Correction to: A central limit theorem associated with a sequence of positive line bundles (The Journal of Geometric Analysis, (2025), 35, 3, (86), 10.1007/s12220-025-01921-9)

Official URL: https://dx.doi.org/10.1007/s12220-025-01973-x

In this article, the following corrections were made: Under Lemma 2.2, the following equation was incorrectly given as: “Let {Uj}j=1N be a finite subcover of X. Locally, on each Uj, we have the following representations (Formula presented.) and (Formula presented.) Here [χkj(z)]kj is a positive definite Hermitian matrix, because in a Kähler manifold, we always have a strictly plurisubharmonic local potential function ψ so that (Formula presented.) Similarly, since line bundles Lp are positive, by the definition of positivity, [αkj(z)]kj=[∂2φp∂zj∂zk¯(z)]kj is a positive definite Hermitian matrix, where φp is the corresponding local weight function for hp. Note that in particular αkj(z)∈R for every 1≤k,j≤n. Let us fix some Ul taken from the subcover. By the diophantine approximation condition (2.7) on Ul¯, for any ϵ>0, there exists some p0=p0(ϵ)∈N such that, for all p≥p0, (Formula presented.) for all z∈Ul¯. Take, for example, (Formula presented.) Then (2.11) gives (Formula presented.) Summing this last inequality over idzk∧dzj¯, we have, for all z∈Ul¯⊂X and for all p≥p0 (Formula presented.) which concludes that (Formula presented.) for p≥p0. This will be useful in the proofs of Theorem 4.1 and Theorem 4.2”. Should have been: “Let {Uj}j=1N be a finite subcover of X. Locally, on each Uj, we have the following representations (Formula presented.) and (Formula presented.) Here [χkj(z)]kj is a positive definite Hermitian matrix, because in a Kähler manifold, we always have a strictly plurisubharmonic local potential function ψ so that (Formula presented.) Similarly, since line bundles Lp are positive, by the definition of positivity, [αkj(z)]kj=[∂2φp∂zj∂zk¯(z)]kj is a positive definite Hermitian matrix, where φp is the corresponding local weight function for hp. By using the diophantine approximation (2.8) and the positivity of the above two forms, one can find some large enough p0∈N such that (Formula presented.) for p≥p0. This will be useful in the proofs of Theorem 4.1 and Theorem 4.2”. In the section heading “4.1 Linearization” the following corrections were made: The below sentence has been included: “Fix a point x in X and let us take the coordinates centered at x≡0, Kahler at x, as provided by Lemma 2.2, with coordinate polydisk Pn(x,R) for R>0. Let R′>0 be arbitrary and choose p≫1 so that R′/Ap<R.” The sentence “Ω={(u,v):u,v∈Pn(0,R)}, where p:=Diag[λ1p,…,λnp]” has been corrected to “Ω={(u,v):u∈Pn(0,R′),v∈Pn(0,R′)}⊂Cun×Cvn in the respective coordinates u, v”. The sentence “Ω0:={(u,u¯):u∈Pn(0,R)}” has been corrected to “Ω0:={(u,u¯):u∈Pn(0,R′)}”. The sentence “as p→∞” has been corrected to “on Pn(0,R′)×Pn(0,R′) as p→∞. Therefore, we have proved the next theorem”. The equation number 4.55 has been removed and the below theorem has been included which gives the local uniform convergence: Let (X,ω) be a compact Kähler manifold of complex dimension n and let a sequence of holomorphic line bundles {(Lp,hp)}p≥1 be given, each equipped with a C3-class Hermitian metric hp, satisfying the condition (2.7). Suppose that ηp=‖hp‖31/3Ap→0 as p→∞. Let x be a fixed point in X. Then in the local Kähler coordinates centered at x provided by Lemma 2.2, we have (Formula presented.) locally uniformly on Cun×Cvn as p→∞. In the section heading “5.2 Asymptotic Normality of Random Zero Currents” the sentence “Finally, by the linearization (4.51) on the neighborhood Uj” has been corrected to “Finally, by the linearization (4.51) on the neighborhood Pn(x,R)”. Under the ‘Theorem 1.2’ the equation ||hp||3Ap has been corrected to ||hp||31/3Ap. Under the ‘Theorem 4.1’ the equation ηp=||hp||3Ap has been corrected to ηp=||hp||31/3Ap. Under the ‘Theorem 4.1’ the sentence “By our hypothesis ηp=||hp||3Ap→0” has been corrected to “By our hypothesis ηp=||hp||31/3Ap→0”. Under the ‘Theorem 4.2’ the sentence “Write ηp=||hp||3Ap→0 when p→∞” has been corrected to “Write ηp=||hp||31/3Ap→0 when p→∞”. Under the ‘Theorem 4.2’ the sentence “and by our assumption that ηp=||hp||3Ap→0 when p→∞” has been corrected to “and by our assumption that ηp=||hp||31/3Ap→0 when p→∞”. The original article has been corrected.