DAN COMAN, GEORGE MARINESCU, AND VIÊT-ANH NGUYÊN

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1 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION DAN COMAN, GEORGE MARINESCU, AND VIÊT-ANH NGUYÊN ABSTRACT. We show that normalized currents of integration along the common zeros of random m-tuples of sections of powers of m singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with singularities we also estimate the speed of convergence. CONTENTS. Introduction 2. Asymptotic behavior of Bergman kernel functions 5 3. Intersection of Fubini-Study currents and Bertini type theorem 6 4. Convergence speed towards intersection of Fubini-Study currents Dinh-Sibony equidistribution theorem Equidistribution of pullbacks of Dirac masses by Kodaira maps 5. Equidistribution with convergence speed for Hölder singular metrics 8 References 22. INTRODUCTION Random polynomials or more generally holomorphic sections and the distribution of their zeros represent a classical subject in analysis [BP, ET, H, K], and they have been more recently used to model quantum chaotic eigenfunctions [BBL, NV]. This area witnessed intense activity recently [BL, BS, BMO, DMS, DS, S, SZ, ST], and especially results about equidistribution of holomorphic sections in singular Hermitian holomorphic bundles were obtained [CM, CM2, CM3, CMM, DMM] with emphasis on the speed of convergence. The equidistribution is linked to the Quantum Unique Ergodicity conjecture of Rudnick-Sarnak [RS], cf. [HS, Mar]. Date: May 25, Mathematics Subject Classification. Primary 32L0; Secondary 32A60, 32U40, 32W20, 53C55, 8Q50. Key words and phrases. Bergman kernel function, Fubini-Study current, big line bundle, singular Hermitian metric, zeros of random holomorphic sections. D. Coman is partially supported by the NSF Grant DMS G. Marinescu is partially supported by DFG funded projects SFB/TR 2, MA 2469/2-2. V.-A. Nguyen is partially supported by Humboldt Foundation and Max-Planck Institute for Mathematics. Funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.

2 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 2 The equidistribution of common zeros of several sections is particularly interesting. Their study is difficult in the singular context and equidistribution with the estimate of convergence speed was established in [DMM] for Hölder continuous metrics. In this paper we obtain the equidistribution of common zeros of sections of m singular Hermitian line bundles under the hypothesis that the metrics are continuous outside analytic sets intersecting generically. We will moreover introduce the notion of Hölder metric with singularities and establish the equidistribution with convergence speed of common zeros. Let (, ω) be a compact Kähler manifold of dimension n and dist be the distance on induced by ω. If (L, h) is a singular Hermitian holomorphic line bundle on we denote by c (L, h) its curvature current. Recall that if e L is a holomorphic frame of L on some open set U then e L 2 h = e 2φ, where φ L loc (U) is called the local weight of the metric h with respect to e L, and c (L, h) U = dd c φ. Here d = +, d c = ( ). We 2πi say that h is positively curved if c (L, h) 0 in the sense of currents. This is equivalent to saying that the local weights φ are plurisubharmonic (psh). Recall that a holomorphic line bundle L is called big if its Kodaira-Iitaka dimension equals the dimension of (see [MM, Definition 2.2.5]). By the Shiffman-Ji-Bonavero- Takayama criterion [MM, Lemma 2.3.6], L is big if and only if it admits a singular metric h with c (L, h) εω for some ε > 0. Let (L k, h k ), k m n, be m singular Hermitian holomorphic line bundles on (, ω). Let H(2) 0 (, Lp k ) be the Bergman space of L2 -holomorphic sections of L p k := L p k relative to the metric h k,p := h p k induced by h k and the volume form ω n on, endowed with the inner product () (S, S ) k,p := S, S hk,p ω n, S, S H(2)(, 0 L p k ). Set S 2 k,p = (S, S) k,p, d k,p = dim H(2) 0 (, Lp k ). For every p we consider the multiprojective space (2) p := PH 0 (2)(, L p )... PH 0 (2)(, L p m) equipped with the probability measure σ p which is the product of the Fubini-Study volumes on the components. If S H 0 (, L p k ) we denote by [S = 0] the current of integration (with multiplicities) over the analytic hypersurface {S = 0} of. Set [s p = 0] := [s p = 0]... [s pm = 0], for s p = (s p,..., s pm ) p, whenever this is well-defined (cf. Section 3). We also consider the probability space (Ω, σ ) := ( p, σ p ). Let us recall the following: p= Definition.. We say that the analytic subsets A,..., A m, m n, of a compact complex manifold of dimension n are in general position if codim A i... A ik k for every k m and i <... < i k m.

3 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 3 Here is our first main result. Theorem.2. Let (, ω) be a compact Kähler manifold of dimension n and (L k, h k ), k m n, be m singular Hermitian holomorphic line bundles on such that h k is continuous outside a proper analytic subset Σ(h k ), c (L k, h k ) εω on for some ε > 0, and Σ(h ),..., Σ(h m ) are in general position. Then for σ -a.e. {s p } p Ω, we have in the weak sense of currents on, p m [s p = 0] c (L, h )... c (L m, h m ) as p. In order to prove this theorem we show in Theorem 4.2 that the currents [s p m p = 0] distribute as p like the wedge product of the normalized Fubini-Study currents of the spaces H(2) 0 (, Lp k ) defined in (0) below. Then in Proposition 3. we prove that the latter sequence of currents converges to c (L, h )... c (L m, h m ). Our second main result gives an estimate of the speed of convergence in Theorem.2 in the case when the metrics are Hölder with singularities. Definition.3. We say that a function φ : U [, ) defined on an open subset U is Hölder with singularities along a proper analytic subset Σ if there exist constants c, ϱ > 0 and 0 < ν such that (3) φ(z) φ(w) c dist(z, w) ν min{dist(z, Σ), dist(w, Σ)} ϱ holds for all z, w U \ Σ. A singular metric h on L is called Hölder with singularities along a proper analytic subset Σ if all its local weights are Hölder functions with singularities along Σ. Hölder singular Hermitian metrics appear frequently in complex geometry and pluripotential theory. Let us first observe that metrics with analytic singularities [MM, Definition 2.3.9], which are very important for the regularization of currents and for transcendental methods in algebraic geometry [BD, CM3, D3, D4, D6], are Hölder metrics with singularities. The class of Hölder metrics with singularities is invariant under pullback and push-forward by meromorphic maps. In particular, this class is invariant under birational maps, e. g. blow-up and blow-down. They occur also as certain quasiplurisubharmonic upper envelopes (e. g. Hermitian metrics with minimal singularities on a big line bundle, equilibrium metrics, see [BB, DMM, DMN], especially [BD, Theorem.4]). Theorem.4. In the setting of Theorem.2 assume in addition that h k is Hölder with singularities along Σ(h k ) and set Σ := Σ(h )... Σ(h m ). Then there exist a constant ξ > 0 depending only on m and a constant c = c(, L, h,..., L m, h m ) > 0 with the following property: For any sequence of positive numbers {λ p } p such that lim inf p λ p log p > ( + ξn)c, there are subsets E p p such that for p large enough, (a) σ p (E p ) cp ξn exp( λ p /c),

