Balanced metrics and chow stability of projective bundles over Riemann surfaces

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group. In this article, we give a simple proof for polarizations $\mathcal{O}_{\mathbb{P}E^*}(d)\otimes \pi^* L^k$, where $d$ is a positive integer, $k \gg 0$ and the base manifold is a compact Riemann surface of genus $g \geq 2$.


Introduction
In [M], Morrison proved that for the projectivization of a rank two holomorphic vector bundle over a compact Riemann surface, Chow stability is equivalent to the stability of the bundle. In [S], We generalized one direction of Morrison's result for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group ( [S]).
Let X be a compact complex manifold of dimension m and π : E → X be a holomorphic vector bundle of rank r with dual bundle E * . This gives a holomorphic fibre bundle PE * over X with fibre P r−1 . One can pull back the vector bundle E to PE * . We denote the tautological line bundle on PE * by O PE * (−1) and its dual by O PE * (1). Let L → X be an ample line bundle on X and ω ∈ 2πc 1 (L) be a Kähler form. Since L is ample, there is an integer k 0 so that for any k ≥ k 0 , O PE * k (1) is very ample over PE * k , where E k = E ⊗ L ⊗k . Note that there is a canonical isomorphism PE * k ∼ = PE * and O PE * k (1) ∼ = O PE * (1) ⊗ π * L k . The main theorem of [S] is the following: Theorem 1.1. Suppose that Aut(X) is discrete and X admits a constant scalar curvature Kähler metric in the class of 2πc 1 (L). If E is Mumford stable, then is Chow stable for k ≫ k 0 .
One of the earliest results in this spirit is the work of Burns and De Bartolomeis in [BD]. They constructed a ruled surface which does not admit any extremal metric in certain cohomology class. In [H1], Hong proved that there are constant scalar curvature Kähler metrics on the projectivization of stable bundles over curves. In [H2] and [H3], he generalized this result to higher dimensions with some extra assumptions. Combining Hong's results with Donaldson's, (PE * , O PE * (n) ⊗ π * L m ) is Chow stable for m, n ≫ 0 when the bundle E is stable. Note that it differs from our result, since it implies the Chow stability of (PE * , O PE * m (n)) for n, m big enough.
In [RT], Ross and Thomas developed the notion of slope stability for polarized algebraic manifolds. As one of the applications of their theory, they proved that if (PE * , O PE * (1) ⊗ π * L k ) is slope semi-stable for k ≫ 0, then E is a slope semistable bundle and (X, L) is a slope semistable manifold.Again note that they look at stability of PE * with respect to polarizations O PE * m (n) for n big enough. For the case of one dimensional base, however they showed stronger results. In this case they proved that if (PE * , O PE * (1) ⊗ π * L) is slope (semi, poly) stable for any ample line bundle L, then E is a slope (semi, poly) stable bundle.
In order to prove Theorem 1.1 we used the concept of balanced metrics. Combining the results of Luo, Phong, Sturm, Wang and Zhang on the relation between balanced metrics and stability, we proved the following.
Theorem 1.2. ( [S]) Suppose that Aut(X) is discrete and X admits a constant scalar curvature Kähler metric in the class of 2πc 1 (L).
In this paper, we give another proof of Theorem 1.1 in the case of one dimensional base X. The proof is simple in this case and can be generalized to polarizations O PE * (d) ⊗ π * L k for any positive integer d and k ≫ 0. The main theorem of this paper is the following. Theorem 1.3. Let X be a compact Riemann surface of genus g ≥ 2 and E → X be a holomorphic vector bundle on X. Let d be a positive integer. If E is Mumford stable, then (PE * , O PE * (d) ⊗ π * L k ) admits balanced metrics g k for k ≫ 0.
