Norming Sets on a Compact Complex Manifold

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, 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: https://doi.org/10.1007/s12220-018- 0071-x NORMING SETS ON A COMPACT COMPLEX MANIFOLD TANAUS´ U AGUILAR-HERN´ ANDEZ Abstract. We describe the norming sets for the space of global holomorphic sections to a k-power of a positive holomorphic line bundle on a compact complex manifold X. We characterize in metric terms the sequence of measurable subsets {Gk}kof Xsuch that there is a constant C > 0 where ksk2≤CZGk|s(z)|2dV (z) for every s∈H0(X, O(L⊗k)) and for all k∈N. 1. Introduction Let Xbe a compact complex manifold which is endowed with a smooth hermitian metric ω. As we know, this metric induces a distance function d(x, y) on X, that will be used to define the balls, and a volume form V, which will be used to integrate over the manifold. We assume a holomorphic line bundle Lon a compact complex manifold Xendowed with a smooth hermitian metric φ. Moreover, we consider that the line bundle (L, φ) is positive, that is, i∂∂φ is a positive form. We denote by H0(X, O(L⊗k)) the space of global holomorphic sections to a kpower L⊗kof a positive line bundle L, where L⊗kis endowed with the product metric kφ. The statement of the main result is the following: Theorem 1.1. For a sequence of measurable subsets {Gk}in Xthe following are equivalent: (1) There is a constant C > 0such that for all k∈N, ZX|s|2dV ≤CZGk|s|2dV for every s∈H0(X, O(L⊗k)). We will say that {Gk}is a norming sequence for the section ring RX(L) := M k∈N H0(X, O(L⊗k)). (2) There is a constant δ > 0and a radius Rsuch that V ol Gk∩Ba, k−1/2R≥δ V ol Ba, k−1/2R for all a∈Xand k∈N. We will say that Gkis relatively dense in X. 1 2 TANAUS ´ U AGUILAR-HERN ´ ANDEZ As we will see below this problem has already proved for some space of functions. In [LS74] it appears the proof of the Logvinenko-Sereda theorem for functions of the Paley-Wiener space PWKfor a fixed K: Theorem 1.2. For a measurable subset G⊂Rnthe following are equivalent: (1) There is a constant C > 0such that ZRn|f|2dm ≤CZG|f|2dm for every f∈PWK. We will say that Gis a norming set for the Paley- Wiener space. (2) There is a cube Q⊂Rnans a constant γ > 0such that m((Q+x)∩G)≥γ for all x∈Rn. We will say that Gis relatively dense in Rn. See also [HJ94, pag 112-115]. Theorem 1.2 has an analogue for functions in the Bergman space Apin the unit disk Din [Lue81]: Theorem 1.3. For a measurable subset G⊂Dthe following are equivalent: (1) There is a constant C > 0such that ZD|f|pdm ≤CZG|f|pdm for every f∈Ap. We will say that Gis a norming set for the Bergman space. (2) There is a constant δ > 0and radius R∈(0,1) such that m(G∩D(a, (1 −|a|)R)) ≥δm(D(a, (1 −|a|)R)) for all a∈D. We will say that Gis relatively dense in the disc D. It should be noted that using similar arguments to those used in [Lue81] one can prove the Fock space version of D.H. Luecking: Theorem 1.4. For a a measurable subset G⊂Cnthe following are equivalent: (1) There is a constant C > 0such that ZCn|f|2e−2|z|2dm(z)≤CZG|f|2e−2|z|2dm(z) for every f∈ F2 2|z|2(Cn). We will say that