4 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 4 (b) if s p p \ E p we have m ( p [s λp m p = 0] c (L k, h k ), φ c p + ) log dist(supp dd c φ, Σ) φ C 2, p k= for any form φ of class C 2 such that dd c φ = 0 in a neighborhood of Σ. Here supp ψ denotes the support of the form ψ. Let P p be the Bergman kernel function of the space H 0 (2) (, Lp ) defined in (4). The proof of Theorem.4 uses the estimate for P p obtained in Theorem 2. in the case when the metric h on L is Hölder with singularities. One can also apply Theorem 2. to study the asymptotics with speed of common zeros of random n-tuples of sections of a (single) big line bundle endowed with a Hölder Hermitian metric with isolated singularities. Let (L, h) be a singular Hermitian holomorphic line bundle on (, ω) and H 0 (2) (, Lp ) be the corresponding spaces of L 2 -holomorphic sections. Consider the multi-projective space p := (PH 0 (2)(, L p )) n endowed with the product probability measure σ p induced by the Fubini-Study volume on PH(2) 0 (, Lp ), and let (Ω, σ ) := ( p, σ p). If s p = (s p,..., s pn ) p we let [s p = 0] = [s p = 0]... [s pn = 0], provided this measure is well-defined. Theorem.5. Let (, ω) be a compact Kähler manifold of dimension n and (L, h) be a singular Hermitian holomorphic line bundle on such that h is Hölder with singularities in a finite set Σ = {x,..., x J }, and c (L, h) εω for some ε > 0. Then there exist C > 0, p 0 N depending only on (, ω, L, h), and subsets E p p such that: (a) σ p(e p ) Cp 2 ; (b) For every p > p 0, every s p p \ E p, and every function χ of class C 2 on, ( ) χ p [s n p = 0] c (L, h) n C log p p χ /3 C 2. In particular, this estimate holds for σ -a.e. sequence {s p } p Ω provided that p is large enough. This paper is organized as follows. In Section 2 we prove a pointwise estimate for the Bergman kernel function in the case of Hölder metrics with singularities. Section 3 is devoted to the study of the intersection of Fubini-Study currents and to a version of the Bertini theorem. In Section 4 we consider the Kodaira map as a meromorphic transform and estimate the speed of convergence of the intersection of zero-divisors of m bundles. We use this to prove Theorem.2. Finally, in Section 5 we prove Theorem.4 and Theorem.5. p= Acknowledgment. We thank Vincent Guedj for useful discussions regarding this paper.

5 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 5 2. ASYMPTOTIC BEHAVIOR OF BERGMAN KERNEL FUNCTIONS In this section we prove a theorem about the asymptotic behavior of the Bergman kernel function in the case when the metric is Hölder with singularities. Let (L, h) be a holomorphic line bundle over a compact Kähler manifold (, ω) of dimension n, where h is a singular Hermitian metric on L. Consider the space H(2) 0 (, Lp ) of L 2 -holomorphic sections of L p relative to the metric h p := h p induced by h and the volume form ω n on, endowed with the natural inner product (see ()). Since H(2) 0 (, Lp ) is finite dimensional, let {S p j }dp j=0 be an orthonormal basis and denote by P p the Bergman kernel function defined by d p (4) P p (x) = S p j (x) 2 h p, Sp j (x) 2 h := p Sp j (x), Sp j (x) hp, x. j=0 Note that this definition is independent of the choice of basis. Theorem 2.. Let (, ω) be a compact Kähler manifold of dimension n and (L, h) be a singular Hermitian holomorphic line bundle on such that c (L, h) εω for some ε > 0. Assume that h is Hölder with singularities along a proper analytic subset Σ of and with parameters ν, ϱ as in (3). If P p is the Bergman kernel function defined by (4) for the space H 0 (2) (, Lp ), then there exist a constant c > and p 0 N which depend only on (, ω, L, h) such that for all z \ Σ and all p p 0 (5) c P p(z) cp 2n/ν dist(z, Σ) 2nϱ/ν Recall that by Theorem 5.3 in [CM] we have lim p p log P p(z) = 0 locally uniformly on \ Σ for any metric h which is only continuous outside of Σ. Theorem 2. refines [CM, Theorem 5.3] in this context, and it is interesting to compare it to the asymptotic expansion of the Bergman kernel function in the case of smooth metrics [B, C, HM, MM, MM2, T, Z]. Proof. The proof follows from [CM, Section 5], which is based on techniques of D ly [D, Proposition 3.], [D6, Section 9]. Let x and U α be a coordinate neighborhood of x on which there exists a holomorphic frame e α of L. Let ψ α be a psh weight of h on U α. Fix r 0 > 0 so that the (closed) ball V := B(x, 2r 0 ) U α and let U := B(x, r 0 ). By [CM, (7)] there exist constants c > 0, p 0 N so that log c p p log P p(z) log(c r 2n ) p + 2 ( max B(z,r) ψ α ψ α (z) holds for all p > p 0, 0 < r < r 0 and z U with ψ α (z) >. For z U \ Σ and r < min{dist(z, Σ), r 0 } we have since ψ α is Hölder that max ψ α ψ α (z) B(z,r) cr ν (dist(z, Σ) r) ϱ, )

