In mathematics, the Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study. A Hermitian form in (the vector space) Cn+1 defines a unitary subgroup U(n + 1) in GL(n + 1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(n + 1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CPn is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n + 1)-sphere. In algebraic geometry, one uses a normalization making CPn a Hodge manifold.
Construction The Fubini–Study metric arises naturally in the quotient space construction of complex projective space. Specifically, one may define CPn to be the space consisting of all complex lines in Cn+1, i.e., the quotient of Cn+1 \ {0} by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group C* = C \ {0}:
C P n = { Z = [ Z 0 , Z 1 , … , Z n ] ∈ C n + 1 ∖ { 0 } } / { Z ∼ c Z , c ∈ C ∗ } . {\displaystyle \mathbf {CP} ^{n}=\left\{\mathbf {Z} =[Z_{0},Z_{1},\ldots ,Z_{n}]\in {\mathbf {C} }^{n+1}\smallsetminus \{0\}\right\}{\big /}\{\mathbf {Z} \sim c\mathbf {Z} ,c\in \mathbf {C} ^{*}\}.}
A point of CPn is thus identified with an equivalence class of (n + 1)-tuples [Z0,...,Zn] modulo nonzero complex rescaling; the Zi are called homogeneous coordinates of the point. Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar z = R eiθ can be uniquely thought of as the composition of a dilation by the modulus R followed by a counterclockwise rotation about the origin by an angle θ {\displaystyle \theta } , the quotient mapping Cn+1 \ {0} → CPn splits into two pieces,
C n + 1 ∖ { 0 } ⟶ ( a ) S 2 n + 1 ⟶ ( b ) C P n {\displaystyle \mathbf {C} ^{n+1}\smallsetminus \{0\}\mathrel {\stackrel {(a)}{\longrightarrow }} S^{2n+1}\mathrel {\stackrel {(b)}{\longrightarrow }} \mathbf {CP} ^{n}}
where step (a) is a quotient by the dilation Z ~ RZ for R ∈ R+, the multiplicative group of positive real numbers, and step (b) is a quotient by the rotations Z ~ eiθZ. The result of the quotient in (a) is the real hypersphere S2n+1 defined by the equation |Z|2 = |Z0|2 + ... + |Zn|2 = 1. The quotient in (b) realizes CPn = S2n+1/S1, where S1 represents the group of rotations. This quotient is realized explicitly by the famous Hopf fibration S1 → S2n+1 → CPn, the fibers of which are among the great circles of S 2 n + 1 {\displaystyle S^{2n+1}} .
As a metric quotient When a quotient is taken of a Riemannian manifold (or metric space in general), care must be taken to ensure that the quotient space is endowed with a metric that is well-defined. For instance, if a group G acts on a Riemannian manifold (X,g), then in order for the orbit space X/G to possess an induced metric, g {\displaystyle g} must be constant along G-orbits in the sense that for any element h ∈ G and pair of vector fields X , Y {\displaystyle X,Y} we must have g(Xh,Yh) = g(X,Y). The standard Hermitian metric on Cn+1 is given in the standard basis by
d s 2 = d Z ⊗ d Z ¯ = d Z 0 ⊗ d Z ¯ 0 + ⋯ + d Z n ⊗ d Z ¯ n {\displaystyle ds^{2}=d\mathbf {Z} \otimes d{\bar {\mathbf {Z} }}=dZ_{0}\otimes d{\bar {Z}}_{0}+\cdots +dZ_{n}\otimes d{\bar {Z}}_{n}}
whose realification is the standard Euclidean metric on R2n+2. This metric is not invariant under the diagonal action of C*, so we are unable to directly push it down to CPn in the quotient. However, this metric is invariant under the diagonal action of S1 = U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished. The Fubini–Study metric is the metric induced on the quotient CPn = S2n+1/S1, where S 2 n + 1 {\displaystyle S^{2n+1}} carries the so-called "round metric" endowed upon it by restriction of the standard Euclidean metric to the unit hypersphere.
