Expected centre of mass of the random Kodaira embedding

Let $X \subset \mathbb{P}^{N-1}$ be a smooth projective variety. To each $g \in SL (N , \mathbb{C})$ which induces the embedding $g \cdot X \subset \mathbb{P}^{N-1}$ given by the ambient linear action we can associate a matrix $\bar{\mu}_X (g)$ called the centre of mass, which depends nonlinearly on $g$. With respect to the probability measure on $SL (N , \mathbb{C})$ induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety.


Introduction and the statement of the main result
Let X be a complex smooth projective variety, and ι : X ֒→ P(H 0 (X, L) ∨ ) ∼ = P N −1 be the Kodaira embedding defined with respect to a very ample line bundle L on X, where N := dim H 0 (X, L). There is a natural SL(N, C)-action on the Kodaira embedding ι → g · ι given by the ambient linear action SL(N, C) P N −1 .
For each g ∈ SL(N, C) we can define an N × N hermitian matrixμ X (g), called the centre of mass of the embedding g · ι : X ֒→ P(H 0 (X, L) ∨ ) (see §2.2 for more details). This plays an important role in Kähler geometry, and depends on g ∈ SL(N, C) in a highly nonlinear manner. For example, when the automorphism group of (X, L) is discrete, there exists g ∈ SL(N, C) such thatμ X (g) is a constant multiple of the identity matrix if and only if the embedding ι : X ֒→ P N −1 is Chow stable [22,26], which is an important yet subtle algebro-geometric property of X ⊂ P N −1 .
The following seems to be a natural question to ask. In spite of its apparent simplicity, this is a nontrivial problem sinceμ X (g) depends nonlinearly on g. The main result of this paper is the following. 3) satisfies all the properties stated in the theorem. We also note that the absolute continuity of the measure on B is meant to be with respect to the Haar measure on B.
The study of Kähler and Fubini-Study metrics in connection to the probability theory, such as the random matrix theory, has been an active area of research. There are works e.g. [3][4][5][6][7] by Berman, and [11-14, 19-21, 25] by Ferrari, Flurin, Klevtsov, Song, Zelditch. On the other hand, probabilistic aspects of the centre of massμ X (g) does not seem to have been actively investigated in the aforementioned works, which is the focus of the present paper.
As pointed out in the above, whetherμ X (g) itself is a constant multiple of the identity matrix depends on the Chow stability of X ⊂ P N −1 by the result of Luo [22] and Zhang [26]. Such subtleties disappear, however, when we take the average over g ∈ SL(N, C) as in Theorem 1.2.
While the main point of Theorem of the centre of mass of X ⊂ P N −1 is a constant multiple of the identity matrix.
We can also define a variantμ X,ν of the centre of mass, as in Definition 2.14, by fixing a volume form dν on X. It turns out that Theorems 1.2 and 1.3 easily extend to this variant, as explained in Remarks 2.15 and 3.1, essentially becauseμ X,ν (g) depends on g ∈ SL(N, C) in a much less nonlinear manner thanμ X (g).
The author is grateful to the anonymous referee for suggesting this point to him.
Remark 1.4. Although we shall only treat SL(N, C) and SU (N ) throughout this paper, the determinant one condition does not play any significant role. We can run exactly the same argument for GL(N, C) and U (N ) to get the same results, in fact with a slightly simpler proof.
This version of the article has been accepted for publication, after peer review (when applicable) but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s12220-021-00778-y 2 Preliminaries

Random matrices
Our aim is to define a class of probability measures on SL(N, C) which has some good properties as in the statement of Theorem 1.2. The precise description of such measures is given in Definition 2.1, but that needs to be accompanied by a review of some elementary results in the theory of random matrices; the details can be found e.g. in [1,8,9,23] or any other standard textbooks on random matrices.
by the projection where g * stands for the hermitian conjugate of g with respect to the hermitian form represented by the identity matrix on C N . Throughout, we shall write e for the identity in SL(N, C) or SU (N ).
Definition 2.1. We set our notational convention, and the definition of the measure dσ on SL(N, C), as follows.
• We write dσ SU for the Haar measure on SU (N ) of unit volume.
• We fix a measure dσ B on B, and assume that dσ B is absolutely continuous, unitarily invariant, and of finite volume.
• Given a measure dσ B on B and dσ SU on SU (N ), the measure defined on SL(N, C) via the fibration structure (1) is denoted by dσ.
Given any measure dσ on SL(N, C) as defined above, it is immediate that dσ is of finite volume (see also Lemma 2.2). Henceforth without loss of generality we shall assume by scaling, i.e. dσ is a probability measure on SL(N, C).
We have a more explicit formula for dσ, which follows immediately from the above definition.