6 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 6 where c > 0 depends only on x. Taking r = dist(z, Σ) ϱ/ν p /ν < dist(z, Σ)/2 we obtain log c log P p (z) log c 2n log r + 2 ϱ+ cpr ν dist(z, Σ) ϱ = c 0 + 2n log ( dist(z, Σ) ϱ/ν p /ν). This holds for all z U \ Σ and p > p 0, with constants r 0, p 0, c 0, c depending only on x. A standard compactness argument now finishes the proof. 3. INTERSECTION OF FUBINI-STUDY CURRENTS AND BERTINI TYPE THEOREM In this section we show that the intersection of the Fubini-Study currents associated to line bundles as in Theorem.2 is well-defined. Moreover, we show that the sequence of wedge products of normalized Fubini-Study currents converges weakly to the wedge product of the curvature currents of (L k, h k ). We then prove that almost all zero-divisors of sections of large powers of these bundles are in general position in the sense of Definition.. Let V be a vector space of complex dimension d +. If V is endowed with a Hermitian metric, then we denote by ω FS the induced Fubini-Study form on the projective space P(V ) (see [MM, pp. 65, 22]) normalized so that ω d is a probability measure. We also FS use the same notations for P(V ). We keep the hypotheses and notation of Theorem.2. Namely, (L k, h k ), k m n, are singular Hermitian holomorphic line bundles on the compact Kähler manifold (, ω) of dimension n, such that h k is continuous outside a proper analytic subset Σ(h k ), c (L k, h k ) εω for some ε > 0, and Σ(h ),..., Σ(h m ) are in general position in the sense of Definition.. Consider the space H(2) 0 (, Lp k ) of L2 -holomorphic sections of L p k endowed with the inner product (). Since c (L k, h k ) εω, it is well-known that H(2) 0 (, Lp k ) is nontrivial for p sufficiently large, see e. g. Proposition 4.7. Let d k,p := dim H 0 (2)(, L p k ). The Kodaira map associated to (L p k, h k,p) is defined by (6) Φ k,p : G(d k,p, H 0 (2)(, L p k )), Φ k,p(x) := { s H 0 (2)(, L p k ) : s(x) = 0}, where G(d k,p, H 0 (2) (, Lp k )) denotes the Grassmannian of hyperplanes in H0 (2) (, Lp k ) (see [MM, p. 82]). Let us identify G(d k,p, H 0 (2) (, Lp k )) with P(H0 (2) (, Lp k ) ) by sending a hyperplane to an equivalence class of non-zero complex linear functionals on H 0 (2) (, Lp k ) having the hyperplane as their common kernel. By composing Φ k,p with this identification, we obtain a meromorphic map (7) Φ k,p : P(H 0 (2)(, L p k ) ). To get an analytic description of Φ k,p, let (8) S k,p j H 0 (2)(, L p k ), j = 0,..., d k,p, be an orthonormal basis and denote by P k,p the Bergman kernel function of the space H 0 (2) (, Lp k ) defined as in (4). This basis gives identifications H0 (2) (, Lp k ) Cd k,p+ and

7 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 7 P(H(2) 0 (, Lp k ) ) P d k,p. Let U be a contractible Stein open set in, let ek be a local holomorphic frame for L k on U, and write S k,p j = s k,p j e p k, where sk,p j is a holomorphic function on U. By composing Φ k,p given in (7) with the last identification, we obtain a meromorphic map Φ k,p : P d k,p which has the following local expression (9) Φ k,p (x) = [s k,p 0 (x) :... : s k,p d k,p (x)] for x U. It is called the Kodaira map defined by the basis {S k,p j } d k,p j=0. Next, we define the Fubini-Study currents γ k,p of H(2) 0 (, Lp k ) by (0) γ k,p U = 2 ddc log s k,p j 2, where the open set U and the holomorphic functions s k,p j are as above. Note that γ k,p is a positive closed current of bidegree (, ) on, and is independent of the choice of basis. Actually, the Fubini-Study currents are pullbacks of the Fubini-Study forms by Kodaira maps, which justifies their name. Let ω FS be the Fubini-Study form on P d k,p. By (9) and (0), the currents γk,p can be described as pullbacks () γ k,p = Φ k,p(ω FS ), k m. We introduce the psh function (2) u k,p := d k,p 2p log s k,p j 2 = u k + 2p log P k,p on U, j=0 where u k is the weight of the metric h k on U corresponding to e k, so e k hk = e u k. Clearly, by (0) and (2), dd c u k,p = p γ k,p. Moreover, note that by (2), log P k,p L (, ω n ) and (3) d k,p j=0 p γ k,p = c (L k, h k ) + 2p ddc log P k,p as currents on. By [CM, Theorem 5., Theorem 5.3] (see also [CM, (7)]) there exist c > 0, p 0 N, such that if p p 0, k m and z \ Σ(h k ), then P k,p (z) c. By (2) it follows that (4) u k,p (z) u k (z) + log c 2p, z U, p p 0, k m. For p consider the following analytic subsets of : { } Σ k,p := x : S k,p j (x) = 0, 0 j d k,p, k m. Hence Σ k,p is the base locus of H 0 (2) (, Lp k ), and Σ k,p U = {u k,p = }. Note also that Σ(h k ) U {u k = } and by (4) we have Σ k,p Σ(h k ) for p p 0.

8 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 8 Proposition 3.. In the hypotheses of Theorem.2 we have the following: (i) For all p sufficiently large and every J {,..., m} the analytic sets Σ k,p, k J, Σ(h l ), l J := {,..., m} \ J, are in general position. (ii) If p is sufficiently large then the currents γ k,p c (L l, h l ) k J l J are well defined on, for every J {,..., m}. (iii) γ p m,p... γ m,p c (L, h )... c (L m, h m ) as p, in the weak sense of currents on. Proof. As noted above we have by (4) that Σ k,p Σ(h k ) for all p sufficiently large. Since Σ(h ),..., Σ(h m ) are in general position this implies (i). Then (ii) follows by [D5, Corollary 2.]. (iii) Let U be a contractible Stein open set as above, u k,p, u k be the psh functions defined in (2), so dd c u k = c (L k, h k ) and dd c u k,p = γ p k,p on U. By [CM, Theorem 5.] we have that log P p k,p 0 in L (, ω n ), hence by (2), u k,p u k in L loc (U), as p, for each k m. Recall that by (4), u k,p u k C holds on U for all p p sufficiently large and some constant C > 0. Then [FS, Theorem 3.5] implies that dd c u,p... dd c u m,p dd c u... dd c u m weakly on U as p. We will need the following version of Bertini s theorem. The corresponding statement for the case of a single line bundle is proved in [CM, Proposition 4.]. Proposition 3.2. Let L k, k m n, be holomorphic line bundles over a compact complex manifold of dimension n. Assume that: (i) V k is a vector subspace of H 0 (, L k ) with basis S k,0,..., S k,dk, base locus Bs V k := {S k,0 =... = S k,dk = 0}, such that d k and the analytic sets Bs V,..., Bs V m are in general position in the sense of Definition.. (ii) Z(t k ) := {x : d k j=0 t k,js k,j (x) = 0}, where t k = [t k,0 :... : t k,dk ] P d k. (iii) ν = µ... µ m is the product measure on P d... P dm, where µ k is the Fubini-Study volume on P d k. Then the analytic sets Z(t ),..., Z(t m ) are in general position for ν-a.e. (t,..., t m ) P d... P dm. Proof. If l <... < l k m let ν l...l k = µ l... µ lk be the product measure on P d l... P d l k. For k m consider the sets U k = {(t l,..., t lk ) P d l... P d lk : dim Z(t l )... Z(t lk ) A j n k j}, where l <... < l k m, j = 0 and A 0 =, or j m k and A j = Bs V i... Bs V ij for some i <... < i j in {,..., m} \ {l,..., l k }. The proposition follows if we prove by induction on k that ν l...l k (U k ) =

