Geometric quantization results for semi-positive line bundles on a Riemann surface

In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.


Introduction
Geometric quantization is a procedure to relate classical observables (smooth functions) on a phase space (a symplectic manifold) to quantum observables (bounded linear operators) on the corresponding quantum space (sections of a line bundle).In the case when the line bundle in question is positive, and consequently the underlying manifold Kahler, a well known quantization recipe is that of Berezin-Toeplitz [4,27,33].Showing the validity of the quantization prodecure involves proving that it has the right properties in the semiclassical limit.Key to the proof is the analysis of the semiclassical limit of the Bergman kernel [13,15,29,28,38].In earlier work [31] the authors proved the Bergman kernel expansion in the case when the underlying line bundle is only semipositive, with curvature vanishing at finite order at each point, on a Riemann surface.It is the purpose of this article to explore the corresponding applications of the expansion therein to results in geometric quantization in the semi-positive case.These include Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.
We now state our results more precisely.Let Y 2 be a compact Riemannian surface with complex structure J and Hermitian metric h T Y .Consider holomorphic, Hermitian line and vector bundles L, h L , F, h F on Y and let ∇ L , ∇ F be the corresponding Chern connections.Denote by R L = ∇ L 2 ∈ Ω 2 (Y ; iR) the corresponding curvature of the line bundle.The order of vanishing of R L at a point y ∈ Y is now defined (1.1) r y − 2 = ord y R L := min l|J l Λ 2 T * Y ∋ j l y R L = 0 , r y ≥ 2, where j l R L denotes the lth jet of the curvature.We shall assume that this order of vanishing is finite at any point of the manifold i.e.
The function y → r y being upper semi-continuous then gives a decomposition of the manifold Y = r j=2 Y j ; Y j := {y ∈ Y |r y = j} with each Y ≤j := j j ′ =0 Y j ′ being open.Furthermore, the curvature is assumed to be semipositive: R L (w, w) ≥ 0, for all w ∈ T 1,0 Y .Associated to the above one has the Kodaira Laplacian q k : Ω 0,q Y ; F ⊗ L k → Ω 0,q Y ; F ⊗ L k , 0 ≤ q ≤ 1, acting on tensor powers.The kernel of the Kodaira Laplacian ker q k = H q X; F ⊗ L k is cohomological and corresponds to holomorphic sections.The Bergman kernel Π q k (y, y ′ ) is the Schwartz kernel of the orthogonal projector Π q k : Ω 0,q Y ; F ⊗ L k → ker q k .Its value on the diagonal is |s j (y)| 2 , N q k := dim H q X; F ⊗ L k , for an orthonormal basis {s j } N q k j=1 of H q X; F ⊗ L k .Under these assumptions one has H 1 X; F ⊗ L k = 0 for k ≫ 0. We now first recall our theorem from [31] on the asymptotics of the Bergman kernel Π k := Π 0 k .Theorem 1 ([31, Theorem 3]).Let Y be a compact Riemann surface and (L, h L ) → Y a semipositive line bundle whose curvature R L vanishes to finite order at any point.Let (F, h F ) → Y be another Hermitian holomorphic vector bundle.Then the Bergman kernel Π k := Π 0 k has the pointwise asymptotic expansion on diagonal Here c j are sections of End (F ), with the leading term c 0 (y) = Π g T Y y ,j ry −2 y R L ,J T Y y (0, 0) > 0 being given in terms of the Bergman kernel of the model Kodaira Laplacian on the tangent space at y (A.8).
To explain our first consequence of the above note that the cohomology H 0 (Y ; F ⊗ L k ) is endowed with an L 2 product induced by h T Y , h L and h F .This induces a Fubini-Study metric ω F S on the projective space P H 0 Y ; F ⊗ L k * .The Kodaira map is now defined It is well known that the map is holomorphic.We now have the semi-positive version of Tian's approximation theorem.
Theorem 2. Let Y be a compact Riemann surface and (L, h L ), (F, h F ) be holomorphic Hermitian line bundles on Y such that (L, h L ) is semi-positive and its curvature vanishes at most at finite order.Then the Fubini-Study forms induced by the Kodaira map (1.4) converge uniformly on Y to the curvature R L of the line bundle with speed k −1/3 as k → ∞.
For the next application we consider the Toeplitz quantization of functions on Y , or more generally sections of F .The Toeplitz operator T f,k operator corresponding to a section f ∈ C ∞ (Y ; End (F )) is defined via where f denotes the operator of pointwise composition by f .Each Toeplitz operator above further maps H 0 Y ; F ⊗ L k to itself.A generalized Toeplitz operator, see 5.6 below, acting on H 0 Y ; F ⊗ L k is defined as one having an asymptotic expansion in k −1 with coefficients being the Toeplitz operators (1.5) as above.Our next result is now as follows.
Theorem 3. Let (L, h L ) and (F, h F ) be Hermitian holomorphic line bundles on a compact Riemann surface Y and assume that (L, h L ) is semi-positive line bundle and its curvature R L vanishes to finite order at any point.Given f, g ∈ C ∞ (Y ; End (F )), the Toeplitz operators (5.1) satisfy Moreover, the space of generalized Toeplitz operators supported on the subset Y 2 where the curvature is positive form an algebra under operator addition and composition.