The Hitchin-Kobayashi correspondence implies that the stable bundle E admits a Hermitian-Einstein metric h ∞ . A simple calculation shows that the Hermitian metric Sym d h ∞ on Sym d E is Hermitian-Einstein. Therefore the vector bundle Sym d E is stable. By a theorem of Wang , we know that there exist balanced metrics H (k) on Sym d E ⊗ L k . This means that there exists a basis s 1 , ..., s N for Using the canonical isomorphism between H 0 (X, Sym d E⊗L k ) and H 0 (PE * , O PE * (d)⊗ L k ), we get a sequence of Hermitian metrics H (k) on O PE * (d) ⊗ L k . We prove that the sequence H (k) is "almost balanced", i.e.
where D (k) → C r,d as k → ∞ (see (2.2) for the definition of C r,d ) and M (k) is a trace-free Hermitian matrix such that The next step is to perturb these almost balanced metrics to get balanced metrics. As pointed out by Donaldson, the problem of finding balanced metric can be viewed also as a finite dimensional moment map problem solving the equation M (k) = 0. Indeed, Donaldson shows that M (k) is the value of a moment map µ D on the space of ordered bases with the obvious action of SU (N ). Now, the problem is to show that if for some ordered basis s, the value of moment map is very small, then we can find a basis at which moment map is zero. The standard technique is flowing down s under the gradient flow of |µ D | 2 to reach a zero of µ D . We need a Lojasiewicz type inequality to guarantee that the flow converges to a zero of the moment map. This was done in [S] by adapting Phong-Sturm proof to our situation. In [S2], we generalize Theorem 1.3 to higher dimensional base manifolds that admits cscK metrics and do not have any nonzero holomorphic vector fields. After this work was completed, we became aware of the preprint [DZ].
Acknowledgements: I am sincerely grateful to Richard Wentworth for many helpful discussions and suggestions and his continuous help, support and encouragement.

Preliminaries
Let V be a Hermitian vector space of dimension r. The projective space PV * can be identified with the space of hyperplanes in V via There is a natural isomorphism between V and H 0 ( For any positive integer d, define an equivalence relation ∼ on V ⊗d by We define Sym d V = V ⊗d / ∼ and simply denote the class of Remark 2.1. Let e 1 , . . . , e r be an orthonormal basis for V with respect to h, then the set forms an orthonormal basis for Sym d V with respect to Sym d h.
There is a natural isomorphism between Sym d V and H 0 ( For any Hermitian inner product H on Sym d V , we define a metricĤ on O PV * (d) by In particular, we have The following lemmas are straight forward.
Lemma 2.2. For any Hermitian inner product h on V , we havê Lemma 2.3. There exists a constant C r,d such that for any v, w ∈ Sym d V and any Hermitian inner product h on V , where ω FS,h = i∂∂ logĥ.
Remark 2.4. Let H be a Hermitian inner product on V . Suppose there exists a constant C such that for any v, w ∈ Sym d V . Then H = Sym d h for some Hermitian inner product h on V .
Lemma 2.6. Let X be a Kähler manifold of dimension n and Ω 0 and Ω be two Kähler forms on X. There exists a constant C depends only on the dimension of Proposition 2.7. Let h be a Hermitian inner product on V and H be a Hermitian inner product on Sym d V such that ||H − Sym d h|| Sym d h < min(ǫ, 1 2 ). Then for any v, w ∈ Sym d V , we have where C is a constant depends only on r and d.
Proof. Let e 1 , . . . e K be an orthonormal basis for Sym d V with respect to Sym d h.
The last inequality follows from Cauchy-Shwartz inequality. By a unitary change of basis we may assume that H ij = 0 if i = j. Therefore the basis Therefore by the same argument, we conclude that Applying (2.3), Lemma 2.3, Lemma 2.5 and Lemma 2.6, we have The last inequality follows from the fact that sup PV * | v| Sym d h = |v| Sym d h .

