It is the projective space associated to the vector space $\mathbb{C}^{n+1}$. It is denoted by
$$ \mathbb{C}\mathbb{P}^n=\mathbb{P}(\mathbb{C}^{n+1}) $$It consists of the different complex lines through the origin of $\mathbb{C}^{n+1}$.
It has the structure of a complex manifold.
It has a natural metric, called the Fubini-Study metric. In $\mathbb{C}^{n+1}$ we have the natural Hermitian inner product. This metric cannot be directly inherited by the quotient since is not invariant by the multiplicative action of $\mathbb{C}-\{0\}$. But there is a way to continue... see for example wikipedia.
With this metric, $\mathbb{C}\mathbb{P}^n$ has the structure of a Hermitian manifold.
The isometry group of $\mathbb{C}\mathbb{P}^n$ with this metric is the projective unitary group $\mathbb{P}U(n+1)$ where the stabilizer of a point is $\mathbb{P}U(n)$. Keep an eye, the projective special unitary group $\mathbb{P}SU(n+1)$ is equal to $\mathbb{P}U(n+1)$. See wikipedia.
It is a tuple of points that can be used to define coordinates in the projective space. If we take a basis of $\mathbb{C}^{n+1}$, $\{e_0,\ldots,e_n\}$, we can express points in coordinates. A scalar multiple of this basis leads to the same coordinates, but this wouldn't be a problem because we eventually will take as a frame the set corresponding projective points $\{[e_0],\ldots,[e_n]\}$. But there is another issue. If we only multiply one of the vectors, we obtain the same collection of projective points, but the coordinates are different.
To avoid this difficulty we define a projective frame as a $n+2$-tuple of projective points $\{[e_0],\ldots,[e_n],[\sum e_i]\}$, with $\{e_i\}$ a basis of $\mathbb{C}^{n+1}$. Two basis lead to the same projective frame if and only if one is a scalar multiple of the other (standard proof).
We call homography to the elements of the projective linear group, that is, maps corresponding to isomorphisms of the underlying vector space. Given two projective frames of a projective space, there is exactly one homography of _P_ that maps the first frame onto the second one.
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Author of the notes: Antonio J. Pan-Collantes
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