Uniform L²-estimates for flat nontrivial line bundles on compact complex manifolds

Theorem 1.1

[3, Theorem 1.1] Let ( X , g ) be a compact Kähler manifold. Then, there exists a constant K > 0 such that, for any element F ∈ P ( X ) \ { I X } and any smooth ∂ ¯ -closed ( 0 , 1 ) -form v with values in F whose Dolbeault cohomology class [ v ] ∈ H 0 , 1 ( X , F ) is trivial, there exists a unique smooth section u of F such that ∂ ¯ u = v and

hold for a flat Hermitian metric h on F.

Theorem 1.2

(Ueda’s lemma, [8, Lemma 4]) Let X be a compact complex manifold and U ≔ { U j } be a finite open covering of X such that each U j is a Stein open set and U j trivializes any F ∈ P ( X ) . Then, there exists a constant K U > 0 such that, for any F ∈ P ( X ) \ { I X } and any Čech 0-cochain f ≔ { ( U j , f j ) } ∈ C ˇ 0 ( U , O X ( F ) ) , the inequality

holds for a flat Hermitian metric h on F, where { ( U j k , f j k ) } ≔ δ f ∈ C ˇ 1 ( U , O X ( F ) ) is the Čech coboundary of f .

Theorem 1.3

Theorem 1.1  holds for a compact complex manifold ( X , g ) with a Hermitian metric g.

Proof of Theorem 1.3

The strategy of the proof is to chase the Dolbeault isomorphism between ∂ ¯ -cohomology and Čech cohomology, using the Hörmander’s L 2 -method, which allows us to apply Ueda’s lemma for the δ -equation.

We work with a fixed Hermitian metric g on X , with all norms and volume forms defined with respect to g . We now recall a foundational result by Hörmander (e.g., [2, Chapter VIII, Theorem 6.9], [1, Théorème 4.1], or [4, Theorem 4.4.2]), which guarantees the following: Let U x * be a Stein open neighborhood of a given point x ∈ X , and let h = e − φ be a singular Hermitian metric on the trivial line bundle with φ psh. Then, for any smooth ( 0 , 1 ) -form v x on U x * such that ∂ ¯ v x = 0 and ‖ v x ‖ L 2 ( U x * ) , h < ∞ , there exists a smooth function u x on U x * such that

with the L 2 estimate

where the constant C x , g > 0 depends only on U x * and g . Note that the solution of u x can be chosen to be smooth if h = e − φ and v x are smooth.

We may assume, without loss of generality and by shrinking each U x * , that U x * trivializes any flat line bundle on X [3, Lemma 2.7]. For each x ∈ X , we pick a relatively compact open subset U x * * ⋐ U x * . We then cover X with the open sets { U x * * } x ∈ X , and by the compactness of X , we may select a finite subcover { U x j * * } j ∈ I . By setting U j ′ ≔ U x j * * and U j ≔ U x j * for each j ∈ I , we obtain two finite covers { U j ′ } j ∈ I and { U j } j ∈ I of X . By construction, we can observe that U j are Stein, U j ′ ⋐ U j , and U j (and thus U j ′ as well) trivializes any flat line bundle on X .

Let F be a flat line bundle endowed with a flat Hermitian metric h , which is unique up to a multiplicative constant (e.g., [3, Lemma 2.4]). Let v be a smooth ∂ ¯ -closed ( 0 , 1 ) -form with values in F , whose Dolbeault cohomology class [ v ] ∈ H 0 , 1 ( X , F ) is trivial. In what follows, all norms are with respect to the fixed flat Hermitian metric h on F and the fixed Hermitian metric g on X . By applying Hörmander’s estimates and trivializing F equipped with the metric h , we find that for each j ∈ I , there exists a smooth function u j on U j such that

for a constant C j , g ( 1 ) > 0 , which depends only on U j and g . The key point of this estimate is that it holds uniformly for all flat line bundles ( F , h ) and for all v representing a trivial Dolbeault class in H 0 , 1 ( X , F ) .

We then find that { u i − u j } i , j ∈ I defines a holomorphic section of F on U i j ≔ U i ∩ U j . The isomorphism between Čech cohomology and Dolbeault cohomology implies that { u i − u j } i , j ∈ I represents the Čech cohomology class corresponding to [ v ] . Since [ v ] is trivial, we find that { u i − u j } i , j ∈ I is a Čech 1-coboundary (after replacing U x * with a smaller polydisk if necessary), meaning that there exist local holomorphic sections { f j } j ∈ I of F such that

on U i j for all i , j ∈ I . We take a partition of unity { ρ j } j ∈ I subordinate to { U j } j ∈ I . The equation f i − f j = u i − u j on U i j for all i , j ∈ I then implies

on U i . Taking the ∂ ¯ -operator of both sides, we have

on U i , as ∑ j ∈ I ρ j ≡ 1 . Since ∂ ¯ u i = v ∣ U i , we obtain

Noting that ∑ j ∈ I ρ j ( u j − f j ) is defined globally on X , we have

and hence, u ≔ ∑ j ∈ I ρ j ( u j − f j ) gives the solution to the equation ∂ ¯ u = v , which is necessarily unique since F is flat and nontrivial (a well-known result, refer to, e.g., [3, Lemma 2.3] for the proof).

The argument above (except for the uniqueness of solutions) is valid for any ( 0 , q ) -forms v with values in F ⊗ ℐ ( h ) , where h is a singular Hermitian metric on F with positive curvature current (refer to [7, Subsection 5.3] for details). In the following discussion, we essentially use the flatness property.

We apply Ueda’s lemma (Theorem 1.2) to the Čech 0-cochain f ≔ { ( U j ′ , f j ∣ U j ′ ) } corresponding to the smaller cover U ′ ≔ { U j ′ } j ∈ I , to obtain

uniformly for all F ∈ P ( X ) \ { I X } , for some constant K U ′ > 0 . Note that we have

for each j ∈ I , and

since u j − u k = f j − f k . Consider a weight φ j of h on U j such that h = e − φ j . Since φ j is pluriharmonic, there exists a holomorphic function w j on U j such that φ j is the real part of w j , which implies that h = e − φ j = ∣ e − w j ∣ . Given that ( u j − u k ) e − w j is holomorphic on U j k (which contains U j k ′ ), we have

for some constant C j , g ( 2 ) > 0 , depending only on U j ′ , U j , and g , by the mean value inequality and the Cauchy-Schwarz inequality. Hence,

by the Hörmander estimate (1) on U j , where we set C g ( 3 ) ≔ 2 max j C j , g ( 2 ) C j , g ( 1 ) > 0 . With (2) and (3), the estimate (4) gives

for any j ∈ I .

We now recall u = ∑ j ∈ I ρ j ( u j − f j ) and evaluate

again by the Hörmander estimate (1) on U j , where C g ( 4 ) > 0 is a constant, which depends only on g . Combined with (5), we thus obtain

Noting that sup F ∈ P ( X ) d ( I X , F ) is finite and I is a finite set, we obtain the required estimate.□