For our next result, we consider the asymptotics of zeroes of random sections associated to tensor powers.To state the result first note that the natural L 2 metric on H 0 Y ; F ⊗ L k gives rise to a probability density µ k on the sphere 15).We now define the product probability space To a random sequence of sections s = (s k ) k∈N ∈ Ω given by this probability density, we then associate the random sequence of zero divisors Z s k = {s k = 0} and view it as a random sequence of currents of integration in Ω 0,0 (Y ).We now have the following.Theorem 4. Let (L, h L ) and (F, h F ) be Hermitian holomorphic line bundles on a compact Riemann surface Y and assume that (L, h L ) is semi-positive line bundle and its curvature R L vanishes to finite order at any point.Then for µ-almost all s = (s k ) k∈N ∈ Ω, the sequence of currents converges weakly to the semi-positive curvature form.
Our final result concerns the asymptotics of holomorphic torsion.
Theorem 5. Let (L, h L ) and (F, h F ) be Hermitian holomorphic line bundles on a compact Riemann surface Y and assume that (L, h L ) is semi-positive line bundle and its curvature R L vanishes to finite order at any point.The holomorphic torsion satisfies the asymptotics All of our results above are well known in the case when the line bundle L is positive.In the positive case, the leading term of the Bergman kernel expansion Theorem 1 was first shown in [37] and thereafter improved to a full expansion in [13,38] as a consequence of the Boutet de Monvel-Sjöstrand parametrix [12] for the Szegő kernel of a strongly pseudoconvex CR manifold.Subsequently a different geometric method for the expansion was developed in [15,28] inspired by the analytic localization method of [6].The application of the Bergman kernel to induced Fubini-Study metrics Theorem 2 is also found in [37] in the positive case.The construction of the full Toeplitz algebra, along with the properties of Toeplitz operators, was first done in [10] as an application of the the Boutet de Monvel-Guillemin calculus of Toeplitz operators [11].The equidistribution result for random sections in the positive case was first done in [36], and [17] gave also the speed of convergence of the zero-divisors.Finally, the asymptotics of holomorphic torsion for positive line bundles is due to Bismut-Vasserot [7].
In the semi-positive case our results are new.The Bergman kernel expansion Theorem 1 was shown by the authors in their earlier work [31].The corresponding problem for the Szegő kernel of a weakly pseudoconvex CR manifold in dimension three was solved by the second author in [23].The expansion proved in [31,Theorem 3] is however only pointwise along the diagonal.In order to obtain the approximation for Fubini-Study metrics Theorem 2 one needs to prove uniform estimates on the Bergman kernel and its derivatives.The composition for Toeplitz operators supported on the subset where the curvature is positive in Theorem 3 was shown earlier by the first author in [22,Theorem 1.4] under the assumption of a small spectral gap for the Kodaira Laplacian.A more general result, than the equidistribution for zeroes of a random holomorphic section of a semipositive line bundle, was obtained in [16, S 4] using L 2 estimates for the ∂-equation of a modified positive metric.
The paper is organized as follows.In Section 2 we begin with some standard preliminaries.These include the relevant spectral gap properties for the Bochner and Kodaira Laplacians in subsections 2.1 and 2.2 respectively.In3 we recall the proof of the pointwise Bergman kernel expansion from [31].In 3.1 we further derive uniform estimates on semipositive Bergman kernels that are necessary for the applications in this article.In 4 we use the uniform Bergman kernel estimates to prove the semipositive version of Tian's theorem Theorem 2. In Section 5 we prove the analogous expansion for the kernel of a Toeplitz operator and the corresponding theorem Theorem 3 on Toeplitz quantization.In 6 we prove the equidistribution result Theorem 4 for random sections.In the final Section 7 we prove the asymptotic result for holomorphic torsion Theorem 5.The final appendix Section A describes facts on model Laplacians and Bergman kernels that are used throughout the article.

Preliminaries
Here we begin with some preliminary notions.Let Y be a compact Riemann surface.It is equipped with an integrable complex structure J and Hermitian metric h T Y on its complex tangent space.Also denote by g T Y the associated Riemannian metric on T Y .Next let (L, h L ), (F, h F ) be an auxiliary pair of Hermitian, holomorphic bundles where L is of rank one.We denote by ∇ L , ∇ F the corresponding Chern connections and R L , R F their corresponding curvatures.The order of vanishing r y of the curvature R L at a point y ∈ Y is now defined as in (1.1).And we assume that the curvature R L vanishes at finite order at any point of Y i.e. (2.1) The curvature R L of ∇ L is a (1, 1) form which is further assumed to be semi-positive We note that semipositivity implies that the order of vanishing r y − 2 ∈ 2N 0 of the curvature R L at any point y is even.Semipositivity and finite order of vanishing imply that there are points where the curvature is positive (the set where the curvature is positive is in fact an open dense set).Hence so that L is ample.
2.1.sR and Bochner Laplacians.Associated to the above data one has the Bochner Laplacian on tensor powers defined by for each k ∈ N, with the adjoint above being taken with respect to the corresponding metrics and the Riemannian volume form.Each Bochner Laplacian (2.3) above is the Fourier mode of a sub-Riemannian (sR) Laplacian on the unit circle bundle of L. To elaborate, denote by X = S 1 L → Y the unit circle bundle of the line bundle L. Further let E := HX ⊂ T X be the horizontal distribution.The distribution carries the metric g E = π * g T Y pulled back from the base.We also denote by the same notation the pullback of F, h F , ∇ F from Y to X.The finite order of vanishing for the curvature R L in (1.2) is equivalent to the bracket generating condition for the distribution E: the Lie brackets in C ∞ (E) generates all vector fields C ∞ (T X) [31,Prop. 6].As such the triple X, E ⊂ T X, g E is a sub-Riemannian (sR) manifold.Furthermore the maximum order of vanishing for the curvature r (1.2) is then the degree of non-holonomy of the distribution E, i.e. the number of brackets required to generate the missing vertical direction.A volume form on X is defined via µ X := µ g T Y ∧ e * with µ g T Y denoting the Riemannian volume form on Y and e * being the dual one form to the generating e ∈ C ∞ (T X) of the circle action on X.