9 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 9 for every set U k with l <... < l k m, 0 j m k and A j as above. Clearly, it suffices to consider the case {l,..., l k } = {,..., k}. To simplify notation we set ν k := ν...k. Let k =. If j = 0, A 0 =, so U = {t P d : dim Z(t ) n } = P d. Assume next that j m and write A j = N l= D l B, where D l are the irreducible components of A j of dimension n j and dim B n j. We have that {t P d : D l Z(t )} is a proper linear subspace of P d. Indeed, otherwise D l Bs V, so dim A j Bs V = n j, which contradicts the hypothesis that Bs V,..., Bs V m are in general position. If t P d \ U then dim Z(t ) A j n j. Since Z(t ) A j is an analytic subset of A j, it follows that D l Z(t ) A j for some l, hence P d \ U = N l= {t P d : D l Z(t )}. Therefore µ (P d \ U ) = 0. We assume now that ν k (U k ) = for any set U k as above. Let U k+ = {(t,..., t k+ ) P d... P d k+ : dim Z(t )... Z(t k+ ) A j n k j}, where 0 j m k, A 0 =, or A j = Bs V i... Bs V ij with k+2 i <... < i j m. Consider the set U = U U, where U = {(t,..., t k ) P d... P d k : dim Z(t )... Z(t k ) A j n k j}, U = {(t,..., t k ) P d... P d k : dim Z(t )... Z(t k ) Bs V k+ A j n k j }. By the induction hypothesis we have ν k (U ) = ν k (U ) =, so ν k (U) =. To prove that ν k+ (U k+ ) = it suffices to show that ν k+ (W ) = 0, where W := (U P d k+ ) \ U k+. To this end we fix t := (t,..., t k ) U, we let Z(t) := Z(t )... Z(t k ), W (t) := {t k+ P d k+ : dim Z(t) A j Z(t k+ ) n k j}, and prove that µ k+ (W (t)) = 0. Since t U U we can write Z(t) A j = N l= D l B, where D l are the irreducible components of Z(t) A j of dimension n k j and dim B n k j. If t k+ W (t) then Z(t) A j Z(t k+ ) is an analytic subset of Z(t) A j of dimension n k j, so D l Z(t) A j Z(t k+ ) for some l. Thus W (t) = N F l (t), where F l (t) := {t k+ P d k+ : D l Z(t k+ )}. l= If D l Bs V k+ then dim Z(t) A j Bs V k+ = n k j, which contradicts the fact that t U. Hence the sections in V k+ cannot all vanish on D l, so we may assume that S k+,dk+ 0 on D l. We have F l (t) {t k+,0 = 0} H l (t) where H l (t) := {[ : t k+, :... : t k+,dk+ ] P d k+ : D l Z([ : t k+, :... : t k+,dk+ ])}. For each (t k+, :... : t k+,dk+ ) C d k+ there exists at most one ζ C with [ : t k+, :... : t k+,dk+ : ζ] H l (t). Indeed, if ζ ζ have this property then S k+,0 + t k+, S k+, t k+,dk+ S k+,dk+ + as k+,dk+ 0 on D l,

10 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 0 for a = ζ, ζ, hence S k+,dk+ 0 on D l, a contradiction. It follows that µ k+ (H l (t)) = 0, so µ k+ (F l (t)) = 0. Hence µ k+ (W (t)) = 0 and the proof is complete. We return now to the setting of Theorem.2. If {S k,p j } d k,p j=0 is an orthonormal basis of H(2) 0 (, Lp k ), we define the analytic hypersurface Z(t k), for t k = [t k,0 :... : t k,dk,p ] P d k,p, as in Proposition 3.2 (ii). Let µk,p be the Fubini-Study volume on P d k,p, k m, p, and let µ p = µ,p... µ m,p be the product measure on P d,p... P dm,p. Applying Proposition 3.2 we obtain: Proposition 3.3. In the above setting, if p is sufficiently large then for µ p -a.e. (t,..., t m ) P d,p... P dm,p the analytic subsets Z(t ),..., Z(t m ) are in general position, and Z(t i )... Z(t ik ) has pure dimension n k for each k m, i <... < i k m. Proof. Let V k,p := H(2) 0 (, Lp k ), so Bs V k,p = Σ k,p. By Proposition 3., Σ,p,..., Σ m,p are in general position for all p sufficiently large. We fix such p and denote by [Z(t k )] the current of integration along the analytic hypersurface Z(t k ); it has the same cohomology class as pc (L k, h k ). Proposition 3.2 shows that the analytic subsets Z(t ),..., Z(t m ) are in general position for µ p -a.e. (t,..., t m ) P d,p... P dm,p. Hence if k m, i <... < i k m, the current [Z(t i )]... [Z(t ik )] is well defined by [D5, Corollary 2.] and it is supported in Z(t i )... Z(t ik ). Since c (L k, h k ) εω, it follows that [Z(t i )]... [Z(t ik )] ω n k = p k c (L i, h i )... c (L ik, h ik ) ω n k > p k ε k ω n. So Z(t i )... Z(t ik ), hence it has pure dimension n k. 4. CONVERGENCE SPEED TOWARDS INTERSECTION OF FUBINI-STUDY CURRENTS In this section we rely on techniques introduced by Dinh-Sibony [DS], based on the notion of meromorphic transform, in order to estimate the speed of equidistribution of the common zeros of m-tuples of sections of the considered big line bundles towards the intersection of the Fubini-Study currents. We then prove Theorem Dinh-Sibony equidistribution theorem. A meromorphic transform F : Y between two compact Kähler manifolds (, ω) of dimension n and (Y, ω Y ) of dimension m is the data of an analytic subset Γ Y (called the graph of F ) of pure dimension m + k such that the projections π : Y and π 2 : Y Y restricted to each irreducible component of Γ are surjective. We set formally F = π 2 (π Γ ). For y Y generic (that is, outside a proper analytic subset), the dimension of the fiber F (y) := π (π2 Γ (y)) is equal to k. This is called the codimension of F. We consider two of the intermediate degrees for F (see [DS, Section 3.]): d(f ) := F (ωy m ) ω k and δ(f ) := F (ω m Y ) ω k+. By [DS, Proposition 2.2], there exists r := r(y, ω Y ) such that for every positive closed current T of bidegree (, ) on Y with T = there is a smooth (, )-form α which depends uniquely on the class {T } and a quasiplurisubharmonic (qpsh) function ϕ such

