^1
  \big)
  \ar@{=}[d]
  \\
  1
  \ar@{=}[r]
  &
  \pi_1
  \big(
    \mathbb{C}P^n
  \big)
  \ar[r]
  &
  \ast
\end{xymatrix}

and

\begin{xymatrix@C=20pt@R=20pt}
  \pi_2
  \big(
    S^{2n+1}
  \big)
  \ar@{=}[d]
  \ar[r]
  &
  \pi_2
  \big(
    \mathbb{C}P^n
  \big)
  \ar@{=}[d]
  \ar[r]
  &
  \pi_1
  \big(
    S^1
  \big)
  \ar@{=}[d]
  \ar[r]
  &
  \pi_1
  \big(
    S^{2n+1}
  \big)
  \ar@{=}[d]
  \\
  0
  \ar[r]
  &
  \pi_2
  \big(
    \mathbb{C}P^n
  \big)
  \ar@{=}[r]
  &
  \mathbb{Z}
  \ar[r]
  &
  0
\end{xymatrix}

and these parts, for all $k \geq 3$:

\begin{xymatrix@C=20pt@R=20pt}
  \pi_{k}
  \big(
    S^1
  \big)
  \ar@{=}[d]
  \ar[r]
  &
  \pi_k
  \big(
    S^{2n+1}
  \big)
  \ar[r]
  \ar@{=}[d]
  &
  \pi_k
  \big(
    \mathbb{C}P^n
  \big)
  \ar[r]
  \ar@{=}[d]
  &
  \pi_{k-1}
  \big(
    S^1
  \big)
  \ar@{=}[d]
  \\
  0
  \ar[r]
  &
  \pi_k
  \big(
    S^{2n+1}
  \big)
  \ar@{=}[r]
  &
  \pi_k
  \big(
    \mathbb{C}P^n
  \big)
  \ar[r]
  &
  0
\end{xymatrix}

In all cases the identification in the bottom line follows from [[exact sequence|exactness]], given that the two outer items are [[trivial group|trivial]].

This gives the list (eq:ListOfHomotopyGroupsOfComplexProjectiveSpace), where we just made explicit that $\pi_{2n+1}(S^{2n+1}) = \mathbb{Z}$ (the [[Hopf degree theorem]], if you wish).
\end{proof}

### Homology and Cohomology
 {#Cohomology}

#### Ordinary
 {#OrdinaryHomologyAndCohomology}

+-- {: .num_prop #OrdinaryCohomologyOfComplexProjectiveSpace}
###### Proposition

For $A \in $ [[Ab]] any [[abelian group]], then the [[ordinary homology]] [[homology groups|groups]] of complex projective space $\mathbb{C}P^n$ with [[coefficients]] in $A$ are

$$
  H_k(\mathbb{C}P^n,A)\simeq
  \left\{
    \array{
       A & for \; k \;even\; and \; k \leq 2n
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

Similarly the [[ordinary cohomology]] [[cohomology groups|groups]] of $\mathbb{C}P^n$ are

$$
  H^k(\mathbb{C}P^n,A)
   \simeq
  \left\{
    \array{
       A & for \; k \;even\; and \; k \leq 2n
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

Moreover, if $A$ carries the structure of a [[ring]] $R = (A, \cdot)$, then under the [[cup product]] the [[cohomology ring]] of $\mathbb{C}P^n$ is the the [[graded ring]]

\[
  \label{CohomologyRing}
  H^\bullet(\mathbb{C}P^n, R)
   \simeq
  R[c_1] / (c_1^{n+1})
  \mathrlap{\,,}
\]

which is the [[quotient]] of the [[polynomial ring]] on a single generator $c_1$ in degree 2, by the relation that identifies [[cup products]] of more than $n$-copies of the generator $c_1$ with zero (see also the disfferential relation at *[[Sullivan model of complex projective space]]*).

Finally, the [[cohomology ring]] of the infinite-dimensional complex projective space is the [[formal power series ring]] in one generator:

$$
  H^\bullet(\mathbb{C}P^\infty, R)
   \simeq
  R[ [ c_1 ] ]
  \,.
$$

(Or else the [[polynomial ring]] $R[c_1]$, depending on how one chooses to extract a ring from a graded ring, see remark \ref{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}.)

=--

+-- {: .proof}
###### Proof

First consider the case that the coefficients are the [[integers]] $A = \mathbb{Z}$.

Since $\mathbb{C}P^n$ admits the structure of a [[CW-complex]] by prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, we may compute its [[ordinary homology]] equivalently as its [[cellular homology]] ([thm.](Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence)). By definition ([defn.](cellular+homology#CellularChainComplex)) this is the [[chain homology]] of the chain complex of [[relative homology]] groups

$$
  \cdots
    \overset{\partial_{cell}}{\longrightarrow}
  H_{q+2}((\mathbb{C}P^n)_{q+2}, (\mathbb{C}P^n)_{q+1})
   \overset{\partial_{cell}}{\longrightarrow}
  H_{q+1}((\mathbb{C}P^n)_{q+1}, (\mathbb{C}P^n)_{q})
   \overset{\partial_{cell}}{\longrightarrow}
  H_{q}((\mathbb{C}P^n)_{q}, (\mathbb{C}P^n)_{q-1})
    \overset{\partial_{cell}}{\longrightarrow}
  \cdots
  \,,
$$

where $(-)_q$ denotes the $q$th stage of the [[CW-complex]]-structure. Using the CW-complex structure provided by prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, then there are cells only in every second degree, so that

$$
  (\mathbb{C}P^n)_{2k+1} = (\mathbb{C}P)_{2k}
$$

for all $k \in \mathbb{N}$. It follows that the cellular chain complex has a zero group in every second degree, so that all differentials vanish. Finally, since prop. \ref{CellComplexStructureOnComplexProjectiveSpace} says that $(\mathbb{C}P^n)_{2k+2}$ arises from $(\mathbb{C}P^n)_{2k+1} = (\mathbb{C}P^n)_{2k}$ by attaching a single $2k+2$-cell it follows that (by passage to [[reduced homology]])

$$
  H_{2k}(\mathbb{C}P^n, \mathbb{Z})
    \simeq
  \tilde H_{2k}(S^{2k})((\mathbb{C}P^n)_{2k}/(\mathbb{C}P^n)_{2k-1})
   \simeq
  \tilde H_{2k}(S^{2k})
    \simeq
  \mathbb{Z}
  \,.
$$

This establishes the claim for ordinary homology with integer coefficients.