The subRiemannian Laplacian on X being the composition of the sR gradient defined via , where h E,F := g E ⊗ h F , with its adjoint taken with respect to µ X .Under the bracket generating condition, the sR Laplacian satisfies the sharp subelliptic estimate of Rothschild and Stein with a gain of 1 r derivatives (2.6) , with ϕ = 1 on spt (ψ), and where r is again given by (1.2) and corresponds to the maximum step size of the distribution E.
Next, the unit circle bundle of L being X, the pullback C ∼ = π * L → X is canonically trivial via the identification π * L ∋ (x, l) → x −1 l ∈ C. Pulling back sections then gives the identification Each summand on the right hand side above corresponds to an eigenspace of ∇ F e with eigenvalue −ik.While horizontal differentiation d H on the left corresponds to differentiation with respect to the tensor product connection∇ L k on the right hand side above.Pick an invariant density µ X on X inducing a density µ Y on Y .This now defines the sR Laplacian ∆ g E ,F,µ X acting on sections of F .By invariance the sR Laplacian commutes ∆ g E ,F,µ X , e = 0 with the generator of the circle action and hence preserves the decomposition (2.7).It acts via on each component where ∆ k is the Bochner Laplacian (2.3) on the tensor powers F ⊗ L k , with adjoint being taken with respect to µ g T Y .
Using the description of the Bochner Laplacian as the Fourier mode of the sR Laplacian (2.8), in [31, Thm.1] a general leading asymptotic result for the first positive eigenvalues was proved.Here we recall a simple argument for its lower bound.Proposition 6.There exist constants c 1 , c 2 > 0, such that one has Spec Proof.The subelliptic estimate (2.6) on the circle bundle is the pullback of an orthonormal eigenfunction s ′ of ∆ k with eigenvalue λ on the base gives k 2/r ≤ C (λ + 1)as required.
2.2.Kodaira Laplacian and its spectral gap.Related to the Bochner Laplacian (2.3) is the Kodaira Laplacian on tensor powers.Namely, with Ω 0, * X; F ⊗ L k ; ∂k denoting the Dolbeault complex the Kodaira Laplace and Dirac operators acting on Ω 0, * X; F ⊗ L k are defined Clearly, D k interchanges while k preserves Ω 0,0/1 .We denote ), v ∈ T Y , and extended to the entire exterior algebra Denote by ∇ T Y , ∇ T 1,0 Y the Levi-Civita and Chern connections on the real and holomorphic tangent spaces as well as by ∇ T 0,1 Y the induced connection on the anti-holomorphic tangent space.Denote by Θ the real (1, 1) form defined by contraction of the complex structure with the metric Θ (., .)= g T Y (J., .).This is clearly closed dΘ = 0 (or Y is Kähler) and the complex structure is parallel With the induced tensor product connection on Λ 0, * ⊗ F ⊗ L k being denoted via ∇ Λ 0, * ⊗F ⊗L k , the Kodaira Dirac operator (2.10) is now given by the formula Next we denote by R F the curvature of ∇ F and by κ the scalar curvature of g T Y .Define the following endomorphisms of in terms of an orthonormal section w of T 1,0 Y .The Lichnerowicz formula for the above Dirac operator ([28] Thm 1.4.7)simplifies for a Riemann surface and is given by (2.12) We now have the following.
Proposition 7. Let Y be a compact Riemann surface, (L, h L ) → Y a semi-positive line bundle whose curvature R L vanishes to finite order at any point.Let (F, h F ) → Y be a Hermitian holomorphic vector bundle.Then there exist constants c 1 , c 2 > 0, such that 2), (2.11).This gives from Proposition 6, (2.12) and (2.13).
We now derive as a corollary a spectral gap property for Kodaira Dirac/Laplace operators D k , k corresponding to Proposition 6.Since L is ample, we know also by the Kodaira-Serre vanishing theorem that H 1 Y ; F ⊗ L k vanishes for k sufficiently large.If F is also a line bundle this follows from the well known fact that for a line bundle E on Y we have

Corollary 8. Under the hypotheses of Proposition 7 there exist constants
It is however interesting to have a direct analytic proof.Of course, the vanishing theorem for a semi-positive line bundle works only in dimension one, see Remark 9 below.
The vanishing by Riemann-Roch, with χ Y ; F ⊗ L k , ch F ⊗ L k , Td (Y ), g denoting the holomorphic Euler characteristic, Chern character, Todd genus and genus of Y respectively.Remark 9.The argument for Proposition 7 breaks down in higher dimensions since there are more components to 2ω R L − τ L in the Lichnerowicz formula (2.12) which semi-positivity is insufficient to control.Indeed, there is a known counterexample to the existence of a spectral gap for semi-positive line bundles in higher dimensions due to Donnelly [18].