Lemma 2.2. Suppose that dσ is a probability measure on SL(N, C) defined as in Definition 2.1. If φ : where Vol(B) := B dσ B is the volume of B with respect to dσ B , and h ∈ SL(N, C) is a hermitian matrix such that π(g) = hh * .
We now recall some basic facts on the Euclidean volume form (or its associated Lebesgue measure) on the N × N hermitian matrices (not necessarily positive definite or of determinant one), induced by the natural Euclidean metric. By unitarily diagonalising a hermitian matrixH asH = u −1 Λu for u ∈ U (N ) where dσ U is the Haar measure on U (N ) and ∆ 2 (λ) is the square of the Vandermonde determinant We consider the volume form on B, which consists of positive definite hermitian matrices H of determinant one, induced by the Euclidean metric as above.
and carrying out the computation exactly as in [15, §2], we find for some smooth positive function γ ∈ C ∞ (R N −1 >0 , R >0 ) on the (N − 1)-fold direct product of positive real numbers R N −1 >0 . The notation δ(log detH)dH using the delta function is also used e.g. in [13, §4.1] to denote dH as in (3). Now returning to our original setting, we note that the measure dσ B on B being absolutely continuous means that we can write where dH is as defined in (3) and ρ : B → [0, +∞) is a measurable function (called the Radon-Nikodym density) which is known to exist by the Radon-Nikodym theorem. Moreover, dσ B being of finite volume Finally, dσ B being unitarily invariant means that dσ B (H) = dσ B (uHu −1 ) for all H ∈ B and u ∈ SU (N ), which is equivalent to saying that ρ(H) depends only on the eigenvalues λ 1 , . . . , ). By abuse of notation we also write ρ(λ 1 , . . . , λ N −1 ) for ρ(H). With this notation, the finite volume condition (5) translates to Recalling (3), the Gaussian unitary ensemble dσ B can be written more explicitly as  (for all u ∈ SU (N )) does not seem to hold for S ν , so Theorem 1.2 does not seem to apply to the case when we use exp(−γS ν (H))dH as a measure on B.
Remark 2.7. Note that the measure dσ B or dσ as discussed in the above depends on the fixed hermitian form on C N , represented by the identity matrix. This corresponds to the choice of the reference basis that we take to identify H 0 (X, L) with C N in §2.2.

Moment maps and the centre of mass
We review the ingredients from complex geometry that we need in this paper. Let X be a complex smooth projective variety of complex dimension n, with a very ample line bundle L and the associated embedding We fix a basis for H 0 (X, L) once and for all and identify P( we also note that the basis we fixed here can be identified with an orthonormal basis for the hermitian form represented by the identity matrix on C N ∼ = H 0 (X, L) (see also Remark 2.7). With respect to such a reference basis, we write [Z 1 : · · · : Z N ] for the homogeneous coordinates for P N −1 . Furthermore, by abuse of terminology, we also write {Z i } N i=1 for the reference basis itself. Pick g ∈ SL(N, C) and write where g ij is the matrix representation of g with respect to the basis where z i := Z i /Z 1 for i = 2, . . . , N . By abuse of terminology, the restriction ofω He to ι(X) ⊂ P N −1 is also called the Fubini-Study metric on ι(X), and written ω He := ι * ω He .
While the above definition is often stated for a fixed hermitian matrix, different hermitian matrices lead to different Fubini-Study metrics; for the hermitian matrix H g , the associated Fubini-Study metricω Hg can be written, on C N −1 = {Z 1 (g) = 0} ⊂ P N −1 , as by replacing z i with z i (g) := Z i (g)/Z 1 (g). While the isometry group ofω Hg is isomorphic to SU (N ), it is not the same SU (N ) that we fixed above; while the SU (N ) as above preserves the hermitian form H e , in general it does not preserve H g if g = e. Recall also that ω Hg := ι * ω Hg ∈ c 1 (L) for all g ∈ SL(N, C). From the above definition, by writing in terms of polar coordinates z i (g) = r i (g)e √ −1θi(g) we havẽ Note also that the restriction ofω n Hg to ι(X) defines a volume form on ι(X), which we write as dν Hg := ω n Hg n! .
The total volume of X with respect to dν Hg can be computed as X dν Hg = X c 1 (L) n /n! =: Vol(X, L), which depends only on (X, L) and is independent of g ∈ SL(N, C).
Recall that ( √ −1 times) the moment map µ SU : P N −1 → √ −1su(N ) for the SU (N )-action on P N −1 is given by where δ ij is the Kronecker delta and the subscript ij stands for the (i, j)-th entry of the N × N matrix. The second term δ ij /N is just to make µ SU trace-free. Observing that SU (N ) acts transitively on P N −1 , we find that µ SU naturally defines a map µ SU,p : SU (N ) → √ −1su(N ) by µ SU,p (u) := µ SU (up) where p ∈ P N −1 is a fixed reference point.