In particular this means that $H_q(\mathbb{C}P^n, \mathbb{Z})$ is a [[free abelian group]] for all $q$. Since free abelian groups are the [[projective objects]] in [[Ab]] ([prop.](projective+object#ProjectiveObjectsInAbAreFreeGroups)) it follows (with the discussion at _[[derived functors in homological algebra]]_) that the [[Ext]]-groups vanishe:

$$
  Ext^1(H_q(\mathbb{C}P^n, \mathbb{Z}),A) = 0
$$

and the [[Tor]]-groups vanishes:

$$
 Tor_1(H_q(\mathbb{C}P^n), A) = 0
  \,.
$$

With this, the statement about homology and cohomology groups with general coefficients follows with the [[universal coefficient theorem]] for ordinary homology ([thm.](universal+coefficient+theorem#TheoremInOrdinaryHomology)) and for ordinary cohomology ([thm.](universal+coefficient+theorem#OrdinaryStatementInCohomology)).

Finally to see the action of the [[cup product]]: by definition this is the composite

$$
  \cup
  \;\colon\;
  H^\bullet(\mathbb{C}P^n, R)
  \otimes
  H^\bullet(\mathbb{C}P^n, R)
    \longrightarrow
  H^\bullet(\mathbb{C}P^n \times \mathbb{C}P^n , R)
    \overset{\Delta^\ast}{\longrightarrow}
  H^\bullet(\mathbb{C}P^n,R)
$$

of the "cross-product" map that appears in the [[Kunneth theorem]], and the pullback along the [[diagonal]] $\Delta\colon \mathbb{C}P^n \to \mathbb{C}P^n \times \mathbb{C}P^n$.

Since, by the above, the groups $H^{2k}(\mathbb{C}P^n,R) \simeq R[2k]$ and $H^{2k+1}(\mathbb{C}P^n,R) = 0$ are free and finitely generated, the [[Kunneth theorem]] in ordinary cohomology applies ([prop.](Künneth+theorem#KunnethInOrdinaryCohomology)) and says that the cross-product map above is an isomorphism. This shows that under cup product pairs of generators are sent to a generator, and so the statement $H^\bullet(\mathbb{C}P^n , R)\simeq R[c_1](c_1^{n+1})$ follows.

This also implies that the projection maps

$$
  H^\bullet((\mathbb{C}P^\infty)_{2n+2}, R)
  =
  H^\bullet(\mathbb{C}P^{n+1}, R)
  \to
  H^\bullet(\mathbb{C}P^{n+}, R)
  =
  H^\bullet((\mathbb{C}P^\infty)_{2n}, R)
$$

are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] ([def.](lim1#MittagLefflerCondition), [exmpl.](lim1#MittagLefflerSatisfiedInParticularForTowerOfSurjections)) and therefore the [[Milnor exact sequence]] for cohomology ([prop.](lim1#MilorSequenceForReducedCohomologyOnCWComplex)) implies the last claim to be proven:

$$
  \begin{aligned}
    H^\bullet(\mathbb{C}P^\infty, R)
    &
     \simeq
    H^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n , R)
   \\
    &\simeq \underset{\longrightarrow}{\lim}_n H^\bullet(\mathbb{C}P^n, R)
   \\
   &\simeq \underset{\longrightarrow}{\lim}_n ( R [c_1^E] / ((c_1)^{n+1})  )
   \\
   & \simeq R[ [ c_1 ] ]
   \,,
  \end{aligned}
$$

where the last step is [this prop.](formal+scheme#FormalPowerSeries).

=--

+-- {: .num_remark #ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}
###### Remark

There is in general a choice to be made in interpreting the [[cohomology groups]] of a [[multiplicative cohomology theory]] $E$ as a [[ring]]:

a priori $E^\bullet(X)$ is a sequence

$$
  \{E^n(X)\}_{n \in \mathbb{Z}}
$$

of [[abelian groups]], together with a system of group homomorphisms

$$
  E^{n_1}(X) \otimes E^{n_2}(X)
    \longrightarrow
  E^{n_1 + n_2}(X)
  \,,
$$

one for each pair $(n_1,n_2) \in \mathbb{Z}\times\mathbb{Z}$.

In turning this into a single [[ring]] by forming [[formal sums]] of elements in the groups $E^n(X)$, there is in general the choice of whether allowing formal sums of only finitely many elements, or allowing arbitrary formal sums.

In the former case the ring obtained is the [[direct sum]]

$$
  \oplus_{n \in \mathbb{N}} E^n(X)
$$

while in the latter case it is the [[Cartesian product]]

$$
  \prod_{n \in \mathbb{N}} E^n (X)
  \,.
$$

These differ in general. For instance if $E$ is [[ordinary cohomology]] with [[integer]] [[coefficients]] and $X$ is complex projective space $\mathbb{C}P^\infty$, then (prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace})

$$
  E^n(X)
  =
  \left\{
    \array{
      \mathbb{Z} & n \; even
      \\
      0 & otherwise
    }
  \right.
$$

and the product operation is given by

$$
  E^{2{n_1}}(X)\otimes E^{2 n_2}(X)
    \longrightarrow
  E^{2(n_1 + n_2)}(X)
$$

for all $n_1, n_2$ (and zero in odd degrees, necessarily).  Now taking the [[direct sum]] of these, this is the [[polynomial ring]] on one generator (in degree 2)

$$
  \oplus_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z}[c_1]
  \,.
$$

But taking the [[Cartesian product]], then this is the [[formal power series ring]]

$$
  \prod_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z} [ [ c_1 ] ]
  \,.
$$

A priori both of these are sensible choices. The former is the usual choice in traditional [[algebraic topology]]. However, from the point of view of regarding [[ordinary cohomology]] theory as a [[multiplicative cohomology theory]] right away, then the second perspective tends to be more natural:

The cohomology of $\mathbb{C}P^\infty$ is naturally computed as the [[inverse limit]] of the cohomolgies of the $\mathbb{C}P^n$, each of which unambiguously has the ring structure $\mathbb{Z}[c_1]/((c_1)^{n+1})$. So we may naturally take the limit in the [[category]] of [[commutative rings]] right away, instead of first taking it in $\mathbb{Z}$-indexed sequences of abelian groups, and then looking for ring structure on the result. But the limit taken in the category of rings gives the [[formal power series ring]] (see [here](formal+scheme#FormalPowerSeries)).

Incidentally, this is the default choice of ring structure for [[generalized (Eilenberg-Steenrod) cohomology|generalized]] [[multiplicative cohomology theories]] evaluated on $\mathbb{C}P^\infty$. In particular in [[complex oriented cohomology]] (see there) this choice is of paramount importance.

See also for instance remark 1.1. in [[Jacob Lurie]]: _[[A Survey of Elliptic Cohomology]]_.

=--

#### Complex-oriented

+-- {: .num_prop #CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}
###### Proposition

Given a [[complex oriented cohomology theory]] $(E^\bullet, c^E_1)$ ([defn.](complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheory)), then there is an [[isomorphism]] of [[graded rings]]

$$
  E^\bullet(\mathbb{C}P^\infty) \simeq E^\bullet(\ast)[ [ c_1^E ] ]
$$

between the $E$-[[cohomology ring]] of infinite-dimensional complex projective space and the [[formal power series]] in one generator of even degree over the $E$-[[cohomology ring]] of the point.