Bergman kernel expansion
In this section we now first recall the expansion for the Bergman kernel proved in [31,Sec 4.1].First recall that the Bergman kernel is the Schwartz kernel Π k (y 1 , y 2 ) of the projector onto the nullspace of k with respect to the L 2 inner product given by the metrics g T Y , h F and h L .Alternately, if We wish to describe the asymptotics of Π k along the diagonal in Y × Y .Consider y ∈ Y , and fix orthonormal bases {e 1 , e 2 (= Je 1 )}, {l}, {f j } rk(F ) j=1 for T y Y , L y , F respectively and let w := 1 √ 2 (e 1 − ie 2 ) be the corresponding orthonormal frame for T 1,0 y Y .Using the exponential map from this basis obtain a geodesic coordinate system on a geodesic ball B 2̺ (y).Further parallel transport these bases along geodesic rays using the connections ∇ T 1,0 Y , ∇ L , ∇ F to obtain orthonormal frames for T 1,0 Y , L, F on B 2̺ (y).In this frame and coordinate system, the connection on the tensor product again has the expression in terms of the curvatures of the respective connections.We now define a modified frame {ẽ 1 , ẽ2 } on R 2 which agrees with {e 1 , e 2 } on B ̺ (y) and with {∂ x 1 , ∂ x 2 } outside B 2̺ (y).Also define the modified metric gTY and almost complex structure J on R 2 to be standard in this frame and hence agreeing with g T Y , J on B ̺ (y).The Christoffel symbol of the corresponding modified induced connection on Λ 0, * now satisfies With r y − 2 ∈ 2N 0 being the order of vanishing of the curvature R L as before, we may Taylor expand the curvature as Further we may define the modified connections ∇F , ∇L via as well as the corresponding tensor product connection ∇Λ 0, * ⊗F ⊗L k which agrees with ∇ Λ 0, * ⊗F ⊗L k on B ̺ (y).Clearly the curvature of the modified connection ∇L is given by RL (3.6) and is semipositive by (3.5).Equation (3.6) also gives RL = R L 0 + O (̺ ry−1 ) and that the (r y − 2)-th derivative/jet of RL is non-vanishing at all points on R 2 for Here c is a uniform constant depending on the C r−2 norm of R L .We now define the modified Kodaira Dirac operator on R 2 by the similar formula agreeing with D k on B ̺ (y) .This has a similar Lichnerowicz formula the adjoint being taken with respect to the metric gTY and corresponding volume form.Also the endomorphisms RF , τ F , τ L and ω (κ) are the obvious modifications of (2.11) defined using the curvatures of ∇F , ∇L and gTY respectively.The above (3.9)again agrees with k on B ̺ (y) while the endomorphisms RF , τ F , ω (κ) all vanish outside B ̺ (y).Being semi-bounded below (3.9) is essentially self-adjoint.A similar argument as Corollary 8 gives a spectral gap Thus for k ≫ 0, the resolvent ˜ k − z −1 is well-defined in a neighborhood of the origin in the complex plane.On account on the local elliptic estimate, the projector Πk from L 2 R 2 ; Λ 0, * y ⊗ F y ⊗ L ⊗k y onto ker ˜ k then has a smooth Schwartz kernel with respect to the Riemannian volume of gTY .
We are now ready to prove the Bergman kernel expansion Theorem 1.
Proof of Theorem 1.
On account of the spectral gap Corollary 8, and as ϕ 1 decays at infinity, we have for a ∈ N. Combining the above with semiclassical Sobolev and elliptic estimates gives (3.13) |ϕ , we may use a finite propagation argument to conclude ϕ (D k ) (., y) = ϕ Dk (., 0) .
By similar estimates as (3.12) for Dk we now have a localization of the Bergman kernel It thus suffices to consider the Bergman kernel of the model Kodaira Laplacian (3.9) on R 2 .
Next with the rescaling/dilation δ k −1/r y = k −1/r y 1 , . . ., k −1/r y n−1 , the rescaled Kodaira Laplacian for ϕ ∈ S (R).Using a Taylor expansion via (3.6), (3.8) the rescaled Dirac operator has an expansion Here each is a (k-independent) self-adjoint, second-order differential operator while each is a k-dependent self-adjoint, second-order differential operator on R 2 .Furthermore the functions appearing in (3.18) are polynomials with degrees satisfying and whose coefficients involve while the coefficients a α j;pq (y; k) , b α j;p (y; k) , c α j (y; k) of (3.19) are uniformly (in k) C ∞ bounded.Using (3.3), (A.4), (A.8) and (A.9) the leading term of (3.17) is computed in terms of the the model Kodaira Laplacian on the tangent space T Y (A.8).
It is now clear from (3.15) that for ϕ supported and equal to one near 0. In light of the spectral gap (3.11), the equation (3.16) specializes to (3.21) Πk (y ′ , y) = k 2/ry Π ⊡ y ′ k 1/ry , yk 1/ry as a relation between the Bergman kernels of ˜ k , ⊡. Next, the expansion (3.17) along with local elliptic estimates gives for each s ∈ R.More generally, we let I j := {p = (p 0 , p 1 , . ..) |p α ∈ N, p α = j}denote the set of partitions of the integer j and define Then by repeated applications of the local elliptic estimate using (3.17) we have for each N ∈ N, s ∈ R. A similar expansion as (3.17) for the operator (⊡ + 1) M (⊡ − z), M ∈ N, also gives .
For M ≫ 0 sufficiently large, Sobolev's inequality gives an expansion for the corresponding Schwartz kernels in (3.24) in C l (R 2 × R 2 ), ∀l ∈ N 0 .Next, plugging the above resolvent expansion into the Helffer-Sjöstrand formula as before gives ∀l, N ∈ N 0 and for some (k-independent) As ϕ was chosen supported near 0, the spectral gap properties (3.11), 25 give The expansion is now a consequence of (3.13), (3.14) and (3.21).Finally, in order to show that there are no odd powers of k −j/ry , one again notes that the operators ⊡ j (3.18) change sign by (−1) j under δ −1 x := −x.Thus the integral expression (3.22) corresponding to C z j (0, 0) changes sign by (−1) j under this change of variables and must vanish for j odd.
Next we show that a pointwise expansion on the diagonal also exists for derivatives of the Bergman kernel.In what follows we denote by j l s/j l−1 s ∈ S l T * Y ⊗ E the component of the l-th jet of a section s ∈ C ∞ (E) of a Hermitian vector bundle E that lies in the kernel of the natural surjection J l (E) → J l−1 (E).