We now consider the "complexified" version of the above moment map, defined for SL(N, C) = SU (N ) C .
We fix a reference point p ∈ P N −1 represented by the homogeneous coordinates [Z 1 : · · · : Z N ], and observe that for each g ∈ SL(N, C) the point gp ∈ P N −1 is represented by [Z 1 (g) : · · · : Z N (g)] in terms of the notation (7). We then define an N × N hermitian matrix µ p (g) ∈ √ −1u(N ) whose (i, j)-th entry is given by This corresponds to the first term of µ SU at the point gp; note that gp is in the SU (N ) C -orbit of p. We choose not to normalise the trace of µ p (g) to be zero, to be consistent with the notation in the literature. The centre of mass, which plays an important role in this paper, is defined for g ∈ SL(N, C) and the embedded variety ι : X ֒→ P N −1 as the integralμ We summarise the above in the following formal definition.
Definition 2.9. The centre of massμ X (g), defined for g ∈ SL(N, C) and ι : X ֒→ P N −1 , is a hermitian matrix of size N whose (i, j)-th entry is given in terms of the notation (7) bȳ where dν Hg is the measure on ι(X) defined by the Fubini-Study metric on P N −1 with respect to H g , and integrates with respect to the variables It is easy to see how µ p (g) in (10) changes when g is pre-multiplied by a unitary matrix u, as in the following lemma. Note, on the other hand, that we do not have an analogous formula for µ p (gu).
Remarks 2.11. We observe some other elementary properties of the centre of mass which immediately follow from the definition.
1. Both µ p (g) andμ X (g) are positive definite as a hermitian matrix for each g ∈ SL(N, C).
2. We observe thatμ X (g) is nothing but the integral of µ p (e) over p ∈ g · ι(X) with respect to dν Hg ; µ X (g) can be regarded as the centre of mass of the Kodaira embedding g · ι(X) ⊂ P N −1 .
3.μ X (g) is independent of the overall scaling of g, so depends only on its class in P SL (N, C). Moreover, we observe that each entry of the integrand µ p (g) of the centre of mass is manifestly bounded as a function of g ∈ SL(N, C) for each p ∈ P N −1 .
Computing the centre of mass is in general difficult sinceμ X (g) depends on g ∈ SL(N, C) (and the embedding ι : X ֒→ P N −1 ) in a highly nonlinear manner and the size N of the matrices is typically large.
However, there are some special cases in which we can explicitly compute it.
Example 2.12. Take X := P N −1 and L := O P N −1 (1). Then, by using (8) and the polar coordinates for C N −1 , we find thatμ P N −1 (g) is a constant multiple of the identity matrix for all g ∈ SL(N, C); this computation is well-known to the experts and reduces to the periodicity of the angle coordinates, but the details can be found e.g. in [18,Lemma 2.7]. In particular, E[μ P N −1 (g)] is a constant multiple of the identity matrix for any probability measure dσ on SL(N, C).
Example 2.13. The above method using the polar coordinates also work for the case when P n is embedded in a higher dimensional projective space by the Veronese embedding, i.e. when L = O P n (m) for m > 1, and is given by the monomial basis for H 0 (P n , O P n (m)), where N = dim C H 0 (P n , O P n (m)). As in the previous example,μ P n (g) can be easily seen to be a diagonal matrix for g ∈ SL(N, is a monomial basis. By appropriately scaling the monomial basis, we find that there exists g ∈ SL(N, C) such thatμ P n (g) is a constant multiple of the identity, and the explicit scaling can be written down as in [2,Example2.4].
We also have a variant of the centre of mass, introduced by Donaldson [10, §2] as follows.
Definition 2.14. Let dν be a fixed volume form on ι(X). We define a variantμ X,ν (g) of (11) by the following formulaμ in which we replaced dν Hg in (11) by the fixed volume form dν.
As we shall see later, it is straightforward to extend the results forμ X (g) to the variantμ X,ν (g); indeed, the volume form dν not depending on g means thatμ X,ν (g) depends on g in a much less nonlinear manner thanμ X (g), and the proof turns out to be simpler.

Proof of Theorem 1.3
The properties of the centre of mass presented in §2.2 are sufficient for the proof of Theorem 1.3, which is elementary. We compute Note first that dν Hu = dν He for all u ∈ SU (N ) since H u = (uu * ) −1 = H e . Lemma 2.10 further implies that the above is equal to We pick and fix an arbitrary η ∈ SU (N ), and observe that the group invariance of the Haar measure implies which implies that we have for any η ∈ SU (N ). Recalling that the centre of massμ X (u) is an N × N hermitian matrix, this implies that E SU [μ X (u)] must be a constant multiple of the identity matrix since it is a hermitian matrix that commutes with all elements of SU (N ). Noting that tr(μ X (u)) = Vol(X, L) for all u ∈ SU (N ), we find more explicitly that which completes the proof of Theorem 1.3.