=--

+-- {: .proof}
###### Proof

Using the [[CW-complex]]-structure on $\mathbb{C}P^\infty$ from prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$. With this structure, the unique 2-cell inclusion $i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty$ is identified with the canonical map $S^2 \to B U(1)$.

Then consider the [[Atiyah-Hirzebruch spectral sequence]] for the $E$-cohomology of $\mathbb{C}P^n$.

$$
  H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \;\Rightarrow\;
  E^\bullet(\mathbb{C}P^n)
  \,.
$$

Since, by prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace}, the [[ordinary cohomology]] with [[integer]] [[coefficients]] of projective space is

$$
  H^\bullet(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[c_1]/((c_1)^{n+1})
  \,,
$$

where $c_1$ represents a unit in $H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}$, and since similarly the [[ordinary homology]] of $\mathbb{C}P^n$ is a [[free abelian group]], hence a [[projective object]] in abelian groups ([prop.](projective+object#ProjectiveObjectsInAbAreFreeGroups)), the [[Ext]]-group vanishes in each degree ($Ext^1(H_n(\mathbb{C}P^n), E^\bullet(\ast)) = 0$) and so the [[universal coefficient theorem]] ([prop.](universal+coefficient+theorem#OrdinaryStatementInCohomology)) gives that the second page of the spectral sequence is

$$
  H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \simeq
  E^\bullet(\ast)[ c_1 ] / (c_1^{n+1})
  \,.
$$

By the standard construction of the [[Atiyah-Hirzebruch spectral sequence]] ([here](Atiyah–Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace)) in this identification the element $c_1$ is identified with a generator of the [[relative cohomology]]

$$
  E^2((\mathbb{C}P^n)_2, (\mathbb{C}P^n)_1)
  \simeq
  \tilde E^2(S^2)
$$

(using, by the above, that this $S^2$ is the unique 2-cell of $\mathbb{C}P^n$ in the standard cell model).

This means that $c_1$ is a permanent cocycle of the spectral sequence (in the kernel of all differentials) precisely if it arises via restriction from an element in $E^2(\mathbb{C}P^n)$ and hence precisely if there exists a complex orientation $c_1^E$ on $E$. Since this is the case by assumption on $E$, $c_1$ is a permanent cocycle. (For the fully detailed argument see ([Pedrotti 16](#Pedrotti16)).)

The same argument applied to all elements in $E^\bullet(\ast)[c]$, or else the $E^\bullet(\ast)$-linearity of the differentials ([prop.](multiplicative+cohomology+theory#LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory)), implies that all these elements are permanent cocycles.

Since the AHSS of a multiplicative cohomology theory is a [[multiplicative spectral sequence]] ([prop.](multiplicative+cohomology+theory#AHSSForMultiplicativeCohomologyTheoryIsMultiplicative)) this implies that the differentials in fact vanish on all elements of $E^\bullet(\ast) [c_1] / (c_1^{n+1})$, hence that the given AHSS collapses on the second page to give

$$
  \mathcal{E}_\infty^{\bullet,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1^{E} ] / ((c_1^E)^{n+1})
$$

or in more detail:

$$
  \mathcal{E}_\infty^{p,\bullet}
  \simeq
  \left\{
    \array{
      E^\bullet(\ast) & \text{if}\; p \leq  2n \; and\; even
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

Moreover, since therefore all $\mathcal{E}_\infty^{p,\bullet}$ are [[free modules]] over $E^\bullet(\ast)$, and since the filter stage inclusions $F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X)$ are $E^\bullet(\ast)$-[[module]] homomorphisms ([prop.](multiplicative+cohomology+theory#RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory)) the [extension problem](spectral+sequence#ExtensionProblem) trivializes, in that all the [[short exact sequences]]

$$
  0 \to F^{p+1}E^{p+\bullet}(X) \longrightarrow F^{p}E^{p+\bullet}(X) \longrightarrow \mathcal{E}_\infty^{p,\bullet} \to 0
$$

[[split exact sequence|split]] (since the [[Ext]]-group $Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0$ vanishes on the [[free module]], hence [[projective module]] $\mathcal{E}_\infty^{p,\bullet}$).

In conclusion, this gives an isomorphism of graded rings

$$
  E^\bullet(\mathbb{C}P^n)
  \simeq
  \underset{p}{\oplus} \mathcal{E}_\infty^{p,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1 ] / ((c_1^{E})^{n+1})
  \,.
$$

A first consequence is that the projection maps

$$
  E^\bullet((\mathbb{C}P^\infty)_{2n+2})
  =
  E^\bullet(\mathbb{C}P^{n+1})
  \to
  E^\bullet(\mathbb{C}P^{n+})
  =
  E^\bullet((\mathbb{C}P^\infty)_{2n})
$$

are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] ([def.](lim1#MittagLefflerCondition), [exmpl.](lim1#MittagLefflerSatisfiedInParticularForTowerOfSurjections)) and therefore the [[Milnor exact sequence]] for generalized cohomology ([prop.](lim1#MilorSequenceForReducedCohomologyOnCWComplex)) finally implies the claim:

$$
  \begin{aligned}
    E^\bullet(B U(1))
     & \simeq
    E^\bullet(\mathbb{C}P^\infty)
    \\
    & \simeq
    E^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n )
   \\
    &\simeq \underset{\longrightarrow}{\lim}_n E^\bullet(\mathbb{C}P^n)
   \\
   &\simeq \underset{\longrightarrow}{\lim}_n ( E^\bullet(\ast) [c_1^E] / ((c_1^E)^{n+1})  )
   \\
   & \simeq E^\bullet(\ast)[ [ c_1^E ] ]
   \,,
  \end{aligned}
$$

where the last step is [this prop.](formal+scheme#FormalPowerSeries).

=--

### Rational homotopy type / Sullivan model
  {#SullivanModel}

The [[Sullivan model of complex projective space]] $\mathbb{C}P^n$ is

$$
  \mathbb{R}
  [
    f_2, h_{2n+1}
  ]
  \big/
  \left(
  \begin{aligned}
    d\,f_2 & = 0
    \\
    d\,h_{2n+1} & = (f_2)^{n+1}
  \end{aligned}
  \right)
$$

(e.g. [Félix-Halperin-Thomas 00, p. 203](Sullivan+model+of+complex+projective+space#FelixHalperinThomas00), [Menichi 13, 5.3](Sullivan+model+of+complex+projective+space#Menichi13))

### Relation to Oka theory

\begin{prop}\label{ComplexProjectiveSpaceIsOkaManifold}
**([[complex projective space]] is [[Oka manifold]])** \linebreak
Every [[complex projective space]] $\mathbb{C}P^n$, $n \in \mathbb{N}$, is an [[Oka manifold]]. More generally every [[Grassmannian]] over the [[complex numbers]] is an Oka manifold.
\end{prop}
(review in [Forstnerič & Lárusson 11, p. 9](Oka+manifold#ForstnericLarusson11), [Forstnerič 2013, Ex. 2.7](Oka+manifold#Forstneric13))

### Relation to topological K-theory
 {#RelationToTopologicalKtheory}

Write $\Sigma^\infty (\mathbb{C}P^\infty)_+ \in Ho(Spectra)$ for the [[H-group ring spectrum]] of $\mathbb{C}P^\infty \simeq B U(1)$ (see there for details).