11 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION that rω Y α rω Y and dd c ϕ T = α. If Y is the projective space P l equipped with the Fubiny-Study form ω FS, then we have r(p l, ω FS ) =. Consider the class Q(Y, ω Y ) := {ϕ qpsh on Y, dd c ϕ r(y, ω Y )ω Y }. A positive measure µ on Y is called a BP measure if all qpsh functions on Y are integrable with respect to µ. When dim Y =, it is well-known that µ is BP if and only if it admits locally a bounded potential. The terminology BP comes from this fact (see [DS]). If µ is a BP measure on Y and t R, we let { } R(Y, ω Y, µ) := sup max ϕ : ϕ Q(Y, ω Y ), ϕ dµ = 0, Y Y { } (Y, ω Y, µ, t) := sup µ(ϕ < t) : ϕ Q(Y, ω Y ), ϕ dµ = 0. Let Φ p be a sequence of meromorphic transforms from a compact Kähler manifold (, ω) into compact Kähler manifolds ( p, ω p ) of the same codimension k, where p is defined in (2). Let ν p be a BP probability measure on p and ν = p ν p be the product measure on Ω := p p. For every p > 0 and ε > 0 let E p (ε) := { xp p : Φ p(δ xp ) Φ p(ν p ), φ d(φp )ε }, φ C 2 where δ xp is the Dirac mass at x p. Note that Φ p(δ xp ) and Φ p(ν p ) are positive closed currents of bidimension (k, k) on, and the former is well defined for the generic point x p p (see [DS, Section 3.]). Now we are in position to state the part which deals with the quantified speed of convergence in the Dinh-Sibony equidistribution theorem [DS, Theorem 4.]. Theorem 4. ([DS, Lemma 4.2 (d)]). In the above setting the following estimate holds: ) ν p (E p (ε)) ( p, ω p, ν p, η ε,p, where η ε,p := εδ(φ p ) d(φ p ) 3R( p, ω p, ν p ) Equidistribution of pullbacks of Dirac masses by Kodaira maps. Let (, ω) be a compact Kähler manifold of dimension n and (L k, h k ), k m n, be singular Hermitian holomorphic line bundles on such that h k is continuous outside a proper analytic subset Σ(h k ), c (L k, h k ) εω on for some ε > 0, and Σ(h ),..., Σ(h m ) are in general position. Recall from Section that p := PH(2)(, 0 L p )... PH(2)(, 0 L p m), (Ω, σ ) := ( p, σ p ), where the probability measure σ p is the product of the Fubini-Study volume on each factor. From now on let p N be large enough. Fix an orthonormal basis {S k,p j } d k,p j=0 as in (8) and let Φ k,p : P d k,p be the Kodaira map defined by this basis (see (9)). By () we have that Φ k,p ω = γ FS k,p, where γ k,p is the Fubini-Study current of the space H(2) 0 (, Lp k ) as defined in (0). Y p=

12 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 2 We consider now the Kodaira maps as meromorphic transforms from to PH(2) 0 (, Lp k ) which we denote still by Φ k,p : PH(2) 0 (, Lp k ). Precisely, this is the meromorphic transform with graph Γ k,p = { (x, s) PH 0 (2)(, L p k ) : s(x) = 0}, k m. Indeed, since dim H(2) 0 (, Lp k ) 2 (see e.g. Proposition 4.7 below), there exists, for every x, a section s H(2) 0 (, Lp k ) with s(x) = 0, so the projection Γ k,p is surjective. Moreover, since L p k is non-trivial, every global holomorphic section of Lp k must vanish at some x, hence the projection Γ k,p PH(2) 0 (, Lp k ) is surjective. Note that Φ k,p (x) = { s PH 0 (2)(, L p k ) : s(x) = 0}, Φ k,p (s) = { x : s(x) = 0 }. Let Φ p be the product transform of Φ,p,..., Φ m,p (see [DS, Section 3.3]). It is the meromorphic transform with graph (5) Γ p = { (x, s p,..., s pm ) p : s p (x) =... = s pm (x) = 0 }. By above, the projection Π : Γ p is surjective. The second projection Π 2 : Γ p p is proper, hence by Remmert s theorem Π 2 (Γ p ) is an analytic subvariety of p. Proposition 3.3 implies that Π 2 (Γ p ) has full measure in p, so Π 2 is surjective and Φ p is a meromorphic transform of codimension n m, with fibers Φ p (s p ) = {x : s p (x) =... = s pm (x) = 0}, where s p = (s p,..., s pm ) p. Considering the product transform of any Φ i,p,..., Φ ik,p, i <... < i k m, and arguing as above it follows that, for s p = (s p,..., s pm ) p generic, the analytic sets {s p = 0},..., {s pm = 0} are in general position. Hence by [D5, Corollary 2.] the following current of bidegree (m, m) is well defined on : Φ p(δ sp ) = [s p = 0] = [s p = 0]... [s pm = 0] = Φ,p(δ sp )... Φ m,p(δ spm ). The main result of this section is the following theorem. Theorem 4.2. Under the hypotheses of Theorem.2 there exist a constant ξ > 0 depending only on m and a constant c = c(, L, h,..., L m, h m ) > 0 with the following property: For any sequence of positive numbers {λ p } p with lim inf p λ p log p > ( + ξn)c, there are subsets E p p such that (a) σ p (E p ) cp ξn exp( λ p /c) for all p large enough; (b) if s p p \ E p we have that the estimate [sp = 0] γ p m,p... γ m,p, φ c λ p p φ C 2 holds for every (n m, n m) form φ of class C 2. In particular, for σ -a.e. s Ω the estimate from (b) holds for all p sufficiently large.