Gis a norming set for the classical Fock space. (2) There is a constant δ > 0and a radius R > 0such that m(G∩B(z, R)) ≥δ m(B(a, R)) for all z∈Cn. We will say that Gis relatively dense in Cn. NORMING SETS ON A COMPACT COMPLEX MANIFOLD 3 2. Peak Sections 2.1. The Bergman Kernel. The space H0(X, O(L⊗k)) admits a Hilbert space structure endowed with the scalar product hu, vi=ZXhu(x), v(x)i, u, v ∈H0(X, O(L⊗k)) where the integration is taken with respect to the volume form V. The Bergman kernel Πk(x, y) associated to this space is a section to the line bundle L⊗kL⊗kover the manifold X×X, defined by Πk(x, y) = N X j=1 sj(x)⊗sj(y) where s1, . . . , sNis an orthonormal basis for H0(X, O(L⊗k)). Moreover, this definition does not depend on the choice of the orthonormal basis. Notice that L⊗kL⊗k= π∗ 1(L⊗k)⊗π∗ 2(L⊗k) where πi=X×X→Xis the projection onto the i-factor. The Bergman kernel Πk(x, y) is in a sense the reproducing kernel for the space H0(X, O(L⊗k)), satisfying the reproducing formula s(x) = ZXhs(y),Πk(x, y)idV (y) for s(y)∈H0(X, O(L⊗k)). The pointwise norm of the Bergman kernel in symmetric, |Πk(x, y)|=|Πk(y, x)|. Moreover, it satisfies |Πk(x, x)|=ZX|Πk(x, y)|2dV (y). Lemma 2.1. Let (L, φ)be a positive line bundle. We have the off-diagonal estimate |Πk(x, y)|.kne−c√k d(x,y) where cis an appropriate positive constant and the diagonal estimate |Πk(x, x)|  kn. Moreover, from this we obtain the estimate of the dimension of H0(X, O(L⊗k)) dim H0(X, O(L⊗k)) kn. See [Ber03]. 2.2. Construction of Peak Sections. The main goal of the lemma is to prove for each ball the existence of a normalized peak-section, that is, a section such that most of the mass is in such ball. For this we will use the reproducing kernel of H0(X, O(L⊗k)). Lemma 2.2. Given ε > 0, there exist R > 0and k0>0such that for all k≥k0 and all y∈Xthere exists a section s=sy,k ∈H0(X, O(L⊗k)) such that • ksk2=RX|s|2dV = 1, and 4 TANAUS ´ U AGUILAR-HERN ´ ANDEZ •RX\B(y,k−1/2R)|s|2dV < ε. Proof. Given ε > 0. First of all, we take any k > k1= max{k∈N:H0(X, O(L⊗k)) = 0}, since there are ample line bundles Lwithout non-zero sections. We consider the reproducing kernel of the Hilbert space H0(X, O(L⊗k)) Πk(x, y) = N X j=1 sj(x)⊗sj(y) where s1, . . . , sNis an orthonormal basis for H0(X, O(L⊗k)). If we fix y∈X, there is a section Φk,y ∈H0(X, O(L⊗k)) such that |Φk,y(x)|=|Πk(x, y)|, x ∈X, by the lemma in [LOC12, pag. 431]. So, we assume the section sk,y(x) = Φk,y(x) p|Πk(y, y)|, x ∈X, where Πkdenotes the Bergman kernel for the k’th power L⊗kof the line bundle L. First, ZX|sk,y(x)|2dV (x) = 1 |Πk(y, y)|ZX|Πk(x, y)|2dV (x) = |Πk(y, y)| |Πk(y, y)|= 1. Next, ZX\B(y,k−1/2R)|sk,y(x)|2dV (x) = 1 |Πk(y, y)|ZX\B(y,k−1/2R)|Πk(x, y)|2dV (x) =1 |Πk(y, y)|Z∞ 0 V ol {x:|Πk(x, y)|> λ}\By, k−1/2R2λ dλ ≤1 |Πk(y, y)|ZknD 0 V ol nx:De−c√k d(x,y)≥k−nλo\By, k−1/2R2λ dλ. where we use that |Πk(x, y)| ≤ knDe−c√k d(x,y)for some constants c, D > 0, as we see in Lemma 2.1. Applying the change of variable λ=knDe−cu. 1 |Πk(y, y)|Z∞ 0 V ol By, k−1/2u\By, k−1/2R(2knDe−cu)|−cknDe−cu|du .1 knZ∞ Ru √k2n (2knDe−cu)|−cknDe−cu|du .Z∞ R u2ne−2cu du =e−2cR Z∞ 0 (µ+R)2ne−2cµ dµ =e−2cR 2n X j=0 2n kR2n−jZ∞ 0 µje−2cµ dµ .e−2cR 2n X j=0 2n kR2n−j (2c)jj!.e−2cR 2n X j=0 2n k(2cR)2n−j=e−2cR(1 + 2cR)2n. NORMING SETS ON A COMPACT COMPLEX MANIFOLD 5 Finally, for Rlarge enough we obtain that ZX\B(y,k−1/2R)|sk,y(x)|2dV (x)< ε. In addition, we choose k0>> R2such that k0> k1and for all k≥k0the balls B(y, k−1/2R) are not the whole space. So, we conclude the proof of the lemma.  