Theorem 10.For each l ∈ N 0 , the l-th jet of the on-diagonal Bergman kernel has a pointwise expansion being given in terms of the l-th jet of the Bergman kernel of the Kodaira Laplacian (A.8) on the tangent space at y.
The derivative expansion on Y 2 is also known to satisfy c 0 = c 1 = . . .= c [ l−1 2 ] = 0 (i.e.begins at the same leading order k) with the leading term given by 3.1.Uniform estimates on the Bergman kernel.The expansions for the Bergman kernel Theorem 1 and its derivatives Theorem 10 are not uniform in the point on the diagonal.For applications in the later sections we need to give uniform estimates on the Bergman kernel.Below we set C r with the o (1) terms being uniform in y ∈ Y .
Proof.Note that theorem Theorem 1 already shows given in terms of the norm of the first non-vanishing jet.The norm of this jet affects the choice of ̺ needed for (3.7); which in turn affects the C ∞ -norms of the coefficients of (3.19) via (3.6).We first show that this estimate extends to a small ( j ry−2 R L (y) -dependent) size neighborhood of y.To this end, for any ε > 0 there exists a uniform constant c ε depending only on ε and R L C r such that (3.31) We begin by rewriting the model Kodaira Laplacian ˜ k (3.9) near y in terms of geodesic coordinates centered at y.In the region as in (3.30).Now, in the region Continuing in this fashion, we are finally left with the region .
In this region we have following (3.31) with the remainder being uniform.A rescaling by δ k −1/ry then giving a similar estimate in this region, we have finally arrived at Finally a compactness argument finds a finite set of points {y j } N j=1 such that the corresponding B cε j ry j −2 R L (y j )'s cover Y .This gives a uniform constant c 1,ε > 0 such that ∀y ∈ Y , ε > 0 proving the lower bound (3.29).The argument for the upper bound is similar.
We now prove a second lemma giving a uniform estimate on the derivatives of the Bergman kernel.Again below, the model Bergman kernel Π g T Y y ,j 1 y R L /j 0 y R L ,J T Y y (0, 0) and its relevant ratio are extended (continuously) by zero from y|j 1 y R L /j 0 y R L = 0 to Y .Lemma 13.The l-th jet of the Bergman kernel satisfies with the o (1) term being uniform in y ∈ Y .
Proof.The proof follows a similar argument as the previous lemma.Given ε > 0 we find a uniform c ε such that (3.31) holds for each y ∈ Y and y ∈ B cε|j ry−2 R L | (y).Then rewrite the model Kodaira Laplacian ˜ k (3.9) near y in terms of geodesic coordinates centered at y.In the region following 11 as r y = 2. Diving the above by (3.32) gives Next, in the region a rescaling of ˜ k by δ k −1/3 centered at y similarly shows as in Theorem 10.Dividing this by (3.34) gives Continuing in this fashion as before eventually gives . By compactness one again finds a uniform c 1,ε such that ∀y ∈ Y , proving the lemma.

Induced Fubini-Study metrics
A theorem of Tian [37], with improvements in [13,38] (see also [28, S 5.1.2,S 5.1.4]),asserts that the induced Fubini-Study metrics by Kodaira embeddings given by kth tensor powers of a positive line bundle converge to the curvature of the bundle as k goes to infinity.In this Section we will give a generalization for semi-positive line bundles on compact Riemann surfaces.
Let us review first Tian's theorem.Let (Y, J, g T Y ) be a compact Hermitian manifold, (L, h L ), (F, h F ) be holomorphic Hermitian line bundles such that (L, h L ) is positive.We endow H 0 (Y ; F ⊗ L k ) with the L 2 product induced by g T Y , h L and h F .This induces a Fubini-Study metric ω F S on the projective space P H 0 Y ; F ⊗ L k * and a Fubini-Study metric h F S on O(1) → P H 0 Y ; F ⊗ L k * (see [28, S 5.1]).Since (L, h L ) is positive the Kodaira embedding theorem shows that the Kodaira maps Φ k : Y → P H 0 Y ; F ⊗ L k * (see (4.7)) are embeddings for k ≫ 0.Moreover, the Kodaira map induces a canonical isomorphism Θ k : F ⊗L k → Φ * k O(1) and we have (see e.g.[28, (5.1.15) This implies immediately (see e.g.[28, (5.1.50 ∂∂ ln Π k (y, y) .
Applying now the Bergman kernel expansion in the positive case one obtains Tian's theorem, which asserts that we have Let us also consider the convergence of the induced Fubini-Study metric Θ * k h F S to the initial metric h L .For this purpose we fix a metric h L 0 on L with positive curvature.We can then express are the global potentials of the metrics h and Θ * k h F S with respect to h L 0 and Then (4.1) can be written as We obtain by (1.3) that (4.5) that is, the normalized potentials of the Fubini-Study metric converge uniformly on Y to the potential of the initial metric h L with speed k −1 ln k.Moreover, (4.6) and we get the same bound O k −1 for higher derivatives, obtaining again (4.3).Note that if g T Y is the metric associated to ω = i 2π R L , then we have a bound O(k −2 ) in (4.3) and (4.6).We return now to our situation and consider that Y is a compact Riemann surface and (L, h L ), (F, h F ) be holomorphic Hermitian line bundles on Y such that (L, h L ) is semi-positive and its curvature vanishes at finite order.An immediate consequence of Lemma 12 is that the base locus with the projective space P H 0 Y ; F ⊗ L k * by sending a non-zero dual element in H 0 Y ; F ⊗ L k * to its kernel.This now gives a well-defined Kodaira map It is well known that the map is holomorphic.