13 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 3 Prior to the proof we need to establish some preparatory results. Let d 0,p = d,p d m,p be the dimension of p and π k be the canonical projection of p onto its k-th factor. Let ω p := c p ( π ω FS π mω FS ), so σp = ω d 0,p p. Here ω FS denotes, as usual, the Fubini-Study form on each factor PH(2) 0 (, Lp k ), and the constant c p is chosen so that σ p is a probability measure on p, thus (6) (c p ) d 0,p = d 0,p! d,p!... d m,p!. Lemma 4.3. There is a constant c 0 > 0 such that c p c 0 for all p. Proof. Fix p large enough. For each k m, let l k := d k,p. Using Stirling s formula l! (l/e) l 2πl it suffices to show that there is a constant c > 0 such that for all l,..., l m, ( l log l log (l l m ) l m log l ) m c. l l m l l m Since the function t t log t, t > 0, is convex, we infer that ) (l log l l m log l m l l m m m This implies the required estimate with c := log m. log l l m m Following subsection 4. we consider two intermediate degrees for the Kodaira maps Φ p : d p = d(φ p ) := Φ p(ω d 0,p p ) ω n m and δ p = δ(φ p ) := Φ p(ω d 0,p p ) ω n m+. The next result gives the asymptotic behavior of d p and δ p as p. Lemma 4.4. We have d p = p m c (L, h )... c (L m, h m ) and m δ p = pm d k,p m c (L l, h l ) Cp m, c p d 0,p k= l=,l k where C > 0 is a constant depending on (L k, h k ), k m. Proof. We use a cohomological argument. For the first identity we replace ω d 0,p p by a Dirac mass δ s, where s := (s,..., s m ) p is such that {s = 0},..., {s m = 0} are in general position, so the current Φ p(δ s ) = [s = 0]... [s m = 0] is well defined (see Proposition 3.3). By the Poincaré -Lelong formula [MM, Theorem 2.3.3], [s k = 0] = pc (L k, h k ) + dd c log s k hk,p, k m. Since the current c (L, h )... c (L m, h m ) is well defined (see Proposition 3.) it follows that Φ p(δ s ) ω n m = p m θ... θ m ω n m = p m c (L, h )... c (L m, h m ) ω n m,.

14 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 4 where θ k is a smooth closed (, ) form in the cohomology class of c (L k, h k ). Thus d p = Φ p(ω d 0,p p ) ω n m = Φ p(δ s ) ω n m = p m c (L, h )... c (L m, h m ). For the second identity, a straightforward computation shows that m ω d c d 0,p 0,p p (d 0,p )! p = d,p!... (d k,p )!... d m,p! π ω d,p... π FS kω d k,p... π FS mω dm,p. FS k= Using (6) and replacing ω d k,p FS line) in PH(2) 0 (, Lp k ), we may replace ωd 0,p p m T := k= (resp. ω d k,p ) by a generic point (resp. a generic complex FS by a current of the form d k,p c p d 0,p [{s }... D k... {s m }]. Here, D k is a generic complex line in PH(2) 0 (, Lp k ) and (s,..., s m ) is a generic point in p. The genericity of D k implies that Φ k,p (D k) =, so m Φ p([{s }... D k... {s m }]) = [s l = 0]. The Poincaré-Lelong formula yields Φ p([{s }... D k... {s m }]) = p m l=,l k m l=,l k c (L l, h l ). Since δ p = Φ p(t ), the second identity follows. Using Lemma 4.3, this yields the upper bound on δ p. Lemma 4.5. For all p sufficiently large we have Φ p(σ p ) = γ,p... γ m,p. Proof. Let us write p =,p... m,p and σ p = σ,p... σ m,p, where k,p = PH(2) 0 (, Lp k ) and σ k,p is the Fubini-Study volume on k,p. Recall that the meromorphic transform Φ p has graph Γ p defined in (5), and Π : Γ p, Π 2 : Γ p p, denote the canonical projections. By the definition of Φ p(σ p ) (see [DS, Sect. 3.]) we have Φ p(σ p ), φ = Π (φ) Π 2(σ p ) = Π 2 Π (φ) σ p = [s p = 0], φ dσ p (s p ), Γ p p p where φ is a smooth (n m, n m) form on. Thanks to Propositions 3. and 3.2, we can apply [CM, Proposition 4.2] as in the proof of [CM, Theorem.2] to show that Φ p(σ p ), φ =... [s p ] = 0]... [s pm = 0], φ dσ,p (s p )... dσ m,p (s pm ) m,p,p =... γ,p [s p2 ] = 0]... [s pm = 0], φ dσ 2,p (s p2 )... dσ m,p (s pm ) m,p 2,p =... = γ,p... γ m,p, φ. This concludes the proof of the lemma.

15 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 5 Lemma 4.6. There exist absolute constants C, α > 0, and constants C 2, α, ξ > 0 depending only on m, such that for all l, l,..., l m and t 0, where R(P l, ω FS, ω l ) FS ( + log l), 2 (P l, ω FS, ω l, t) FS C l e αt, r(p l... P lm, ω MP ) d r(l,..., l m ) := max, k m l k R(P l... P lm, ω MP, ω d ) MP C 2r(l,..., l m )( + log d), (P l... P lm, ω MP, ω d MP, t) C 2d ξ e α t/r(l,...,l m), d = l l m, ω MP := c ( π (ω FS ) π m(ω FS ) ), c d = so ω d MP is a probability measure on Pl... P lm. d! l!... l m!, Proof. The first two inequalities are proved in Proposition A.3 and Corollary A.5 from [DS]. If T is a positive closed current of bidegree (, ) on P l... P lm with T = then T is in the cohomology class of α = a π(ω FS ) a m πm(ω FS ), for some a k 0. Hence Now ( 0 α max k m ) a k ω c MP. = T = α ω d = m P l... P lm MP k= a k l k cd, so a k /c d/l k. Thus r(p l... P lm d, ω MP ) max k m l k. The last two inequalities follow from these estimates by applying [DS, Proposition A.8, Proposition A.9]. We will also need the following lower estimate for the dimension d k,p. Proposition 4.7. Let (, ω) be a compact Kähler manifold of dimension n. Let (L, h) be a singular Hermitian holomorphic line bundle such that c (L, h) εω for some ε > 0 and h is continuous outside a proper analytic subset of. Then there exists C > 0 and p 0 N such that dim H 0 (2)(, L p ) Cp n, p p 0. Proof. Let Σ be a proper analytic set such that h is continuous on \ Σ. We fix x 0 \ Σ and r > 0 such that B(x 0, 2r) Σ =. Let 0 χ be a smooth cutoff function which equals on B(x 0, r) and is supported in B(x 0, 2r). We consider the function ψ : [, ), ψ(x) = ηχ(x) log x x 0, where η > 0. Consider the metric h 0 = h exp( ψ) on L. We choose η sufficiently small such that c (L, h 0 ) ε ω on. 2

