A characterization of finite vector bundles on Gauduchon astheno-Kahler manifolds

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact complex manifold admitting a Gauduchon astheno-Kahler metric.


Introduction
For a polynomial f (x) = n i=0 a i x i , where a i are nonnegative integers, and a holomorphic vector bundle E on a complex projective variety or more generally on a compact complex manifold X, the vector bundle n i=0 (E ⊗i ) ⊕a i on X is denoted by f (E) (the vector bundle E 0 is defined to be the trivial line bundle O X ). An algebraic (respectively, holomorphic) vector bundle E on a projective variety (respectively, compact complex manifold) X is called finite if there are two distinct polynomials f , g with nonnegative integral coefficients, such that the vector bundle f (E) is isomorphic to g(E). It is known that a vector bundle E is finite if and only if there is a finite collection of vector bundles {F j } m j=1 and nonnegative integers {a i,j } m j=1 such that No1,p. 35,Lemma 3.1], [No2,p. 80,Lemma 3.1]. When X is a compact Riemann surface, it was noted by Weil that E is finite if it admits a flat connection with finite monodromy that is compatible with holomorphic structure of E [We]. Note that the condition that E admits a flat connection with finite monodromy is equivalent to the condition that the pullback of E to some finite étale Galois covering of X is holomorphically trivial. In a foundational work, Nori proved that a vector bundle E on a complex projective variety X is finite if and only if there is a finite étale Galois covering q : X −→ X and a Gal(q)-module V , such that E is isomorphic to the quotient of X × V by the twisted diagonal action of Gal(q) [No1], [No2]. Therefore, the monodromy homomorphism of the flat connection on E, given by the trivial connection on X × V −→ X, coincides with the one given by the action on V of Gal(q) (it is a quotient of the fundamental group of X).
Our aim here is to prove the following: Note that a flat connection with finite monodromy is a unitary connection for a suitable Hermitian structure. Theorem 1.1 was proved earlier in [BHS] under the assumption that X is a compact Kähler manifold. The proof in [BHS] crucially used a well-known theorem of Corlette-Simpson [Si]. However, that theorem is available only in the Kähler context. Here we have been able to avoid using the Corlette-Simpson theorem.

Vanishing of Chern classes
In whatever follows, for a compact complex manifold X we use the Bott-Chern cohomology defined as where A p,q (X) consists of smooth (p, q)-forms on X. These groups are finite dimensional. They coincide with the Dolbeault cohomology groups for Kähler manifolds, as a consequence of the ∂∂ lemma. If h is a Hermitian metric on a holomorphic vector bundle E over X, and Θ h is the curvature of its Chern connection, then for any characteristic class Φ, its Chern- Take a finite vector bundle E and vector bundles V j , 1 ≤ j ≤ m, as in Theorem 2.1. We may assume that (2) the isomorphism classes of V j are distinct, and (3) for each 1 ≤ j ≤ m, there is some j ≥ 1 such that a j ,j in Theorem 2.1 is nonzero (if a i,j = 0 for all i then discard V j ).

Lemma 2.2. Each of the above vector bundles
Proof. Take any V j . Take any k ≥ 1 such that a k,j ≥ 1. Then for every n ≥ 1, the vector bundle V ⊗n j is a direct summand of (2.1) If W be a holomorphic vector bundle on X, and where all A α and B α are indecomposable holomorphic vector bundles, then a theorem of Atiyah says that • a = b, and • there is a permutation σ of {1, · · · , a} such that the holomorphic vector bundle A α is holomorphically isomorphic to B σ (α) for all α ∈ {1, · · · , a}.
(See [At,p. 315,Theorem 3].) In view of this theorem, it follows from (2.1) that the direct summand V ⊗n j is a direct sum of copies of V , 1 ≤ ≤ m. Therefore, V i is finite.

A characterization of numerically flat bundles on Gauduchon astheno-Kähler manifolds
We recall the definition of a nef vector bundle as in [DPS]. A line bundle L over a compact complex manifold M, equipped with a Hermitian metric ω, is called nef if given any real number > 0, there is a smooth Hermitian metric h on L such that Θ(h ) + ω is a nonnegative Hermitian form, where Θ(h ) is the Chern curvature form. This definition actually does not depend on the choice of ω.
We note that Lemma 2.4 of [BHS], which was obtained as a corollary of Theorem 1.12 of [DPS], is stated in the Kähler set-up, because [BHS] is entirely in the Kähler set-up. However, Theorem 1.12 of [DPS] is proved for compact complex manifolds. Hence Lemma 2.4 of [BHS] remains valid for compact complex manifolds.