For $X \in Ho(Top)$ the [[homotopy type]] of a [[topological space]] in the [[classical homotopy category]], write

$$
  [\Sigma^\infty X_+ , \Sigma^\infty, \mathbb{C}P^\infty_+]
   \simeq
  [X, \Omega^\infty \Sigma^\infty \mathbb{C}P^\infty_+]
  \in Ab
$$

for the [[hom-group]] in the [[stable homotopy category]], which, by [[adjunction]], is equivalently computed in the [[classical homotopy category]] as shown on the right.

Write

$$
  i \;\colon\; \mathbb{C}P^\infty \simeq B U(1) \simeq B U(1) \times \{1\}
 \hookrightarrow B U \times \mathbb{Z}
$$

for the inclusion into the [[classifying space]] for complex [[topological K-theory]] which classifies the inlusion of [[complex line bundles]] $E$ as  [[virtual vector bundles]] $[E] - 0$.

+-- {: .num_prop #HGroupRingSpectrumSurjectsOntoTopologicalKTheory}
###### Proposition

For any $X \in Ho(Top)$ the [[ring homomorphism]]

$$
  i_\ast \;\colon\;
  [\Sigma^\infty X_+, \Sigma^\infty \mathbb{C}P_+]
    \longrightarrow
  K_{\mathbb{C}}(X)
$$

to [[topological K-theory]] is [[surjective function|surjective]].

=--

This is due to ([Segal 73, prop. 1](#Segal73)).

Prop. \ref{HGroupRingSpectrumSurjectsOntoTopologicalKTheory} is sharpened by _[[Snaith's theorem]]_. See there for more. The version for [[real projective space]] is called the _[[Kahn-Priddy theorem]]_.

## Related concepts

* [[real projective space]]

* [[complex oriented cohomology]]

* [[CP^N sigma-model|$\mathbb{C}P^N$-sigma model]]

* [[classifying space]]

* [[Fubini-Study metric]]

* [[projective G-space]]

  [[infinite complex projective G-space]]

* [[symplectic formulation of quantum mechanics]]

## References

Textbook accounts:

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Exp. 14.22 of: _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0))

In the context of [[quantum state spaces]]:

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], Chapter 4 of: *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048)]

See also

* [[Topospaces]], _[Complex projective space](http://topospaces.subwiki.org/wiki/Complex_projective_space)_

* Wikipedia, _[Complex projective space](https://en.wikipedia.org
/wiki/Complex_projective_space)_

[[Topological complexity]] of complex projective space:

* {#FarberTabachnikovYuzvinsky02} [[Michael Farber]], [[Serge Tabachnikov]], [[Sergey Yuzvinsky]], _Topological robotics: motion planning in projective spaces_ (2002), [arXiv:math/0210018](https://arxiv.org/abs/math/0210018);

Computation of [[Cohomotopy]]-sets of complex projective spaces:

* {#West71} Robert West, _Some Cohomotopy of Projective Space_, Indiana University Mathematics Journal Indiana University Mathematics Journal Vol. 20, No. 9 (March, 1971), pp. 807-827 ([jstor:24890146](https://www.jstor.org/stable/24890146))

Computation of the [[stable homotopy groups]] of $\mathbb{C}P^\infty$:

* [[Robert E. Mosher]],  *Some stable homotopy of complex projective space*, Topology **7** (1968) 179-193 [<a href="https://doi.org/10.1016/0040-9383(68)90026-8">doi:10.1016/0040-9383(68)90026-8</a>, [pdf](https://core.ac.uk/download/pdf/82547916.pdf)]

* {#Segal73} [[Graeme Segal]], _The stable homotopy of complex of projective space_, The quarterly journal of mathematics (1973) 24 (1): 1-5. ([[Segal72.pdf:file]], [doi:10.1093/qmath/24.1.1]( https://doi.org/10.1093/qmath/24.1.1))

(See also at _[[Snaith's theorem]]_.)

Detailed review of the [[Atiyah-Hirzebruch spectral sequence]] for [[complex oriented cohomology]] is in

* {#Pedrotti16} [[Riccardo Pedrotti]], _Complex oriented cohomology, generalized orientation and Thom isomorphism_, 2016, 2018 ([[PedrotticECohomology2018.pdf:file]])

Computation of the [[ordinary cohomology]] of [[cyclic loop spaces]] of complex projective spaces (their *[[string cohomology]]*):

* [[Marcel Bökstedt]], [[Iver Ottosen]], *String cohomology groups of complex projective spaces*, Algebr. Geom. Topol. 7(4): 2165-2238 (2007). ([arXiv:math/0605754](https://arxiv.org/abs/math/0605754), [doi:10.2140/agt.2007.7.2165](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-7/issue-4/String-cohomology-groups-of-complex-projective-spaces/10.2140/agt.2007.7.2165.full))

[[!redirects complex projective spaces]]

[[!redirects complex projective n-space]]
[[!redirects complex projective n-spaces]]

[[!redirects infinite complex projective space]]
[[!redirects infinite complex projective spaces]]


complex structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Complex structure on vector spaces
 {#ComplexStructureOnAVectorSpace}

\begin{definition}
\label{LinearComplexStructure}
Given a [[ground field]] $\mathbb{K}$,
a _complex structure_ on a $\mathbb{K}$-[[vector space]] $V$ is a $\mathbb{K}$-[[linear map|linear]] [[automorphism]]
$$
  J
    \,\colon\,
  V \longrightarrow V
$$
that squares to minus the [[identity morphism|identity]]:
$$
  J \circ J = - Id_V
  \,.
$$
\end{definition}

The idea here is that $J$ acts by multiplication with the would-be [[imaginary unit]] $\mathrm{i}$, and indeed: A [[real vector space]] ($\mathbb{K}$ the [[real numbers]]) equipped with a complex structure $J$ as in Def. \ref{LinearComplexStructure} becomes a [[complex vector space]] by declaring that $\mathrm{i}$ acts via $J$:
$$
  \array{
    \mathbb{C} \times V &\longrightarrow& V
    \\
    \big(
      x + \mathrm{i} \cdot y
      ,\;
      v
    \big)
    &\mapsto&
    x \cdot v + y \cdot J(v)
  }
$$

Here a real-[[linear map]] $\phi \colon V \to V$ is complex-linear iff it commutes with the complex structure in that:
$$
  \phi \circ J \,=\, J \circ \phi
  \,.
$$

### Complex structure on manifolds

More generally, an almost complex structure on a [[smooth manifold]] is a smoothly varying [[fiber]]wise complex structure in the sense of Def. \ref{LinearComplexStructure} on its [[tangent spaces]] (which *a priori* are [[real vector spaces]]):

+-- {: .num_defn}
###### Definition

An _almost complex structure_ on a [[smooth manifold]] $X$ (of even [[dimension]]) is a rank $(1,1)$-[[tensor field]] $J$, hence a smooth [[section]] $J \in \Gamma(T X \otimes T^* X)$, such that, over each point $x \in X$, $J$ is a linear complex structure, def. \ref{LinearComplexStructure}, on that [[tangent space]] $T_x X$ under the canonical identification $End T_x X \simeq T_x X\otimes T_x^* X$.