Remark 2.15. Note that the above proof applies word by word to prove for the variant in Definition 2.14, by noting that dν is fixed and remains invariant under the SU (N )-action.

Proof of Theorem 1.2
Observe first that the definition of the centre of mass (11) implies where µ x is as defined in (10) and we endow SL(N, C) × ι(X) with the product measure dσ × dν He . We swap the order of the above integrals by Fubini's theorem to find We first fix x ∈ ι(X), pick a hermitian h ∈ SL(N, C) such that π(g) = hh * , and compute the second integral in the above as by using Lemma 2.2, where we note that each entry of µ x (g) is bounded (Remark 2.11) and that π −1 (hh * ) = h · SU (N ). Observe that we may write h = ηΛη * for some η ∈ SU (N ) and a diagonal matrix Λ = diag(Λ 1 , . . . , Λ N ) which we can identify with a vector in R N . With this notation we may write where u ∈ SU (N ). We also note which implies that ω n Hg (x)/ω n He (x) is bounded over SL(N, C), since an overall scaling of Λ leaves the above metric invariant.
Thus, by writingΛ := Λ 2 , the above integral may be written as by (3) and (4), where we set , ρ(Λ) is the Radon-Nikodym density of dσ B , and ∆, γ are as in (3). By the group invariance of the Haar measure, we have by recalling Lemma 2.10 and noting that does not depend on η, where the homogeneous coordinates [Z 1 : · · · : Z N ] are evaluated at x ∈ ι(X).
We are thus reduced to first computing We claim that the off-diagonal entries of the above integral are zero. Since any x ∈ ι(X) ⊂ P N −1 can be moved to p 0 = [1 : 0 : · · · : 0] by the SU (N )-action, for the moment we assume without loss of generality that x = p 0 , by using the SU (N )-invariance of the Haar measure. Since p 0 is fixed by the subgroup S(U (1)×U (N −1)) of SU (N ), the integral (13) is in fact an integral over P N −1 = SU (N )/S(U (1)×U (N −1)).
We now recall that a group invariant measure on a homogeneous space (if exists) is unique up to an overall positive multiplicative constant by [24, Chapter III, §4, Theorem 1], which is a result credited to Weil in [24].
Thus, the measure on P N −1 induced by the Haar measure dσ SU agrees, up to an overall constant multiple, with the SU (N )-invariant Fubini-Study measureω N −1 He . Thus, by using the homogeneous coordinate system [Z 1 : · · · : Z N ] given by the reference basis, we find that the (i, j)-th entry of (13) is equal to Thus we find for some maps α i : R N → R ≥0 (i = 1, . . . , N ); observe that each α i depends smoothly on Λ and is bounded over R N , since the (i, i)-th entry of the integrand is wherex ∈ C N is any nonzero lift (i.e. the homogeneous coordinates) of x ∈ ι(X) ⊂ P N −1 . We further observe that each α i does not depend on x ∈ ι(X), since for any x ′ ∈ ι(X) there exists u ′ ∈ SU (N ) such that x ′ = u ′ x (as SU (N ) acts transitively on the ambient P N −1 ) and hence the dependence on x is integrated out by the group invariance of the Haar measure. Moreover, the above formula and (12) imply that each α i can be naturally regarded as a function ofΛ = Λ 2 , and hence by abuse of notation we shall write α i (Λ) for α i (Λ), which can be considered as a smooth bounded function on R N >0 .
Let C N := {η 1 , . . . , η N } be the group of cyclic permutations of N letters, which is naturally a subgroup of U (N ). We then find with α(Λ) := N i=1 α i (Λ); we also note that in the above we may assume η i ∈ SU (N ) for i = 1, . . . , N by dividing them by an N -th root of det(η i ) ∈ U (1) which leaves the above integral invariant. Thus we get, again by the group invariance of the Haar measure, and hence, by recalling that α(Λ) does not depend on x ∈ X as pointed out in the above, we find since tr(μ X (g)) = Vol(X, L) for all g ∈ SL(N, C) and dσ is assumed to have unit volume as in (2 by definition. Noting that dν is fixed and does not depend on g ∈ SL(N, C), we again apply Fubini's theorem to SL(N, C) × ι(X) with the product measure dσ × dν, to find E[μ X,ν (g)] = x∈ι(X) dν SL(N,C) µ x (g)dσ(g) (15) and repeat the argument presented above.