Balanced Metrics On Holomorphic Vector Bundles
Let (X, ω 0 ) be a compact Kähler manifold of dimension n and (L, g) be an ample holomorphic Hermitian line bundle over X such that i∂∂ log g = ω 0 . Let E be a holomorphic vector bundle of rank r over X. By possibly tensoring with high power of the ample line bundle L, we may assume that E is very ample. Therefore we can embed X into G(r, H 0 (X, E) * ), the Grassmanian of r-planes in H 0 (X, E) * . Indeed, for any x ∈ X, we have the evaluation map H 0 (X, E) → E x , which sends s to s(x). Since E is globally generated, this map is a surjection. So its dual is an inclusion of E * x ֒→ H 0 (X, E) * , which determines an r-dimensional subspace of H 0 (X, E) * . Therefore we get a map ι : X → G(r, H 0 (X, E) * ). Since E is very ample, ι is an embedding. Clearly we have ι * U r = E * , where U r is the tautological vector bundle on G(r, H 0 (X, E) * ), i.e. at any r-plane in G(r, H 0 (X, E) * ), the fibre of U r is exactly that r-plane. A choice of basis for H 0 (X, E) gives an isomorphism between G(r, H 0 (X, E) * ) and the standard Grassmanian G(r, N ), where N = dim H 0 (X, E). We have the standard Fubini-Study Hermitian metric on U r , so we can pull it back to E and get a Hermitian metric on E.
Definition 3.1. The embedding is called balanced if X s i , s j ι * hFS ω n n! = Cδ ij .
Notice that being balanced depends on the choice of the Kähler form. A Hermitian metric on E is called balanced (more precisely ω-balanced) if it is the pull back ι * h FS , where ι is a balanced embedding.
Equivalently, we can formulate the definition of balance metrics in terms of Bergman kernels.
Definition 3.2. Let h be a Hermitian metric on E and s 1 , ..., s N be an orthonmal basis for H 0 (X, E) with respect to the inner product Note that B(h, ω 0 ) does not depend on the choice of the orthonmal basis. A Hermitian metric h on E is balanced if and only if B(h, ω 0 ) = CI E for a positive constant C.
Theorem 3.3. ( [C], [Z]) Let (X, ω 0 ) be a compact Kähler manifold of dimension n and (L, g) be an ample holomorphic Hermitian line bundle over X such that i∂∂ log g = ω 0 . For any Hermitian metric h on the vector bundle E, there exist smooth endomorphisms A i (h) ∈ Γ(X, End(E)) such that the following asymptotic expansion holds as k → ∞ There is a close relationship between stability of vector bundles and the existence of balanced metrics given by the following theorem of Wang.
is the scalar curvature of ω ∞ and s is the average of the scalar curvature. Conversely, if h ∞ solves (3.2), then there exists a sequence of balanced metrics h (k) on E ⊗ L k for k ≫ 0 and h k → h ∞ in C ∞ .
In the case that the base manifold X has dimension one and the Kähler metric ω ∞ has constant curvature, we prove that the rate of convergence of h k to h ∞ is O(k −∞ ).
Theorem 3.5. Let X be a compact Riemann surface and ω ∞ be a Kähler form of constant curvature on X. Let a be a positive integer. Suppose that the Hermitian metric h ∞ on E satisfies the Hermitian-Einstein equation Let h (k) be a sequence of balanced metric on E⊗L k for k ≫ 0 and h k = h (k) ⊗g The proof follows from Theorem 3.4, lemma 3.6 and lemma 3.7.
Lemma 3.6. Let h be a Hermitian metric on E. Suppose that E is stable and coefficients A 1 , . . . , A q in the asymptotic expansion (3.1) are constant endomorphisms of E. If q is big enough, then there exists a sequence of balanced metrics h (k) on Proof. First we claim that where ||σ k || C a = O(k n−q−1 ). In order to prove this, we observe that there exists a smooth section A(x) of End(E) such that The bundle E is stable and A j 's are constant sections of End(E). Therefore there exist numbers a 1 , ..., a q such that A j = a j I E . On the other hand where ||σ k || C a = O(k n−q−1 ). Now Wang's argument ([W2, page 276]) concludes the proof.
Lemma 3.7. In the situation of Theorem 3.5, all coefficients A i 's are constant.
Proof. The coefficients of the asymptotic expansion of the Bergman kernel are polynomials of the curvature tensor on the base manifold, curvature tensor on the bundle and their covariant derivatives. The whole curvature tensors on the base manifold and on the bundle are constant. Therefore all coefficients are constant.