=--

Equivalently, stated more intrinsically:

+-- {: .num_defn}
###### Definition

An _almost complex structure_ on a [[smooth manifold]] $X$ of [[dimension]] $2 n$ is a [[reduction of the structure group]] of the [[tangent bundle]] to the [[complex numbers|complex]] [[general linear group]] along $GL(n,\mathbb{C}) \hookrightarrow GL(2n,\mathbb{R})$.

=--

+-- {: .num_remark #InTermsOfSmoothStacks}
###### Remark

In terms of modulating maps of bundles into their [[smooth infinity-groupoid|smooth]] [[moduli stacks]], this means that an almost complex structure is a lift in the following [[diagram]] in [[Smooth∞Grpd]]:

$$
  \array{
    && \mathbf{B} GL(n,\mathbb{C})
    \\
    & {}^{\mathllap{alm.compl.str.}}\nearrow & \downarrow^{\mathrlap{}}
    \\
    X
    &\underoverset{\tau}{tang.\,bund.}{\to}&
    \mathbf{B} GL(2n,\mathbb{R})
  }
  \,.
$$

By further [[reduction of the structure group|reduction]] along the [[maximal compact subgroup]] inclusion of the [[unitary group]] this yields an [[almost Hermitian structure]]

$$
  \array{
    && \mathbf{B} U(n)
    \\
    & {}^{\mathllap{herm.alm.compl.str.}}\nearrow & \downarrow^{\mathrlap{}}
    \\
    X
    &\underoverset{\tau}{tang.\,bund.}{\to}&
    \mathbf{B} GL(2n,\mathbb{R})
  }
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

A _complex structure_ on a [[smooth manifold]] $X$ is the structure of a [[complex manifold]] on $X$. Every such defines  an almost complex structure and almost complex structures arising this way are called _integrable_  (see also at _[[integrability of G-structures]]_ the section _[Examples -- Complex structure](integrability+of+G-structures#ExampleComplexStructure)_).

=--

## Properties

### Characterizations of integrability

The _[[Newlander-Nirenberg theorem]]_ states that an almost complex structure $J$ on a smooth manifold is integrable (see also at _[[integrability of G-structures]]_) precisely if its [[Nijenhuis tensor]] vanishes, $N_J = 0$.

See also at _[[integrability of G-structures]]_ the section _[Examples -- Complex structure](integrability+of+G-structures#ExampleComplexStructure)_.

### On 2-dimensional manifolds

+-- {: .num_prop}
###### Proposition

Every [[Riemannian metric]] on an [[orientation|oriented]] 2-[[dimension|dimensional]] [[manifold]] induces an almost complex structure given by forming orthogonal tangent vectors.

=--

+-- {: .num_prop #In2dAnyAlmostComplexStructureIsIntegrable}
###### Proposition

Every almost complex structure on a 2-[[dimension|dimensional]] [[manifold]] is integrable, hence is a complex structure.

=--

In the special case of [[real analytic manifolds]] this fact was known to [[Carl Friedrich Gauss]]. For the general case see for instance [Audin, remark 3 on p. 47](#Audin).

### Relation to $Spin^c$-structures

Every almost complex structure canonically induces a [[spin^c-structure]] by postcomposition with the universal [[characteristic map]] $\phi$ in the [[diagram]]

$$
  \array{
    \mathbf{B}U(n) &\stackrel{\phi}{\to}& \mathbf{B}Spin^c &\to& \mathbf{B}U(1)
    \\
    &\searrow& \downarrow && \downarrow^{\mathrlap{}}
    \\
    && \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
$$

See at _[[spin^c-structure]]_ for more.

### Relation to Hermitian and Kähler structure

* An almost complex structure equipped with a compatible [[Riemannian metric]] is a _[[Hermitian structure]]_.

* An almost complex structure equipped with a compatible [[Riemannian structure]] and [[symplectic structure]] is a _[[Kähler structure]]_.

| [[complex structure]] | + [[Riemannian structure]] | + [[symplectic structure]] |
|--|--|--|
[[complex structure]] | [[Hermitian structure]] | [[Kähler structure]] |

### Moduli stacks of complex structures

One may consider the [[moduli stack of complex structures]] on a given manifold. For 2-dimensional manifolds these are famous as the Riemann [[moduli stacks of complex curves]]. They may also be expressed as moduli stacks of almost complex structures, see [here](moduli+space+of+curves#OverTheComplexNumbers).

## Related concepts

* [[stable complex structure]]

* [[real structure]]

* [[quaternionic structure]]

* [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]

* [[para-complex structure]]

## References

### Complex structure on vector spaces

Textbook accounts:

* [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], p. 114 in: _Foundations of differential geometry_, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprinted by Wiley Classics Library (1996) [[ISBN:978-0-470-55558-3](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+2+Volume+Set-p-9780470555583), [Wikipedia entry](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry)]

See also:

* Wikipedia, *[Linear complex structure](https://en.wikipedia.org/wiki/Linear_complex_structure)*

Fiberwise complex structure on [[real vector bundles]]:

* Emery Thomas, *Complex Structures on Real Vector Bundles*, American Journal of Mathematics **89** 4 (1967) 887-908 [[arXiv;10.2307/2373409](https://doi.org/10.2307/2373409), [jstor:2373409](https://www.jstor.org/stable/2373409)]

### Complex structure on manifolds

Lecture notes include

* {#Audin} Michèle Audin, _Symplectic and almost complex manifolds_ ([pdf](http://www-irma.u-strasbg.fr/~maudin/HolomorphicChapII.pdf))

Discussion from the point of view of [[integrable G-structures]] includes

* [[Robert Bryant]], _Remarks on the geometry of almost complex 6-manifolds_, The Asian Journal of Mathematics, vol. 10 no. 3 (September, 2006), pp. 561--606. ([arXiv:math/0508428](http://arxiv.org/abs/math/0508428))

A discussion of deformations of complex structures is in

* [[Domenico Fiorenza]], _[[domenicofiorenza:The periods map of a complex projective manifold. An oo-category perspective]]_