Lemma 3.1. ([BHS, Lemma 2.4]) If E is finite, then
Now fix smooth Hermitian structures g j on V j . This produces a Hermitian structure on any direct sum of copies of V j such that the direct sum decomposition is orthogonal. Consequently, we get a Hermitian structure h i on Sym i (E) using the above isomorphism.
Let Θ j be the curvature of the Chern connection on the Hermitian vector bundle (V , g j ). So the curvature Θ i of the Chern connection for (Sym i (E), h i ) satisfies the equation According to Theorem 1.12 of [DPS], the vector bundle E is nef if for every > 0 and m ≥ m 0 ( ) we have in the sense of Griffiths. It is straight-forward to check that this condition is satisfied. Hence the finite vector bundle E is nef.
We now prove an analogue of Theorem 1.18 of Demailly-Peternell-Schneider [DPS] in our special case of astheno-Kähler manifolds.
Theorem 3.2. Suppose (X, ω) is a compact complex manifold equipped with a Hermitian metric ω that is both, Gauduchon (∂∂ω n−1 = 0), as well as astheno-Kähler (∂∂ω n−2 = 0). Suppose a holomorphic vector bundle E of rank r over X satisfies [c 2 (E)] BC = 0. Then it is numerically flat if and only if the following three conditions hold: (2) there is a Jordan-Hölder filtration

consisting of subbundles, and
(3) for all 1 ≤ i ≤ p, the ω-stable bundle E i /E i−1 is Hermitian flat, i.e., given by an irreducible unitary representation of π 1 (X) in U(rank(E i /E i−1 )).

Remark 3.3.
It can be shown that [c 1 (E)] BC = [c 1 (det(E))] BC = 0 whenever E is numerically flat. Indeed, applying Proposition 1.14 (ii) of [DPS] to the determinant representation of E and E * we see that det(E) is numerically flat. Now Corollary 1.5 of [DPS] implies that det(E) is Hermitian flat and hence [c 1 (det(E))] BC = 0. We are grateful to the anonymous referee for pointing this out.

Proof of Theorem 3.2.
A Hermitian flat bundle is clearly numerically flat. Now from [DPS,Proposition 1.15(ii)] it follows that any holomorphic vector bundle admitting a filtration of holomorphic subbundles such that each successive quotient admits a flat Hermitian structure is also numerically flat. Consequently, E is numerically flat if the three conditions in the theorem hold. We prove the converse below. Assume that E is numerically flat. Our proof inducts on the rank of E. Note that if S ⊂ E is a holomorphic subbundle then the vector bundle E/S is nef because E is nef [DPS,p. 308,Proposition 1.15(i)]. If S is nef, then applying [DPS,p. 308,Proposition 1.15(iii)] to the exact sequence we conclude that (E/S) * is nef. Therefore, E/S is numerically flat if S is so. These observations are crucial for the induction step.
Since ω is assumed to be Gauduchon, we see that for every holomorphic line bundle L, is independent of the choice of Chern connection on L. Moreover, because ω is astheno-Kähler, for any is independent of the choice of Chern connection, i.e., independent of the Bott-Chern representative. Let W be a coherent analytic subsheaf of O(E) such that the quotient O(E)/W is torsion-free. This implies that W is reflexive [Ko,p. 153,Proposition 4.13]. The rank of W will be denoted by q w . Then the arguments of Step 1 in the proof of Theorem 1.18 in [DPS] go through to show that X [c 1 (W )] BC ∧ω n−1 ≤ 0. Now assume that Then we have the following: • det(W ), which is a holomorphic line bundle, is Hermitian flat, As noted before, F is a holomorphic subbundle of E satisfying We also note that F is stable, because it is of minimal rank satisfying (3.2). Therefore, the Kobayashi-Lübke inequality says that see [LYZ,p. 107]. We will now show that in (3.4) the equality holds. It was noted earlier that W is a subbundle of E if it satisfies the condition in (3.2). Consequently, we get a filtration of holomorphic subbundles The arguments in step 2 of the proof of Theorem 1.18 in [DPS] show that it is numerically flat. Therefore, we may induct on the rank of E to get the above filtration. Let q i be the rank of F i /F i−1 . We again have the Kobayashi-Lübke inequality By the cup product formula (and [c 1 (E) 2 ] BC = 0) we see that Recall that [c 2 (E)] BC = 0 by assumption; therefore, we see that equality holds in all of the above inequalities.
Since F is ω-stable, by Li-Yau and Lübke-Teleman's generalization ( [LY], [LT]) of the Uhlenbeck-Yau theorem [UY], it follows that F admits a Hermite-Einstein metric. Since the inequality in (3.4) is an equality, the Hermite-Einstein metric on F is flat and hence has vanishing Chern classes in the Bott-Chern cohomology. Now we take the quotient E/F and use the induction hypothesis. Note that the remarks made earlier show that the quotient is numerically flat, and the cup product formula shows that it has vanishing [c 2 ] BC .