In local affine coordinates Corresponding to a point in CPn with homogeneous coordinates [ Z 0 : ⋯ : Z n ] {\displaystyle [Z_{0}:\dots :Z_{n}]} , there is a unique set of n coordinates ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} such that
[ Z 0 : ⋯ : Z n ] ∼ [ 1 , z 1 , … , z n ] , {\displaystyle [Z_{0}:\dots :Z_{n}]\sim [1,z_{1},\dots ,z_{n}],}
provided Z 0 ≠ 0 {\displaystyle Z_{0}\neq 0} ; specifically, z j = Z j / Z 0 {\displaystyle z_{j}=Z_{j}/Z_{0}} . The ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} form an affine coordinate system for CPn in the coordinate patch U 0 = { Z 0 ≠ 0 } {\displaystyle U_{0}=\{Z_{0}\neq 0\}} . One can develop an affine coordinate system in any of the coordinate patches U i = { Z i ≠ 0 } {\displaystyle U_{i}=\{Z_{i}\neq 0\}} by dividing instead by Z i {\displaystyle Z_{i}} in the obvious manner. The n + 1 coordinate patches U i {\displaystyle U_{i}} cover CPn, and it is possible to give the metric explicitly in terms of the affine coordinates ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} on U i {\displaystyle U_{i}} . The coordinate derivatives define a frame { ∂ 1 , … , ∂ n } {\displaystyle \{\partial _{1},\ldots ,\partial _{n}\}} of the holomorphic tangent bundle of CPn, in terms of which the Fubini–Study metric has Hermitian components
g i j ¯ = h ( ∂ i , ∂ ¯ j ) = ( 1 + | z | l 2 ) δ i j ¯ − z ¯ i z j ( 1 + | z | l 2 ) 2 . {\displaystyle g_{i{\bar {j}}}=h(\partial _{i},{\bar {\partial }}_{j})={\frac {\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)\delta _{i{\bar {j}}}-{\bar {z}}_{i}z_{j}}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}.}
where |z|2 = |z1|2 + ... + |zn|2. That is, the Hermitian matrix of the Fubini–Study metric in this frame is
[ g i j ¯ ] = 1 ( 1 + | z | l 2 ) 2 [ 1 + | z | 2 − | z 1 | 2 − z ¯ 1 z 2 ⋯ − z ¯ 1 z n − z ¯ 2 z 1 1 + | z | 2 − | z 2 | 2 ⋯ − z ¯ 2 z n ⋮ ⋮ ⋱ ⋮ − z ¯ n z 1 − z ¯ n z 2 ⋯ 1 + | z | 2 − | z n | 2 ] {\displaystyle {\bigl [}g_{i{\bar {j}}}{\bigr ]}={\frac {1}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}\left[{\begin{array}{cccc}1+|\mathbf {z} |^{2}-|z_{1}|^{2}&-{\bar {z}}_{1}z_{2}&\cdots &-{\bar {z}}_{1}z_{n}\\-{\bar {z}}_{2}z_{1}&1+|\mathbf {z} |^{2}-|z_{2}|^{2}&\cdots &-{\bar {z}}_{2}z_{n}\\\vdots &\vdots &\ddots &\vdots \\-{\bar {z}}_{n}z_{1}&-{\bar {z}}_{n}z_{2}&\cdots &1+|\mathbf {z} |^{2}-|z_{n}|^{2}\end{array}}\right]}
Note that each matrix element is unitary-invariant: the diagonal action z ↦ e i θ z {\displaystyle \mathbf {z} \mapsto e^{i\theta }\mathbf {z} } will leave this matrix unchanged. Accordingly, the line element is given by
d s 2 = g i j ¯ d z i d z ¯ j = ( 1 + | z | l 2 ) | d z | 2 − ( z ¯ ⋅ d z ) ( z ⋅ d z ¯ ) ( 1 + | z | l 2 ) 2 = ( 1 + z i z ¯ i ) d z j d z ¯ j − z ¯ j z i d z j d z ¯ i ( 1 + z i z ¯ i ) 2 . {\displaystyle {\begin{aligned}ds^{2}&=g_{i{\bar {j}}}\,dz^{i}\,d{\bar {z}}^{j}\\[4pt]&={\frac {\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)|d\mathbf {z} |^{2}-({\bar {\mathbf {z} }}\cdot d\mathbf {z} )(\mathbf {z} \cdot d{\bar {\mathbf {z} }})}{\left(1+|\mathbf {z} |{\vphantom {l}}^{2}\right)^{2}}}\\[4pt]&={\frac {(1+z_{i}{\bar {z}}^{i})\,dz_{j}\,d{\bar {z}}^{j}-{\bar {z}}^{j}z_{i}\,dz_{j}\,d{\bar {z}}^{i}}{\left(1+z_{i}{\bar {z}}^{i}\right)^{2}}}.\end{aligned}}}
In this last expression, the summation convention is used to sum over Latin indices i,j that range from 1 to n. The metric can be derived from the following Kähler potential:
K = ln ( 1 + z i z ¯ i ) = ln ( 1 + δ i j ¯ z i z ¯ j ) {\displaystyle K=\ln(1+z_{i}{\bar {z}}^{i})=\ln(1+\delta _{i{\bar {j}}}z^{i}{\bar {z}}^{j})}
as
g i j ¯ = K i j ¯ = ∂ 2 ∂ z i ∂ z ¯ j K {\displaystyle g_{i{\bar {j}}}=K_{i{\bar {j}}}={\frac {\partial ^{2}}{\partial z^{i}\,\partial {\bar {z}}^{j}}}K}
Using homogeneous coordinates An expression is also possible in the notation of homogeneous coordinates, commonly used to describe projective varieties of algebraic geometry: Z = [Z0:...:Zn]. Formally, subject to suitably interpreting the expressions involved, one has
d s 2 = | Z | 2 | d Z | 2 − ( Z ¯ ⋅ d Z ) ( Z ⋅ d Z ¯ ) | Z | 4 = Z α Z ¯ α d Z β d Z ¯ β − Z ¯ α Z β d Z α d Z ¯ β ( Z α Z ¯ α ) 2 = 2 Z [ α d Z β ] Z ¯ [ α d Z ¯ β ] ( Z α