The _[[moduli space of complex structures]]_ on a manifold is discussed for instance from page 175 on of

* [[Yongbin Ruan]], _Symplectic topology and complex surfaces_ in _Geometry and analysis on complex manifolds_ (1994)

and in

* Yurii M. Burman, _Relative moduli spaces of complex structures: an example_ ([arXiv:math/9903029](http://arxiv.org/abs/math/9903029))

[[!redirects complex structures]]

[[!redirects almost complex structure]]
[[!redirects almost complex structures]]

[[!redirects almost complex manifold]]
[[!redirects almost complex manifolds]]

[[!redirects linear complex structure]]
[[!redirects linear complex structures]]


complex supermanifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are two different things that one might mean by a "complex supermanifold", and the term is in fact used for two different notions in the literature (the terminology is a mess!):

1. In the first sense, a complex supermanifold generalizes the notion of a smooth manifold with its sheaf of smooth complex-valued functions, just as an
ordinary [[supermanifold]] is a generalization of an ordinary manifold with its sheaf
of smooth real-valued functions.
However, considering ordinary smooth manifolds as ringed spaces with either their sheaves of real or complex smooth functions gives two equivalent categories, whereas this is not true in the case of real and complex supermanifolds; the corresponding functor is neither essentially surjective nor fully faithful.
(For $X$ a complex supermanifold in this sense, the underlying reduced manifold $X_{red}$ is **not** a [[complex manifold]] but just a [[smooth manifold]] regarded as a [[ringed space]] with [[structure sheaf]] taken to be the sheaf of $\mathbb{C}$-valued smooth functions on the ordinary real manifold.)

1. In the second sense, a complex supermanifold is a super(complex manifold), a super-version of [[complex manifold]].

## complex {super manifold}

A **complex supermanifold** is a [[ringed space]] $X = (|X|, O_X)$ such that

* the [[structure sheaf]] $O_X$ a [[sheaf]] of commutative complex [[super algebra]]s

*  locally $O_X$ is [[isomorphism|isomorphic]] to $
  C^\infty(\mathbb{R}^d) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$

Write [[cSDiff]] for the [[category]] of complex supermanifolds.

**Example**
The functor $ \Pi : \{real vector bundles\} \to SDiff$
has a complex analogue $ \Pi : \{complex vector bundles\} \to cSDiff $.

Let $E \to X$ be a complex [[vector bundle]] of rank $\delta$.
This gives rise to the complex supermanifold $\Pi E$, in the [[supermanifold|same way]] as a real vector bundle gives rise to a real supermanifold:
the structure sheaf is given by sections of the exterior algebra
of the dual of $E$.

**Remark**

$C^\infty(X) := O_X(X)$ does not in general have a $\mathbb{C}$-antilinear involution  $\bar{-} : C^\infty(X) \to C^\infty(X)$ but there does exist a canonical complex conjugation on the quotient $C^\infty(X)$ by the ideal of nilpotent sections, which is $C^\infty(X_{red}; \mathbb{C})$.
So on a complex supermanifold we have complex conjugation only on the reduced manifold.

As for ordinary [[supermanifold]]s (and with same proof as in the real case)
we have the following two statements:

**Theorem**

1. Every complex supermanifold is isomorphic to one of the form $\Pi E$.

1. $ cSDiff(X,Y) \simeq  ComplexSuperAlg(C^\infty(Y), C^\infty(X)). $

**Remark**
It turns out that a $\mathbb{C}$-super algebra homomorphism
$
  \phi : C^\infty(Y) \to C^\infty(X)
$
automatically satisfies $\phi_{red}(\overline{f_{red}}) = \overline{\phi_{red}(f_{red})}$.

Define the complex supermanifold $\mathbb{R}_{cs}^{d|\delta}$  as $\mathbb{R}^d$
with structure sheaf $U\mapsto C^\infty(U) \otimes_{\mathbb{C}} \wedge^\bullet \mathbb{C}^\delta$.

Then for $S$ an arbitrary complex supermanifold we have

$$
  \mathbb{R}_{cs}^{d|\delta}(S)
  =
   cSDiff(S, \mathbb{R}_{cs}^{d|\delta})
  =
  \{
    (x_1, \cdots, x_d, \theta_1, \cdots, \theta_{\delta})|
    x_i \in C^\infty(S)^{ev} ,
    \theta_j \in C^\infty(S)^{odd};
    with x_i real in that \overline{(x_i)_{red}} = (x_i)_{red}
  \}
$$

**Example**

For

$$
  \mathbb{R}^{2|1}(S)
  =
  \{
    (x,y,\theta) |
    x,y \in C^\infty(S)^{ev},
    \theta \in C^\infty(S)^{odd}; x,y real
  \}
$$

we shall write

$$
  \simeq \{
     (z,\bar z, \theta)
     | z, \bar z \in C^\infty(S)^{ev}; \overline{z_{red}}=(\overline z )_{red}
  \}.
$$

## Related concepts

* [[super Riemann surface]]

[[!redirects complex supermanifolds]]


complex surface

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## Idea

A _complex surface_ is a _[[surface]]_ in [[complex analytic geometry]], hence a [[complex analytic space]] of complex [[dimension]] 2 (hence of real dimension 4).

## Examples

* [[K3 surface]]

## Related concepts

* [[algebraic surface]]

* [[complex manifold]], [[almost complex manifold]]

  [[complex surface]], [[complex curve]]

  [[Kähler manifold]], [[hyper-Kähler manifold]]

  [[Calabi-Yau manifold]]

## References

* {#FriedmanMorgan94} [[Robert Friedman]], [[John Morgan]], _Smooth Four-Manifolds and Complex Surfaces_, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (1994) ([doi:10.1007/978-3-662-03028-8](https://www.springer.com/gp/book/9783540570585))

See also

* Wikipedia, _[Complex surfaces](https://en.wikipedia.org/wiki/Category:Complex_surfaces)_

Discussion of patterns in the [[characteristic classes]] ([[Chern classes]]) of complex surfaces, together with some speculations about an anlogy with [[atomic number]] of [[atomic nuclei]] is in

* {#AtiyahManton16} [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_ ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816))

[[!redirects complex surfaces]]


complex system

[[!redirects complex systems]]

## References

* Wikipedia, _[Complex system](http://en.wikipedia.org/wiki/Complex_system)_


complex tone

\tableofcontents

\section{Idea}

In [[acoustics]], [[signal processing]], and [[music]], a **complex tone** is a [[sound]] or [[signal]] whose [[waveform]] is a [[superposition]] of multiple [[sine function|sinusoidal]] [[waveforms]].