Remark 2.3. Notice that taking the same section of the previous lemma and using the Cauchy-Schwarz inequality we obtain that |s(x)|2=|hs(y),Πk(x, y)i|2≤ ksk2|Π(x, x)|.kn. 3. Proof of the Main Theorem We are ready to prove Theorem 1.1. The proof that (1) implies (2) is the easiest. In this proof, we consider a section with the properties of Lemma 2.2. So, given ε≤1/2Cand applying the Lemma 2.2 there is a radius Rand k0>0 such that for all balls Ba, k−1/2Rand k≥k0there is a section s∗=s∗ a,k verifying the properties of the lemma. Hence, applying the Remark 2.3 we obtain V ol Gk∩Ba, k−1/2R V ol (B(a, k−1/2R)) &1 R2nZGk∩B(a,k−1/2R)|s∗|2dV and using the other properties with (1) we have for k≥k0that V ol Gk∩Ba, k−1/2R V ol (B(a, k−1/2R)) &1 R2n ZGk|s∗|2dV −ZX\B(a,k−1/2R)|s∗|2dV ! ≥1 R2n1 C−ε≥1 2CR2n. Therefore, we have proved that (1) implies (2). The proof that (2) implies (1) is the difficult one. We will use a similar argument as in [MOC08, Chapter 4]. We will partition Xin two pieces. The first piece, denoted by As, is the set of points where the section is much smaller that its average and the second is the complementary X\As. The following lemma proves that the integral over Asis irrelevant as most of the mass is carried by X\As. Lemma 3.1. Let s∈H0(X, O(L⊗k)). Define the set As=a∈X| |s(a)|2<1 2V ol(B(a, R/√k)) ZB(a,k−1/2R)|s|2dV . Then ZAs|s|2dV < 1 2ZX|s|2dV. 6 TANAUS ´ U AGUILAR-HERN ´ ANDEZ Proof. For a∈ Aswe have |s(a)|2<1 2V ol(B(a, k−1/2R)) ZB(a,k−1/2R)|s(z)|2dV (z) = 1 2ZX|s(z)|2χB(a,k−1/2R)(z) V ol(B(a, k−1/2R)) dV (z). Integrating with respect to aand applying Fubini’s theorem ZAs|s|2dV < 1 2ZX|s(z)|2ZAs χB(a,k−1/2R)(z) V ol(B(a, k−1/2R)) dV (a)dV (z) ≤V ol(B(a, k−1/2R)) V ol(B(a, k−1/2R)) 1 2ZX|s(z)|2dV (z) = 1 2ZX|s|2dV. Therefore, we obtain ZAs|s|2dV < 1 2ZX|s|2dV.  Let Fs=X\As=na∈X:|s(a)|2≥1 2V ol(B(a,k−1/2R)) RB(a,k−1/2R)|s|2dV o. We have ZX|s|2dV < 2ZFs|s|2dV since ZX|s|2dV =ZFs|s|2dV +ZAs|s|2dV < ZFs|s|2dV +1 2ZX|s|2dV. Thus it is enough to show that ZFs|s|2dV .ZGk|s|2dV. All we need to prove is the existence of a constant C > 0 such that for all k >> 0, all s∈H0(X, O(L⊗k)) and all w∈ Fs |s(w)|2≤C V ol (B(w, k−1/2R)) ZB(w,k−1/2R)∩Gk|s|2dV. (3.1) Indeed, if this is the case then ZFs|s|2dV ≤CZGk|s(z)|2 ZFs χB(w,k−1/2R)(z) V ol (B(w, k−1/2R)) dV (w)!dV (z) ≤CZGk|s|2dV. To prove (3.1) we argue by contradiction. If (3.1) is false then for any q∈Nthere are sections sq∈H0(X, O(L⊗kq)) and wq∈ Fssuch that |sq(wq)|2>q V ol Bwq, k−1/2 qRZB(wq,k−1/2 qR)∩Gkq|sq(z)|2dV (z) (3.2) NORMING SETS ON A COMPACT COMPLEX MANIFOLD 7 By compactness of Xwe can choose a convergent subsequence {w∗ q}of wqto some w∗∈Xsuch that there is a local chart (U, ϕ) where Bw∗ q, k−1/2 qR⊂U, ∀q∈N and ϕ(w∗) = 0. As φis given by the negative logarithm