Theorem 14.Let Y be a compact Riemann surface and (L, h L ), (F, h F ) be holomorphic Hermitian line bundles on Y such that (L, h L ) is semi-positive and its curvature vanishes at most at finite order.Then the normalized potentials of the Fubini-Study metric converge uniformly on Y to the potential of the initial metric h L with speed k −1 ln k as in (4.5).Moreover, uniformly on Y .On compact sets of Y 2 the estimates (4.3) and (4.6) hold.
Proof.The proof follows from (4.2), (4.4) and the uniform estimate of Lemma 13 on the derivatives of the Bergman kernel.
As we noted before, the bundle L satisfying the hypotheses of Theorem 14 is ample, so for k ≫ 0 the Kodaira map is an embedding and the induced Fubini-Study forms 1 k Φ * k ω F S are indeed metrics on Y .Due to the possible degeneration of the curvature R L the rate of convergence in (4.10) is slower than in the positive case (4.3).
One can easily prove a generalization of Theorem 14 for vector bundles (F, h F ) of arbitrary rank (see [28, S 5.1.4]for the case of a positive bundle (L, h L )).We have then Kodaira maps Φ k : Y → G rk (F ) ; H 0 Y ; F ⊗ L k * into the Grassmanian of rk (F )-dimensional linear spaces of H 0 Y ; F ⊗ L k * and we introduce the Fubini-Study metric on the Grassmannian as the curvature of the determinant bundle of the dual of the tautological bundle (cf.[28, (5.1.6)]).Then by following the proof of [28, Theorem 5.1.17]and using Lemma 13 we obtain (4.11) uniformly on Y .

Toeplitz operators
A generalization of the projector (3.1) and Bergman kernel (3.2) is given by the notion of a Toeplitz operator.The Toeplitz operator T f,k operator corresponding to a section f ∈ C ∞ (Y ; End (F )) is defined via where f denotes the operator of pointwise composition by f .Each Toeplitz operator above further maps H 0 Y ; F ⊗ L k to itself.
We now prove the expansion for the kernel of a Toeplitz operator generalizing Theorem 1.For positive line bundles the analogous result was proved in [14, Theorem 2] for compact Kähler manifolds and F = C and in [28,Lemma 7.2.4 and (7.4.6)], [30,Lemma 4.6], in the symplectic case.
Theorem 15.Let Y be a compact Riemann surface, (L, h L ) → Y a semi-positive line bundle whose curvature R L vanishes to finite order at any point.Let (F, h F ) → Y be a Hermitian holomorphic vector bundle.Then the kernel of the Toeplitz operator (5.1) has an on diagonal asymptotic expansion where the coefficients c j (f, •) are sections of End(F ) with leading term Proof.Firstly from the definition (5.1) and the localization/rescaling properties (3.14), (3.21) one has Next as in Section A, ϕ (⊡) (., 0) ∈ S (V ) for ϕ ∈ S (R) in the Schwartz class.Thus plugging (3.25) and a Taylor expansion into (5.2) above gives the result with the leading term again coming from (3.20).Finally and as in the proof of Theorem 1, there are no odd powers of k −j/ry as the corresponding coefficients are given by odd integrals (the integrands change sign by (−1) j under δ −1 x := −x) which are zero.
We now show that the Toeplitz operators (5.1) can be composed up to highest order generalizing the results of [10] Proof.The first part of (5.3) is similar to the positive case.Firstly, T f,k ≤ f ∞ is clear from the definition (5.1).For the lower bound, let us consider y ∈ Y 2 where the curvature is non-vanishing and u ∈ F y , |u| h F = 1.It follows from the proof of [28,Theorem 7.4.2](see also [2, Proposition 5.2, (5.40), Remark 5.7]) that Y is open and dense one may find for any ε > 0 Since ε > 0 is arbitrary, this implies f ∞ ≤ lim inf k→∞ T f,k proving the lower bound.
Next, to prove the composition expansion (5.4) it suffices to prove a uniform kernel estimate To this end we again compute in geodesic chart centered at y dy 1 Bε(y) dy 2 k 2/ry Π ⊡ (., y 1 ) with all remainders being uniform in y ∈ Y .Above we have again used the localization/rescaling properties (3.14), (3.21) as well as the first order Taylor expansion f y Remark 17.Similar to the previous remark 11, we can recover the usual algebra properties of Toeplitz operators when f, g are compactly supported on the set Y 2 where the curvature R L is positive.In particular we define a generalized Toeplitz operator to be a sequence of operators Then this class is closed under composition and one may define a formal star product on (cf. [10,14,30]).Furthermore •} being the Poisson bracket on the Kähler manifold (Y 2 , iR L ).Finally we address the asymptotics of the spectral measure of the Toeplitz operator (5.1), called Szegő-type limit formulas [11,20].The spectral measure of T f,k is defined via We now have the following asymptotic formula.
Theorem 18.The spectral measure (5.7) satisfies 3), the equation (5.8) is equivalent to We first prove that the trace of a Toeplitz operator (5.1) satisfies the asymptotics (5.9) tr To this end first note that the expansion of Theorem 15 is uniform on compact subsets K ⊂ Y 2 while |T f,k (y, y)| = O (k) uniformly in y ∈ Y as in Lemma 12. Further, as with [31,Proposition 7], Y ≥3 is a closed subset of a hypersurface and has measure zero.Then with K j ⊂ Y 2 , j = 1, 2, . . ., being a sequence of compact subsets satisfying K j ⊂ K j+1 , ∩ ∞ j=1 K j = Y ≥3 , one may then breakup the trace integral from which (5.9) follows on knowing Following this one has for all l ∈ N from (2.15), (5.4).A polynomial approximation of the compactly supported by (5.9) as required.