\section{See also}

* [[superposition]]

* [[overtone series]]

* [[sine function]]

* [[pure tone]]

\section{References}

See also:

* Wikipedia, _[Pure tone](https://en.wikipedia.org/wiki/Pure_tone)_

* Wikipedia, _[Overtone series](https://en.wikipedia.org/wiki/Overtone_series)_

* Wikipedia, _[Musical tone](https://en.wikipedia.org/wiki/Musical_tone)_


complex torus

* [[complex manifold|complex]] [[torus]]

* [[elliptic curve]] over the [[complex numbers]]

## Examples

* [[intermediate Jacobian]]

[[!redirects complex tori]]


complex variety

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## Idea

A [[variety]] over the [[complex numbers]].

## Related concepts

* [[complex analytic variety]]

* [[Hodge cohomology]]

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complex vector bundle

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## Definition

A _complex vector bundle_ is a [[vector bundle]] whose fibers are [[complex vector spaces]].

A complex vector bundle with complex 1-dimensional [[fibers]] is a [[complex line bundle]].

More precisely, a __complex vector bundle__ is a [[real vector bundle]] $E\to M$ together with a lifting of its $\mathbf{R}$-module structure $\mathbf{R}\to End(E)$ to a homomorphism of $\mathbf{R}$-algebras $\mathbf{C}\to End(E)$.

The latter lifting is also known as a __complex structure__ on a [[real vector bundle]].

In terms of [[cocycles]], complex vector bundles can be described using cocycle data where the transition maps are complex-linear maps.

## Connections to holomorphic vector bundles

Any [[holomorphic vector bundle]] over a [[complex manifold]] has an underlying complex vector bundle.

Conversely, given a complex vector bundle over a [[complex manifold]], if its transition maps are holomorphic, then it is a [[holomorphic vector bundle]].

## Properties

### Oka-Grauert principle

The _[[Oka-Grauert principle]]_ states that for any [[Stein manifold]] $X$ the holomorphic and the topological classification of [[complex vector bundles]] on $X$ coincide. The original reference is ([Grauert 58](#Grauert58)).

### Relation to holomorphic vector bundles

See at _[[Koszul-Malgrange theorem]]_.

## Related concepts

* [[complex vector space]]

  * [[complex line]]

* **complex vector bundle

  * [[universal complex vector bundle]]

  * [[stable vector bundle]]

  * [[complex line bundle]]

  * [[holomorphic vector bundle]]

  * [[complex oriented cohomology theory]]

* [[real vector bundle]]

* [[quaternionic vector bundle]]

## References

* Emery Thomas, _Complex structures on real vector bundles_ ([JSTOR]( http://www.jstor.org/stable/2373409))

In the context of [[GAGA]]:

* {#Demailly09} [[Jean-Pierre Demailly]], section 3 of _Analytic methods in algebraic geometry_, lecture notes 2009  ([pdf](http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth.pdf))
[[!redirects complex vector bundles]]


complex vector space

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## Definition

A [[vector space]]/[[module]] over the [[field]] of [[complex numbers]].

## Related concepts

* [[rational vector space]]

* [[real vector space]]

* [[real structure]], [[complex structure]]

* [[antilinear map]]

* [[complexification]]

* [[complex cohomology]]

* [[complex vector bundle]]

* [[complex line bundle]]

[[!redirects complex vector spaces]]


complex volume

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## Idea

The _Cheeger-Simons classes_ are complexified [[secondary invariants]].

Under identifying the [[fundamental class]] of a [[hyperbolic 3-manifold]] $X$ as an element in the [[Bloch group]], the corresponding degree-3 Cheegers-Simons invariant is the _complex volume_ of the 3-manifold, namely the linear combination

$$
  CS + i vol
$$

of its [[Chern-Simons form|Chern-Simons invariant]] and its [[volume]] (e.g. [Neumann 11, section 2.3](#Neumann11), [Garoufalidis-Thurston-Zickert 11](#GaroufalidisThurstonZickert11)).

This appears as the action in [[analytically continued Chern-Simons theory]].

(It is, incidentally, also the contribution of a corresponding [[membrane instanton]] [[wrapped brane|wrapping]] a hyperbolic 3-cycle.)

## Properties

### Volume conjecture

The [[volume conjecture]] for the [[Reshetikhin-Turaev construction]] states that in the [[classical limit]] it [[convergence of a sequence|converges]] to the [[complex volume]] ([MMOTY 02, Conjedtcure 1.2](#MMOTY02), see also [Chen-Yang 15](#ChenYang15))

## Related concepts

* [[Borel regulator]]

* [[SnapPy]]

## References

* [[Jeff Cheeger]], [[Jim Simons]], _Differential characters and geometric invariants_, in _Geometry and Topology_, Proceedings of the Special Year, University of Maryland 1983-84, eds. J. Alexander and J. Harer, Lecture Notes in Math. 1167, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 50–80.

* [[Johan Dupont]], [[Richard Hain]], [[Steven Zucker]], _Regulators and characteristic classes of flat bundles_ ([arXiv:alg-geom/9202023](http://arxiv.org/abs/alg-geom/9202023))

* {#Neumann04} [[Walter Neumann]], _Extended Bloch group and the Cheeger-Chern-Simons class_, Geom. Topol. 8 (2004) 413-474 ([arXiv:math/0307092](http://arxiv.org/abs/math/0307092))

* {#Neumann11} [[Walter Neumann]], _Realizing arithmetic invariants of hyperbolic 3-manifolds_, Contemporary Math 541, Amer. Math. Soc. 2011, 233--246 ([arXiv:1108.0062](http://arxiv.org/abs/1108.0062))

* {#GaroufalidisThurstonZickert11} [[Stavros Garoufalidis]], [[Dylan Thurston]], [[Christian Zickert]], _The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds_ ([arXiv:1111.2828](http://arxiv.org/abs/1111.2828), [Euclid](http://projecteuclid.org/euclid.dmj/1259332507))

The [[volume conjecture]] (Kashaev's conjecture) for complex volume is due to

* {#MMOTY02} [[Hitoshi Murakami]], [[Jun Murakami]], M. Okamoto, T. Takata, and Y. Yokota, Conjecture 1.2 in _Kashaev's Conjecture And The Chern-Simons Invariants Of Knots And Links_, Experiment. Math. 11 (2002) 427-435 ([arXiv:math/0203119](https://arxiv.org/abs/math/0203119))

see also

* {#ChenYang15} [[Qingtao Chen]], [[Tian Yang]], _A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary_ ([arXiv:1503.02547](http://arxiv.org/abs/1503.02547))

Relation to [[analytic torsion]] is discussed in

* [[Varghese Mathai]], section 6 of _$L^2$-analytic torsion_, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386

* [[John Lott]], _Heat kernels on covering spaces and topological invariants_, J. Diff. Geom. 35 no 2 (1992) ([pdf](https://math.berkeley.edu/~lott/covering.pdf))

[[!redirects complex volumes]]

[[!redirects Cheeger-Simons class]]
[[!redirects Cheeger-Simons classes]]


complexification

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## Idea

For $V$ a [[vector space]] over the [[real numbers]] or some other $\mathbb{R}$-[[module]], its _complexification_ is the [[tensor product]] over $\mathbb{R}$ with the [[complex numbers]] $\mathbb{C}$: the _[[extension of scalars]]_ along the canonical inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$.