16 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 6 Let us denote by I(h p ) the multiplier ideal sheaf associated to h p. Note that H 0 (2) (, Lp ) = H 0 (, L p I(h p )). The Nadel vanishing theorem [D6, N] shows that there exists p 0 N such that (7) H (, L p I(h p 0)) = 0, p p 0. Note that I(h p 0) = I(h p ) I(pψ). Consider the exact sequence (8) 0 L p I(h p ) I(pψ) L p I(h p ) L p I(h p ) O /I(pψ) 0. Thanks to (7) applied to the long exact cohomology sequence associated to (8) we have (9) H 0 (, L p I(h p )) H 0 (, L p I(h p ) O /I(pψ)) 0, p p 0. Now, for x x 0, I(pψ) x = O,x hence O,x /I(pψ) x = 0. Moreover I(h p ) x0 = O,x0 since h is continuous at x 0. Hence (20) so H 0 (, L p I(h p ) O /I(pψ)) = L p x 0 I(h p ) x0 O,x0 /I(pψ) x0 = L p x 0 O,x0 /I(pψ) x0, (2) H 0 (, L p I(h p )) L p x 0 O,x0 /I(pψ) x0 0, p p 0. Denote by M,x0 the maximal ideal of O,x0 (that is, germs of holomorphic functions vanishing at x 0 ). We have I(pψ) x0 M [pη] n+,x 0 and dim O,x0 /M k+,x 0 = ( ) k+n k, which together with (2) implies the conclusion. Proof of Theorem 4.2. We will apply Theorem 4. to the meromorphic transforms Φ p from to the multi-projective space ( p, ω p ) defined above, and the BP measures ν p := σ p on p. For t R and ε > 0 let (22) R p := R( p, ω p, σ p ), p (t) := ( p, ω p, σ p, t), E p (ε) := { s p : [s = 0] γ,p... γ m,p, φ dp ε }. φ C 2 It follows from Siegel s lemma [MM, Lemma 2.2.6] and Proposition 4.7 that there exists C 3 > 0 depending only on (, L k, h k ) k m and p 0 N such that p n /C 3 d k,p C 3 p n, p p 0, k m. By the last two inequalities in Lemma 4.6 we obtain for p p 0 and t 0, (23) R p mc 2 C3( 2 + log(mc 3 p n )) C 4 log p, ( ) α p (t) C 2 (mc 3 p n ) ξ t exp C 4 p ξn e t/c 4, mc 2 3 where C 4 is a constant depending only on (, L k, h k ) k m. Now set ε p := λ p /p, η p := ε p d p /δ p 3R p. Lemma 4.4 implies that d p p m, δ p p m, so η p C 5 λ p 3C 4 log p, p p 0,

17 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 7 where C 5 is a constant depending only on (, L k, h k ) k m. Note that for all p sufficiently large, η p > C 5 2 λ λ p p, provided that lim inf p log p > 6C 4/C 5. If E p = E p (ε p ) then it follows from Theorem 4. and Lemma 4.5 that for all p sufficiently large ( σ p (E p ) p (η p ) C 4 p ξn C5 λ ) p exp, 2C 4 where for the last estimate we used (23). Let ( 6C 4 c = max C 5 ( + ξn), 2C ) 4, C 4, c (L, h )... c (L m, h m ). C 5 If lim inf p (λ p / log p) > ( + ξn)c then for all p sufficiently large ( σ p (E p ) C 4 p ξn C5 λ ) ( p exp c p ξn λp ) exp. 2C 4 c On the other hand we have by the definition of E p that if s p p \ E p and φ is a (n m, n m) form of class C 2 then p [s m p = 0] γ,p... γ m,p, φ d p λ p p m p φ C 2 c λ p p φ C 2. In the last inequality we used the fact that d p c p m by Lemma 4.4. For the last conclusion of Theorem 4.2 we proceed as in [DMM, p. 9]. The assumption on λ p / log p and (a) imply that σ p (E p ) c p < η p= for some c > 0 and η >. Hence the set E := { s = (s, s 2,...) Ω : s p E p for infinitely many p } satisfies σ (E) = 0. Indeed, for every N, E is contained in the set { s = (s, s 2,...) Ω : s p E p for at least one p N }, p= whose σ -measure is at most σ p (E p ) c p=n p=n The proof of the theorem is thereby completed. 0 as N. pη Proof of Theorem.2. Theorem.2 follows directly from Theorem 4.2 and Proposition 3. (iii).

18 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 8 5. EQUIDISTRIBUTION WITH CONVERGENCE SPEED FOR HÖLDER SINGULAR METRICS In this section we prove Theorems.4 and.5. We close with more examples of Hölder metrics with singularities. Theorem.4 follows at once from Theorem 4.2 and the next result. Theorem 5.. In the setting of Theorem.4, there exists a constant c > 0 depending only on (, L, h,..., L m, h m ) such that for all p sufficiently large the estimate m m ( log p γ p m k,p c (L k, h k ), φ c + ) log dist(supp dd c φ, Σ) φ C 2 p p k= k= holds for every (n m, n m) form φ of class C 2 which satisfies dd c φ = 0 in a neighborhood of Σ. Proof. If φ is as in the statement, then using Proposition 3. and (3) we can write m m m γ p m k,p c (L k, h k ), φ = I k, where I k = = k= k= ( γk,p ) c (L, h )... c (L k, h k ) p c (L k, h k ) γ k+,p... γ m,p p p, φ c (L, h )... c (L k, h k ) ddc log P k,p γ k+,p... γ m,p 2p p p, φ. Since log P k,p is continuous on supp dd c φ we have I k = c (L, h )... c (L k, h k ) γ k+,p... γ m,p p p, log P k,p dd c φ. 2p By (3) the masses of γ l,p and c p (L l, h l ) are equal. Moreover, by Theorem 2. there exist a constant c > and p 0 N such that for all z \ Σ and all p p 0 one has k= c log P k,p c log p + c log dist(z, Σ), k m. It follows that I k c ( log p + ) log dist(supp dd c φ, Σ) φ C 2, p p 0, k m, m p p with some constant c = c(, L, h,..., L m, h m ) > 0, and the proof is complete. Finally, we turn our attention to the proof of Theorem.5. To this end we need the following: Theorem 5.2. Let (, ω) be a compact Kähler manifold of dimension n and (L, h) be a singular Hermitian holomorphic line bundle on such that h is Hölder with singularities in a finite set Σ = {x,..., x J }, and c (L, h) εω for some ε > 0. Let γ p be the