Proof of Theorem 1.1
Any representation W of a finite group G has the property that there are two distinct polynomials f and g with nonnegative integral coefficients such that the two G-modules f (W ) and g(W ) are isomorphic. Indeed, this follows from the following two facts • any finite dimension G-module is a direct sum of irreducible G-modules, and • there are only finitely many isomorphism classes of irreducible G-modules.
Therefore, if a holomorphic vector bundle E on X is given by a linear representation of π 1 (X) with finite image, then E is finite.
To prove the converse, let E be a finite vector bundle over a compact complex manifold X admitting a Gauduchon astheno-Kähler metric. Therefore, there exists integers m ≥ 1, a ij ≥ 0, r j ≥ 1 with 1 ≤ j ≤ m, i ∈ N >0 , and holomorphic vector bundles V j of ranks r j such that (4.1) We may assume that the collection of vector bundles {V j } m j=1 satisfies the three conditions stated prior to Lemma 2.2.
From Theorem 2.1, Lemma 3.1, and Theorem 3.2 we know that E has a filtration of holomorphic subbundles such that for all 1 ≤ i ≤ p, the quotient bundle E i /E i−1 is given by an irreducible unitary representation of π 1 (X).
Lemma 4.1. For every 1 ≤ i ≤ p, the vector bundle E i /E i−1 in (4.2) is finite.
Proof. From Lemma 2.2, Theorem 2.1, Theorem 3.2, and Lemma 3.1 we see that each V j admits a filtration of holomorphic subbundles such that each successive quotient is given by an irreducible unitary representation of π 1 (X). Fix such a Jordan-Hölder filtration for each 1 ≤ j ≤ m, and consider the collection of vector bundles Every vector bundle in the collection in (4.4) is given by an irreducible unitary representation of π 1 (X). We will show that every vector bundle in the collection in (4.4) is finite. More precisely, we will show that for any V j,i /V j,i−1 in (4.4), and any integer α ≥ 1, the tensor power (V j,i /V j,i−1 ) α is a direct sum of copies of vector bundles from the collection in (4.4).
To prove this, note that the filtration of V j in (4.3) produces a holomorphic filtration of subbundles of V α j . The filtration V α j in (4.5) has the following properties: (1) each successive quotient is of the form where n β ≥ 0 and p j β=1 n β = α, and (2) (V j,i /V j,i−1 ) α is one of the successive quotients.
Therefore, each successive quotient in (4.5) is a direct sum of stable vector bundles of degree zero, because it admits a flat unitary connection given by the flat unitary connections on V j,β /V j,β−1 , 1 ≤ β ≤ p j . It should be clarified that each successive quotient in (4.5) is polystable, but need not be stable, because the flat unitary connection on V j,β /V j,β−1 need not be irreducible.
On the other hand, from the proof of Lemma 2.2 we know that Fix an isomorphism of V α j with the direct sum m γ=1 e γ,j V γ . Then fix an ordering of these direct summands of V α j . Now using the filtrations of V j , 1 ≤ j ≤ m, in (4.3), and the above ordering of the direct summands of V α j , we get a filtration , is a vector bundle from the collection in (4.4). We recall that for a Jordan-Hölder filtration of a semi-stable holomorphic vector bundle F with successive quotients Q i reflexive, the isomorphism class of the coherent sheaf i Q i is unique (see [BTT,Proposition 2.1 (p. 998) and Proposition A.2 (p. 1019)]). Consider