## Related concepts

* [[complex Lie algebra]], [[complex Lie group]]

## References

Textbook accounts:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §4.3 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

See also:

* Wikipedia, *[Complexification](https://en.wikipedia.org/wiki/Complexification)*

[[!redirects complexifications]]
[[!redirects complexified]]


complexity class

# Complexity class
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## Idea

In [[computer science]], the [[algorithm|algorithms]] for solving computational problems use resources proportional to the size of their inputs. A **complexity class** is a collection of problems which all can be solved with similar resource usage.

More generally, many [[model of computation|computers]] are restricted in what they can compute, even with unlimited resources. A complexity class for a computer is the collection of all problems which can be solved by a [[program]] on that computer.

A complexity class might be defined for [[decision problem|decision problems]], which only return true-or-false answers, but there are also definitions for [[function problem|function problems]].

## Definition

### Classical

Fix a [[model of computation]]. Traditionally, we use the theory of [[Turing machine|Turing machines]]. The complexity class of this theory is the collection of all problems which can be [[decidability|decided]] by a term of the theory -- a [[program]] of the model of computation.

When the theory is Turing-complete, then we have the complexity class [[R]]. This class is composed of all problems whose answers form a [[recursive set]].

Let a polynomial time bound on a program be a [[polynomial]] in natural numbers which takes the size of an input and returns the number of [[beta-reduction|reductions]] required to compute the output. The theory of Turing machines with at least two tapes, restricted by polynomial time bounds, is associated with the complexity class [[PTIME]]. Similarly, there are exponential time bounds, and a corresponding complexity class [[EXPTIME]].

For an example of a complexity class naturally generated by a type of computer, consider the theory of deterministic finite-state [[automaton|automata]], or DFAs. Each DFA corresponds to a [[regular language]], and the DFAs form a complexity class [[REG]]. In terms of resource usage, REG corresponds to Turing machines using a constant amount of space.

### Descriptive

Fix a [[fragment]] of [[higher-order logic]]. A query is a predicate on finite structures which can be expressed in the fragment. The descriptive complexity class of the fragment is the collection of queries, and the size of each input is the size of the corresponding finite structure.

For the fragment of [[first-order logic]] with an added least [[fixed point]] operator, the descriptive complexity class is [[PTIME]]. Similarly, the fragment of [[second-order logic]] with a least fixed point operator gives the complexity class [[EXPTIME]].

Second-order logic with [[existential quantification]] but not [[universal quantification]] yields the complexity class [[NPTIME]].

## Properties

(...)

## Examples

* The complexity class [[R]] corresponds to the height of computability according to the [[Church-Turing thesis]].

* The classes [[PTIME]] and [[NPTIME]] are clearly related, but the exact details are elusive. The question of [[P vs NP]] is central to computer science.

* The complexity class [[REG]] describes both deterministic and non-deterministic finite-state automata (DFAs and NFAs) because DFAs and NFAs are equivalent in computational power. In practice, this has led to the popularity of regular languages and [[regular expression|regular expressions]] in computer science. The [McNaughton-Yamada-Thompson algorithm](https://en.wikipedia.org/wiki/Thompson%27s_construction) and [Glushkov's algorithm](https://en.wikipedia.org/wiki/Glushkov%27s_construction_algorithm) send regular languages to equivalent NFAs, and [Kleene's algorithm](https://en.wikipedia.org/wiki/Kleene%27s_algorithm) sends NFAs to equivalent regular languages.

* [[quantum complexity theory|quantum complexity classes]]:

  [[BQP]] for default [[quantum computers]]

  [[PostBQP]]

  [[DQC1]] ("one clean qbit"),

  (...)

## Related entries

* [[descriptive complexity theory]]
* [[timed set]]

## References

* Wikipedia, _[Complexity class](https://en.wikipedia.org/wiki/Complexity_class)_, _[P versus NP problem](https://en.wikipedia.org/wiki/P_versus_NP_problem)_

[[!redirects complexity classes]]


complexity theory

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## Idea

**Computational complexity theory** is the study of how much and how many resources are required to perform [[computations]]. Unlike other systems theories, computational complexity theory is purely mathematical and focuses on analysis of [[formal logic]] and abstract machines.

### History

Historically, computational complexity theory arose from questions in [[computability theory]] and [[cryptography]]. In computability theory, we wanted to know what might be uncomputable relative to certain models of computation. In [[cryptography]], we wanted to know which ciphers could be broken for a given amount of computational resources.

### Current Directions

Today, the main focus of computational complexity theory is analysis of the [[complexity class|complexity classes]] [[P]] and [[NP]]. Their relationship determines which of [[Impagliazzo's five worlds]] is physical reality.

Additionally, [[descriptive complexity]] theory is an active research programme which characterizes complexity classes via formal logic rather than abstract machines.

## Related concepts

* [[NP]]

* [[Kolmogorov complexity]]

* [[Chaitin's incompleteness theorem]]

* [[computational complexity and physics]]

* [[quantum complexity theory]]

## References

Textbook accounts:

* [[Christos Papadimitriou]] (1994), _Computational Complexity_ (1st ed.), Addison Wesley.

* [[Michael Sipser]], Part III of: *Introduction to the Theory Of Computation*, 3rd ed: Cengage Learning (2012) [ISBN:978-1-133-18779-0,[pdf](https://www.mog.dog/files/SP2019/Sipser_Introduction.to.the.Theory.of.Computation.3E.pdf), [pdf](https://fuuu.be/polytech/INFOF408/Introduction-To-The-Theory-Of-Computation-Michael-Sipser.pdf)]

See also:

* Wikipedia, _[Computational complexity theory](http://en.wikipedia.org/wiki/Computational_complexity_theory)_

* Wikipedia, _[P vs NP problem](https://en.wikipedia.org/wiki/P_versus_NP_problem)_

Discussion of complexity classes via [[linear logic]]:

* Pierre Boudes, Damiano Mazza, Lorenzo Tortora de Falco, _An Abstract Approach to Stratification in Linear Logic_ ([arXiv:1206.6504](http://arxiv.org/abs/1206.6504))

Discussion for [[quantum computation]] and [[quantum supremacy]]:

* [[Scott Aaronson]], [[Lijie Chen]], *Complexity-Theoretic Foundations of Quantum Supremacy Experiments*, CCC '17: Proceedings of the 32nd Computational Complexity Conference (2017) **22** 1–67 [[arXiv:1612.05903](https://arxiv.org/abs/1612.05903), [doi;10.5555/3135595.3135617](https://dl.acm.org/doi/10.5555/3135595.3135617)]

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