of the norm of some frame ξof L, that is, φ=−log |ξ|2. We can choose a frame properly such that φ(ϕ−1(z)) = |z|2+q(z) in the local ball B(0, τ) where q(z) = o(|z|2). Hence, we have that |2φ(ϕ−1(z)) −2|z|2| ≤ c(τ)τ2 in B(0, τ) for nondecreasing continuous function c(τ) such that c(0) = 0. Notice that we can use the local chart (ϕ−1(B(0, τ∗)), ϕ) instead since there exists ν∈N such that for all q≥ν Bw∗ q, k−1/2 qR⊂ϕ−1(B(0, τ∗)) and τ∗≤τ. Now, we consider the sequence of holomorphic functions Fq(z) = fq(ϕ−1(z)), q ∈N where sq=fqξ⊗kwith |sq|2=|fq|2e−kφ and we need to see that |sq(w∗ q)|2.|Fq(ηq)|2(3.3) where ϕ(w∗ q) = ηq, |sq(ϕ−1(z))|2&|Fq(z)|2(3.4) for a certain ball and |Fq(λq)|2&q m(B(ηq, δq)) ZB(ηq,δq)∩ϕ(Gkq)|Fq(z)|2dm(z).(3.5) First, for the sequence {w∗ q}we have |sq(w∗ q)|2=|fq(ϕ−1(ηq))|2e−kφ(ϕ−1(ηq)) ≤ |fq(ϕ−1(ηq))|2ek 2c(τ∗)τ∗2.|Fq(ηq)|2. Furthermore for z∈B(0, τ∗) we obtain |Fq(z)|2=|fq(ϕ−1(z))|2.|sq(ϕ−1(z))|2, z ∈B(0, τ∗). Finally, using (3.2), (3.3), (3.4) and taking the local chart (ϕ−1(B(0, τ∗)), ϕ), it follows that |Fq(ηq)|2&|sq(w∗ q)|2&q V ol Bw∗ q, k−1/2 qRZBw∗ q,k−1/2 qR∩Gkq|sq|2dV &q V ol Bw∗ q, k−1/2 qRZϕBw∗ q,k−1/2 qR∩Gkq|fq(ϕ−1(z))|2e−kφ(ϕ−1(z)) dm(z) &q m(B(ηq, δkq)) ZB(ηq,δkq)∩G∗ kq|Fq|2dm 8 TANAUS ´ U AGUILAR-HERN ´ ANDEZ where G∗ kq=ϕ(Gkq) and there is a ball B(ηkq, δkq)⊂ϕ(B(w∗ q, k−1/2 q)) such that m(B(ηq, δkq)) V ol Bw∗ q, k−1/2 qR, because as ϕ, ϕ−1∈ C1we have that ϕis bi-Lipschitz. Next, by means of a dilation and a translation we send ηqto the origin of Cn, the ball B(ηq, δkn) to B(0,1) ⊂Cnand the set G∗ kqto G∗ q. We will denote the set B(0,1)∩G∗ qby Hq. Moreover, we apply the above transformations to the function Fq and multiply by a constant such that the resulting function, denoted by ˜ Fq, verifies ZB(0,1) |˜ Fq(z)|2dm = 1. The subharmoniticity of |˜ Fq|2and the fact that w∗ q∈ Fstells us that 1 2.|˜ Fq(0)|2.1.(3.6) Notice that we can modify the inequality of the definition of Fsin the same way as the expression (3.5). This property (3.6) together with (3.5) yields 1 q&ZHq|˜ Fq(u)|2dm(u).(3.7) Now, we have that {˜ Fq}is a locally bounded sequence of holomorphic functions defined in B(0,1) and therefore using Montel theorem there exist a subsequence converging locally uniformly on B(0,1) to some holomorphic function G. We observe that the relative dense hypothesis yields inf qm(Hq)>0. The Banach-Alaoglu theorem [Rud06, pag. 68] guarantees the existence of a weak- ∗limit µof a subsequence of µq=χHqmsuch that σ6≡ 0. Condition (3.7) implies that ˜ Fq= 0 µ-a.e. and therefore suppµ ⊂ {˜ Fq= 0}. We want to show that suppµ cannot lie on a complex (n−1)-dimensional submanifold S⊂Cn. We argue by contradiction. By definition of µqwe have µq(B(x, r)) ≤m(B(x, r)) .r2n. As this inequality is true for all µq, the same holds for µ. But, using the Frostman lemma [Mat99, pag. 112] we obtain that dimHaus(suppτ)≥2n > 0. Therefore, we have proved that (2) implies (1).  4. Sampling Sequences Definition 4.1. Let {µk}k⊂M(X) be a sequence of measures. We will say that it defines uniformly standard norms for the section ring RX(L) if there is a constant C > 0 such that ZX|s|2dµk≤CZX|s|2dV