the two filtrations of subbundles of V α j in (4.5) and (4.6). For the filtration in (4.5), each successive quotient is a direct sum of stable vector bundles of degree zero, while for the filtration in (4.6), each successive quotient is a stable vector bundle of degree zero. Therefore, from Theorem 3 of [At] and the above result of [BTT] we conclude that each successive quotient for the filtration in (4.5) is a direct sum of some successive quotients for the filtration in (4.6). We noted above that all the successive quotients W b j /W b−1 j , 1 ≤ b ≤ 1 , in (4.6) are vector bundles from the collection in (4.4). Therefore, we now conclude that each successive quotient for the filtration in (4.5) is a direct sum of copies of vector bundles from the collection in (4.4). In particular, the vector bundle (V j,i /V j,i−1 ) α is a direct sum of copies of vector bundles from the collection in (4.4). Therefore, the vector bundle (V j,i /V j,i−1 ) α is finite.
Next, from (4.1) we have Fix an isomorphism of E with m j=1 a 1j V j . Then fix an ordering of these direct summands of E. Now using the filtrations of V j , 1 ≤ j ≤ m, in (4.3), and the above ordering of the direct summands of E, we get a filtration such that each successive quotient W b /W b−1 , 1 ≤ b ≤ 0 , is a vector bundle from the collection in (4.4). Consider the two filtrations of subbundles of E in (4.2) and (4.7). For both of them, the successive quotients are stable of degree zero. Therefore, from the above mentioned result of [BTT], namely [BTT,Proposition 2.1 (p. 998) and Proposition A.2 (p. 1019)], we conclude that the graded sheaves for filtrations in (4.2) and (4.7) are isomorphic. We noted above that all the successive quotients W b /W b−1 , 1 ≤ b ≤ 0 , in (4.7) are vector bundles from the collection in (4.4). Therefore, using Theorem 3 of [At] we now conclude that each successive quotient E i /E i−1 , 1 ≤ i ≤ p, in (4.2) is a vector bundle from the collection in (4.4).
We have already proved that every vector bundle in (4.4) is finite. Hence each successive quotient E i /E i−1 , 1 ≤ i ≤ p, in (4.2) is a finite vector bundle.
The following lemma show that E i /E i−1 can be trivialized by pulling back to some finite unramified covering of X.
Lemma 4.2. If X is a compact complex manifold admitting a Gauduchon astheno-Kähler metric, and E is a finite holomorphic vector bundle on X, then there is a finite unramified connected Galois cover π : Y −→ X such that the pull back π * E admits a filtration of holomorphic subbundles with the property that each successive quotient is a holomorphically trivial vector bundle.
Proof. Lemma 4.1 shows that each stable vector bundle E i /E i−1 in (4.2) is finite. Theorem 3.2 shows that the E i /E i−1 has an irreducible unitary flat connection. We claim that the Hermite-Einstein flat connection on E i /E i−1 actually has finite monodromy. Indeed, this follows from Proposition 2.8 of [BHS]. The monodromy representation of the Hermite-Einstein flat connection on E i /E i−1 produces a finite unramified Galois covering of X. Now take Y to be a connected component of the fiber product of the p unramified Galois coverings of X given by . The pullback to Y of the filtration in (4.2) has the required property. The following lemma shows that the holomorphic vector bundle π * E in Lemma 4.2 is trivial.