The analogous result for projective manifolds endowed with the restriction of the hyperplane bundle was originally proved in [11,Theorem 13.13], [20] and for arbitrary positive line bundles in [5], see also [26].In [22,Theorem 1.6] the asymptotics (5.9) are proved for a semi-classical spectral function of the Kodaira Laplacian on an arbitrary manifold.5.1.Branched coverings.We now consider Toeplitz operators and their composition in a particular case of semipositive line bundles.Namely, those that arise from pullbacks along branched coverings.Here f : Y → Y 0 is a branched covering of a Riemann surface Y 0 with branch points {y 1 , . . ., y M } ⊂ Y .The Hermitian holomorphic line bundle on Y is pulled back L, h L = f * L 0 , f * h L 0 from one on Y 0 .If L 0 , h L 0 is assumed positive, then L, h L is semi-positive with curvature vanishing at the branch points.In particular, near a branch point y ∈ Y of local degree r 2 one may find holomorphic geodesic coordinate such that the curvature is given by R L = r 2 4 (zz) r/2−1 R L 0 f (y) + O (y r−1 ).We denote for simplicity R 0 := R L 0 f (y) .The leading term of (1.3) is given by the model Bergman kernel Π ⊡ 0 (0, 0) of the operator We first compute this model Bergman kernel.
Lemma 19.The model Bergman kernel corresponding to the model operator (5.10) at a branch point is given by 0 zz ′ where (5.12) Φ (z) := 1 4 (zz) r/2 R 0 and (5.13) is given in terms of the incomplete gamma function.
Proof.From the formulas (5.11), an orthonormal basis for ker (⊡ 0 ) is easily found to be From here the model Bergman kernel is computed 0 zz ′ , completes the proof.
This gives the first term of the expansion 0 at the vanishing/branch point y in this example.

Random sections
In this section we generalize the results of [36] to the semi-positive case considered here.Let us consider Hermitian holomorphic line bundles (L, h L ) and (F, h F ) on a compact Riemann surface Y .To state the result first note that the natural metric on H 0 Y ; F ⊗ L k arising from g T Y , h F and h L gives rise to a probability density µ k on the sphere 15).We now define the product probability space To a random sequence of sections s = (s k ) k∈N ∈ Ω given by this probability density, we then associate the random sequence of zero divisors Z s k = {s k = 0} and view it as a random sequence of currents of integration in Ω 0,0 (Y ).We now have the following.
Theorem 20.Let (L, h L ) and (F, h F ) be Hermitian holomorphic line bundles on a compact Riemann surface Y and assume that (L, h L ) is semi-positive line bundle and its curvature R L vanishes to finite order at any point.Then for µ-almost all s = (s k ) k∈N ∈ Ω, the sequence of currents converges weakly to the semi-positive curvature form.
Proof.The proof follows [28] Thm 5.3.3 with some modifications which we point out below.With Φ k denoting the Kodaira map (4.7), we first have as in [28] Thm 5.3.1.For a given ϕ ∈ Ω 0,0 (Y ), one has following (4.10) and it thus suffices to show Y ϕ (s k ) → 0, µ-almost surely with being the given random variable.But (6.1) gives The above result may be alternatively obtained using L 2 estimates for the ∂-equation of a modified positive metric as in [16, S 4].
which has two vanishing points at the north/south poles of order r−2.This is the curvature form on the hyperplane line bundle L = O (1) for the metric with potential ϕ = ln (|w 0 | r + |w 1 | r ).
An orthogonal basis for H 0 X, L k is given by s α := z α , 0 ≤ α ≤ k, in terms of the affine coordinate z = w 0 /w 1 on the chart {w 1 = 0} and a C * invariant trivialization of L. The normalization is now given by given in terms of the Gamma function.We have now arrived at the following.
Corollary 22.For each even r ≥ 2, let be a random polynomial of degree k with the coefficients c α being standard i.i.d.Gaussian variables.The distribution of its roots converges in probability The above theorem interpolates between the case of SU (2)/elliptic polynomials (r = 2) [9] and the case of Kac polynomials (r = ∞) [21,25,35].For recent results on the distribution of zeroes of more general classes of random polynomials we refer to [3,8,24].

Holomorphic torsion
In this section we give an asymptotic result for the holomorphic torsion of the semi-positive line bundle L generalizing that of [7] (see also [28, S 5.5]).First recall that the holomorphic torsion of L is defined in terms of the zeta function The above converges absolutely and defines a holomorphic function of s ∈ C in this region.It possesses a meromorphic extension to C with no pole at zero and the holomorphic torsion is defined to be T Note that the above defines a smooth endomorphism R t (y) ∈ C ∞ (Y ; End (Λ 0, * )).Further, let A j ∈ C ∞ (Y ; End (Λ 0, * )) be such that We now prove the following uniform small time asymptotic expansion for the heat kernel.
Proposition 23.There exist A k,j ∈ C ∞ (Y ; End (Λ 0, * )), j = −1, 0, 1, . .., satisfying A k,j −A j = O (k −1 ), such that for each t > 0 Proof.We again work in the geodesic coordinates and local orthonormal frames centered at y ∈ Y introduced in Section 3.With Dk as in (3.8), similar localization estimates as in Section 3 (cf.also Lemma 1.6.5 in [28]) give uniformly in t > 0, y ∈ Y and k.It then suffices to consider the small time asymptotics of e − t k D2 k (0, 0).We again introduce the rescaling δ k −1/2 y = k −1/2 y, under which where A j denotes an analogous term in the small time expansion of e −t⊡ 0 satisfying A j (0, 0) = A j (7.3) ([28, (1.6.68)]).Finally, the remainders in (7.3), (7.5) being given by We now prove the the asymptotic result for holomorphic torsion.Below we denote by x ln x the continuous extension of this function from R >0 to R ≥0 (i.e.taking the value zero at the origin).− a k,−1 T −1 + Γ ′ (1) a k,0 (7.9) following (7.4).