Constructing Almost Balanced Metrics
The goal of this section is to prove Theorem 1.3. In order to prove Theorem 1.3, we construct a sequence of almost balanced metrics on O PE * (d) ⊗ L k (Theorem 4.5). We start with definition of balanced metrics on polarized manifolds.
Let (Y, ω) be a compact Kähler manifold of dimension n and O(1) → Y be a very ample line bundle on Y equipped with a Hermitian metric σ such that i∂∂ log σ = ω.
Since O(1) is very ample, using global sections of O(1), we can embed Y into P(H 0 (Y, O(1)) * ). A choice of ordered basis s = (s 1 , ..., s N ) of H 0 (Y, O(1)) gives an isomorphism between P(H 0 (Y, O(1)) * ) and P N −1 . Hence for any such s, we have an embedding ι s : Y ֒→ P N −1 such that ι * s O P N (1) = O(1). Using ι s , we can pull back the Fubini-Study metric and Kähler form of the projective space to O(1) and Y respectively.
where V = Y ω n /n!. A Hermitian metric (resp. a Kähler form) is called balanced if it is the pull back ι * s h FS (resp. ι * s ω FS ) where ι s is a balanced embedding. Remark 4.2. The concepts of balanced metric on holomorphic vector bundles (Definition 3.1) and balanced metric on polarized manifolds (Definition 4.1) are different. In order to find a balanced metric on a holomorphic vector bundle E → X, we need to fix a Kähler form ω 0 on X. A Hermitian metric h on E is balanced (more pre- For the rest of this section, let X be a compact Riemann surface and L be an ample line bundle on X. Let g be a positive Hermitian metric on L and ω ∞ = i∂∂ log g be a Kähler form on X. Let E be a holomorphic vector bundle on X of rank r and slope µ. The slope of E is defined by µ = deg(E) r . Similar to the case of vector spaces, we have the natural isomorphism Suppose that H is a Hermitian metric on Sym d E and s 1 , ..., s N is an orthonormal basis for H 0 (X, Sym d E ⊗ L k ) with respect to L 2 (H k , ω ∞ ), where H k = H ⊗ g ⊗k . Letŝ 1 , ...,ŝ N be the corresponding basis for H 0 (PE * , O PE * (d)).
We prove that the matrix [ PE * s i , s j H dvol H ] is close to a scalar matrix. More precisely, we prove the following.
where F (∂E ,h∞) is the curvature of the chern connection of h ∞ and µ is the slope of E. Then there exists a constant C depends only on r and d such that if , Here H k = H ⊗ g ⊗k .
Proof. In this proof C denotes a constant depends only on r and d that might change from line to line. Define H ∞ = Sym d h ∞ , ω 0 = i∂∂ log H ∞ and ω k = ω 0 + kω ∞ .
On the other hand, Lemma 2.5 implies that ||ω − ω 0 || C 0 (ω0) ≤ Cǫ. Therefore, since ω ∞ is a semipositive (1, 1)-form on PE * . Applying Lemma 2.6 implies that We have On the other hand, (4.5) implies Note that N k = O(k) by Riemann-Roch theorem. Therefore for any positive integer q, ||M (k) || = O(k −q−1 ) which means that the sequence of Hermitian metrics H (k) on O E * (d) ⊗ L k and ordered bases s (k) = ( s Proof of Theorem 1.3. Fix a positive integer a ≥ 4. Let ω ∞ be the kähler form on X with constant curvature. Since E is a stable bundle, there exists a Hermitian metric h ∞ on E satisfies the Hermitian-Einstein equation (4.1). Therefore Theorem 3.4 and Theorem 3.5 imply that there exists a sequence of balanced metrics H (k) on Sym d E ⊗ L k such that where R is the rank of Sym d E. Hence (4.7) Define ω 0 = i∂∂ log H ∞ and ω k = i∂∂ log H (k) . Thus (4.6) implies On the other hand, Theorem 4.5 and (4.7) imply that the sequence of Her Since PE * has no nontrivial holomorphic vector fields, we can perturb these almost balanced metrics to get balanced metrics on O PE * (d) ⊗ π * L k for k ≫ 0 (see [S,Theorem 4.6]).