19 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 9 Fubini-Study current of H(2) 0 (, Lp ) defined in (0). Then there exist a constant C > 0 and p 0 N which depend only on (, ω, L, h) such that ( ) χ p n γn p c (L, h) n C log p p χ /3 C 2, for every p > p 0 and every function χ of class C 2 on. Proof. Throughout the proof we will denote by C > 0 a constant that depends only on (, ω, L, h) and may change from an estimate to the next. Let r (0, ) be a number to be chosen later such that there exist coordinate balls B(x j, 2r) centered at x j, j J, which are disjoint. Let 0 χ j, j J, be a smooth function with supp χ j B(x j, 2r), χ j = on B(x j, r), and set χ 0 = J j= χ j. Then (24) χ j C 2 C/r 2, 0 j J, for some constant C > 0. Let now χ be a function of class C 2 on. Using its Taylor expansion near x j we have χ(x) χ(x j ) Cr χ C for x B(x j, 2r), hence (25) χ(x)χ j (x) χ(x j )χ j (x) Cr χ C 2 χ j (x), x. If P p is the Bergman kernel function of the space H(2) 0 (, Lp ) defined in (4) then using (3) we obtain p n γn p c (L, h) n = n ( ) n l γp 2p ddc log P p R p, where R p = c (L, h) l. p Using a similar argument to that in the proof of Proposition 3. one shows that all of these currents are well defined (see also [CM, Lemma 3.3 and Remark 3.5]). Moreover, R p = R p ω = n c (L, h) n ω, p n γn p = c (L, h) n. Set r = \ J j= B(x j, r). By Theorem 2. there exist C > 0 and p 0 depending only on (, ω, L, h) such that for p > p 0, (26) log P p L ( r) C(log p log r). Since R p is closed we have ( ) (27) χ p n γn p c (L, h) n = 2p J j=0 (log P p )R p dd c (χχ 0 ) + l=0 χχ j ( p n γn p c (L, h) n ) = J j= χχ j ( p n γn p c (L, h) n ). Note that χ 0 = 0 on J j= B(x j, r), so we deduce by (24) and (26) that (28) 2p (log P p )R p dd c (χχ 0 ) C χ C 2 (log p log r). pr 2

20 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 20 For j we obtain using (25), ( ) ( ) χχ j p n γn p c (L, h) n χ(x j )χ j p n γn p c (L, h) n + ( ) + χχ j χ(x j )χ j p n γn p + c (L, h) n χ(x j) 2p (log P p )R p dd c χ j + Cr χ C 2. Since dd c χ j is supported on r we use again (24) and (26) to get ( ) (29) χχ j p n γn p c (L, h) n C χ C 2 (log p log r) + Cr χ pr 2 C 2, for j J. By (27), (28), (29) we conclude that ( ) ( log p log r χ p n γn p c (L, h) n C χ C 2 pr 2 The proof is finished by choosing r = p /3 in the last estimate. ) + r. Proof of Theorem.5. This follows directly from Theorem 5.2 and Theorem 4.2. Indeed, one applies Theorem 4.2 with m = n, (L k, h k ) = (L, h), and for the sequence λ p = (2 + ξn)c log p. Let us close the paper with more examples of Hölder metrics with singularities. () Consider a projective manifold and a smooth divisor Σ. By [Ko, TY], if L = K O (Σ) is ample, there exist a complete Kähler-Einstein metric ω on M := \Σ with Ric ω = ω. This metric has Poincaré type singularities, describe as follows. We denote by D the unit disc in C. Each x Σ has a coordinate neighborhood U x such that U x = D n, x = 0, U x Σ = {z = (z,..., z n ) : z = 0}, U x M = D D n. Then ω = i n 2 j,k= g jkdz j dz k is equivalent to the Poincaré type metric ω P = i dz dz 2 z 2 (log z 2 ) + i n dz 2 j dz j. 2 Let σ be the canonical section of O (Σ) (cf. [MM, p. 7]) and denote by h σ the metric induced by σ on O (Σ) (cf. [MM, Example 2.3.4]). Note also that c (O (Σ), h σ ) = [Σ] by [MM, (2.3.8)]. Consider the metric (30) h M,σ := h K M h σ on L M = K M O (Σ) M = KM. Note that L is trivial over U x and the metric h M,σ has a weight ϕ on U x M = D D n given by e 2ϕ = z 2 det[g jk ]. So dd c ϕ = Ric 2π ω > 0 and ϕ is psh on U x M. We see as in [CM, Lemma 6.8] that ϕ extends to a psh function on U x, and h M,σ extends uniquely to a positively curved metric h L on L. By construction, h L is a Hölder metric with singularities. j=2

21 HÖLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 2 (2) Let us specialize the previous example to the case of Riemann surfaces. Let be a compact Riemann surface of genus g and let Σ = {p,..., p d }. It is well-known that the following conditions are equivalent: (i) U = \ Σ admits a complete Kähler-Einstein metric ω with Ric ω = ω, (ii) 2g 2 + d > 0, (iii) L = K O (Σ) is ample, (iv) the universal cover of U is the upper-half plane H. If one of these equivalent conditions is satisfied, the Kähler-Einstein metric ω is induced by the Poincaré metric on H. In local coordinates z centered at p Σ we have ω = i gdz dz where g has singularities of type 2 z 2 (log z 2 ) 2. Note that ω extends to a closed strictly positive (, )-current on. By [CM, Lemma 6.8] there exists a singular metric h L on L such that c (L, h L ) = ω on. The weight of 2π hl near a point p Σ has the form ϕ = 2 log( z 2 g), which is Hölder with singularities. (3) Let be a complex manifold, (L, h L 0 ) a holomorphic line bundle on with smooth Hermitian metric such that c (L, h L 0 ) is a Kähler metric. Let Σ be a compact divisor with normal crossings. Let Σ,..., Σ N be the irreducible components of Σ, so Σ j is a smooth hypersurface in. Let σ j be holomorphic sections of the associated holomorphic line bundle O (Σ j ) vanishing to first order on Σ j and let j be a smooth Hermitian metric on O (Σ j ) so that σ j j < and σ j j = /e outside a relatively compact open set containing Σ. Set Θ δ = Ω + δdd c F, where δ > 0, F = N log( log σ j j ). 2 For δ sufficiently small Θ δ defines the generalized Poincaré metric [MM, Lemma 6.2.], [CM, Section 2.3]. For ε > 0, j= h L ε = h L 0 N ( log σ j j ) ε j= is a singular Hermitian metric on L which is Hölder with singularities. The curvature c (L, h L ε ) is a strictly positive current on, provided that ε is sufficiently small (cf. [MM, Lemma 6.2.]). When is compact the curvature current of h ε dominates a small multiple of Θ δ on \ Σ. (4) Let be a Fano manifold. Fix a Hermitian metric h 0 on K such that ω := c (K, h 0) is a Kähler metric. We denote by P SH(, ω) the set of ω-plurisubharmonic functions on. Let Σ be a smooth divisor in the linear system defined by K l, so there exists a section s H 0 (, K l ) with Σ = Div(s). Fix a smooth metric h on the bundle O (Σ) and let β [0, ). A conic Kähler metric ω on with cone angle β along Σ, cf. [Do, T2], is a current ω = ω ϕ = ω + dd c ϕ c () where ϕ = ψ + s 2β h P SH(, ω) and ψ C () P SH(, ω). In a neigbourhood of a point of Σ where Σ is given by z = 0 the metric ω is equivalent to the cone metric i ( z 2 2β 2 dz dz + n j=2 dz j dz j ).

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