Choosing T = k 1−2/r , gives with again the extension of the function x ln x to the origin being given by continuity to be zero as before.The proposition now follows from putting together (7.7), (7.8), (7.9), (7.10) and (7.11).

Appendix A. Model operators
Here we define certain model Bochner/Kodaira Laplacians and Dirac operators acting on a vector space V .First the Bochner Laplacian is intrinsically associated to a triple V, g V , R V with metric g V and tensor 0 = R V ∈ S r−2 V * ⊗ Λ 2 V * , r ≥ 2. We say that tensor R V is nondegenerate if Above i s denotes the s-fold contraction of the symmetric part of R V .For v 1 ∈ V , v 2 ∈ T v 1 V = V , contraction of the antisymmetric part (denoted by ι) of R V gives ι v 2 R V ∈ S r−2 V * ⊗ V * .The contraction may then be evaluated ι v 2 R V (v 1 ) at v 1 ∈ V , i.e. viewed as a homogeneous degree r − 1 polynomial function on V .The tensor R V now determines a one form a R V ∈ Ω 1 (V ) via which we may view as a unitary connection ∇ R V = d + ia R V on a trivial Hermitian vector bundle E of arbitrary rank over V .The curvature of this connection is clearly R V now viewed as a homogeneous degree r − 2 polynomial function on V valued in Λ 2 V * .This now gives the model Bochner Laplacian An orthonormal basis {e 1 , e 2 , . . ., e n }, determines components R pq,α := R V (e ⊙α ; e p , e q ) = 0, α ∈ N n−1 0 , |α| = r − 2, as well as linear coordinates (y 1 , . . ., y n ) on V .The connection form in these coordinates is given by a R V p = i r y q y α R pq,α .While the model Laplacian (A.3) is given As in (2.8), the above may now be related to the (nilpotent) sR Laplacian on the the product S 1 θ × V given by and corresponding to the sR structure S 1 θ × V, ker dθ + a R V , π * g V , dθvolg V where the sR metric corresponds to g V under the natural projection π : S 1 θ × V → V .Note that the above differs from the usual nilpotent approximation of the sR Laplacian since it acts on the product with S 1 .As (??), the heat kernels of (A.3), (A.5) are now related (A.6) e −t∆ g V ,R V (y, y ′ ) = e −iθ e −t ∆g V ,R V (y, 0; y ′ , θ) dθ.
Next, assume that the vector space V of even dimension and additionally equipped with an orthogonal endomorphism J V ∈ O (V ); J V 2 = −1.This gives rise to a (linear) integrable almost complex structure on V , a decomposition V ⊗ C = V 1,0 ⊕ V 0,1 into ±i eigenspaces of J and a Clifford multiplication endomorphism c : V → End (Λ * V 0,1 ).We further assume that R V is a (1, 1) form with respect to J (i.e. S k V * ∋ R V (w 1 , w 2 ) = 0, ∀w 1 , w 2 ∈ V 1,0 ).The (0, 1) part of the connection form (A.2) then gives a holomorphic structure on the trivial Hermitian line bundle C with holomorphic derivative ∂C = ∂ + a V 0,1 .One may now define the Kodaira for some R ε > 0, using Π P Plugging this last inequality into (A.12)gives B Rε (0) dx P e −tP (x, x) B Rε (0) dx e −tP (x, x) ≤ 4ε + ce −εt R n−1 ε from which the theorem follows on choosing t large.
0 giving the second part of the corollary.Moreover, the eigenspaces of D 2 k | Ω 0,0/1 with non-zero eigenvalue being isomorphic by Mckean-Singer, the first part also follows.

Proof.
The proof is a modification of the previous.First note that a similar localization(3.27)Π k (y, y) − Πk (y, y) = O k −∞ , to (3.14) is valid in C l , ∀l ∈ N 0 ,and for y in a uniform neighborhood of y.Next differentiating (3.21) with y = y ′ gives (3.28) ∂ α y Πk (y, y) = k (2+|α|)/ry ∂ α y Π ⊡ yk 1/ry , yk 1/ry , ∀α ∈ N 2 0 .Finally, the expansion (3.25) being valid in C l , ∀l ∈ N 0 , maybe differentiated and plugged into the above with y = 0 to give the theorem.Remark 11.The expansion (1.3) is the same as the positive case on Y 2 (points where r y = 2) and furthermore uniform in any C l -topology on compact subsets of Y 2 cf.[28, Theorem 4.1.1].In particular the first two coefficients for y ∈ Y 2 are given by Furthermore, the Bergman kernel Π g T Y y ,j 0 y R L ,J T Y y (0, 0) of the model operator (A.8) is extended (continuously) by zero from Y 2 to Y .Lemma 12.The Bergman kernel satisfies (3.29) in the Kähler case and F = C and [28, Theorems 7.4.1-2],[30, Theorems 1.1 and 4.19] in the symplectic case.Theorem 16.Given f, g ∈ C ∞ (Y ; End (F )), the Toeplitz operators (5.1) satisfy

Example 21 .
(Random polynomials) The last theorem has an interesting specialization to random polynomials.To this end, let Y = CP 1 = C 2 w \ {0} /C * with homogeneous coordinates [w 0 : w 1 ].A semi-positive curvature form for each even r ≥ 2, is given by