Lemma 4.3. If E is a finite holomorphic vector bundle of rank r on a compact complex manifold Y such that there is a filtration of holomorphic subbundles
satisfying the conditions that rank(E i ) = i, and the holomorphic line bundle Then E is holomorphically trivial.
Proof. Since E is finite, there are finitely many indecomposable holomorphic vector bundles V 1 , · · · , V m such that for all i ≥ 1, where c i,j are nonnegative integers. By Lemma 2.2 the V i can be chosen to be finite bundles themselves. Given that E i /E i−1 is trivial for all 1 ≤ i ≤ r, each vector bundle E ⊗i admits a filtration of holomorphic subbundles such that each successive quotient of it is the trivial line bundle. Akin to the proof of Lemma 4.1, by the (weak) uniqueness of the graded object of a Jordan-Hölder filtration in the sense of Proposition 2.1 of [BTT] and from Theorem 3 of [At] we see that the V i also admit such filtrations. When r = 1, there is nothing to prove. Next we consider the case of rank two. Assume that (1) r = rank(E) = 2, and (2) E is not trivial.
Therefore, E fits in a short exact sequence of holomorphic vector bundles (4.12) We will now show that dim H 0 (Y , Sym n (E)) = 1 , ∀ n ≥ 1 . (4.13) be the extension class for the short exact sequence in (4.10). We have τ 0 because (4.10) does not split holomorphically. Let be the long exact sequence of cohomologies for the short exact sequence in (4.10). Since the homomorphism ϕ in (4.14) sends 1 ∈ H 0 (Y , O Y ) = C to τ ∈ H 1 (Y , O Y ) (this follows from the construction of the extension class), we conclude that α in (4.14) is an isomorphism. Therefore, (4.13) holds for m = 1. Now we will prove (4.13) using induction on m. Suppose (4.13) holds for all n ≤ n 0 − 1, with n 0 ≥ 2. To prove (4.13) for n = n 0 , consider the short exact sequence of holomorphic vector bundles where γ is the projection in (4.10). The above inclusion of Sym n 0 −1 (E) in Sym n 0 (E) is given by the natural inclusion E ⊗(n 0 −1) ⊗ E 1 → E ⊗(n 0 −1) ⊗ E = E ⊗n 0 (see (4.10)). Let be the long exact sequence of cohomologies associated to the above short exact sequence of vector bundles. Now consider the composition where ϕ is the homomorphism in (4.15), and is the homomorphism given by γ in (4.10). This (γ ⊗(n 0 −1) ) * , which is nonzero because τ 0. So, the homomorphism (γ ⊗(n 0 −1) ) * • ϕ is injective, implying that ϕ is injective. Consequently, the homomorphism β in (4.15) is an isomorphism. This completes the proof of (4.13). From (4.11), (4.13) and (4.12) it follows that m j=1 c i,j ≤ 1 for all i ≥ 0. This implies that In view of this contradiction we conclude that the lemma holds for r = 2.
We will now prove the general case. Assume that (1) r = rank(E) > 2, and (2) E is not trivial.
We will show that there is a subbundle F ⊂ E with 2 ≤ rank(F) = s ≤ r = rank(E), and a filtration of holomorphic subbundles satisfying the following conditions: (1) rank(F i ) = i for all 1 ≤ i ≤ s, The corresponding extension class δ := (δ 1 , · · · , δ s−1 ) ∈ H 1 (Y , , because E s is nontrivial, implying that the above short exact sequence of vector bundles does not split holomorphically. Choose an integer j 0 ∈ [1, s − 1] such that δ j 0 0. Now in (4.16), take • F i = E i for all 0 ≤ i ≤ j 0 − 1 (see (4.8)), • F s−1 = E j 0 , and This filtration clearly satisfies all the three conditions.
Note that for all i ≥ 1, the symmetric product Sym i (F) is a subbundle of Sym i (E), and Sym i (F/F s−2 ) is a quotient of Sym i (F). Since F/F s−2 is a nontrivial extension of O Y by O Y , we have dim H 0 (Y , Sym i (F/F s−2 )) = 1 , ∀ i ≥ 1 , (4.17) as proved in (4.13); note that the proof of (4.13) only uses that the extension in (4.10) is nontrivial, and does not use that the rank two vector bundle is finite. Since Sym i (F/F s−2 ) admit a filtration of holomorphic subbundles such that each successive quotient of it is the trivial line bundle, from (4.17) it follows that Sym i (F/F s−2 ) is indecomposable for all i ≥ 1. Consequently, the indecomposable components (direct summands) of Sym i (E) are of arbitrarily large rank as i goes to infinity. Since Sym i (E) is a direct summand of E ⊗i , the indecomposable components of E ⊗i are also of arbitrarily large rank as i goes to infinity. But from (4.9) it follows that these ranks are bounded by max 1≤j≤m rank(V j ).
In view of the above contradiction, the lemma is proved.
The vector bundle π * E in Lemma 4.2 is finite because E is finite. So from Lemma 4.3 we know that π * E is holomorphically trivial. Let D be a connection on π * E given by a trivialization of it (this connection does not depend on the choice of the trivialization). Let Γ := Gal(π) be the Galois group for the Galois covering π. For any g ∈ Γ , the vector bundle g * π * E is trivial, and any isomorphism between π * E and g * π * E takes the connection D on π * E to the connection g * D on g * π * E. Therefore, D descends to a connection on E; this descended connection will be denoted by D E . We note that D E is flat because D is so. The monodromy representation for D E clearly factors through the quotient π 1 (X) −→ Γ . Therefore, E admits a flat connection with finite monodromy group. This completes the proof of Theorem 1.1.