Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface

We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.


Introduction
The Bergman kernel of a holomorphic line bundle L on a complex manifold Y is the Schwartz kernel of the projector from smooth sections of L onto holomorphic ones. The analysis of the Bergman kernel, Kodaira Laplacian and holomorphic sections associated to tensor powers L k := L ⊗k has important applications in complex geometry (see for e.g. [21,44]). When L is positive, the diagonal asymptotic expansion for the Bergman kernel was first proved in [17,61], motivated by [60], and subsequently by a different geometric method in [20,44]. In the present article we prove the Bergman kernel expansion at points where the curvature of L is allowed to vanish at finite order, in complex dimension one.
A related problem is the asymptotics of the spectrum of the Bochner (magnetic) Laplacian on tensor powers. Besides geometric applications, the lowest eigenvalue (ground state) of the Bochner Laplacian and the Bergman kernel expansion are important in the mathematical physics of superconductors and the Quantum Hall effect [25,40].
We now state our results more precisely. Let Y n−1 be a compact Riemannian manifold of dimension n − 1 with complex Hermitian line bundle L, h L and vector bundle F, h F . We equip these with unitary connections ∇ L , ∇ F to obtain the Bochner Laplacian on the tensor powers F ⊗ L k , where the adjoint above is taken with respect to the natural L 2 metric. As the above is elliptic, self-adjoint and positive, one has a complete orthonormal basis ψ k j ∞ j=1 of L 2 Y ; F ⊗ L k consisting of its eigenvectors ∆ k ψ k j = λ j (k) ψ k j , 0 ≤ λ 0 ≤ λ 1 . . .. Denote by R L = ∇ L 2 ∈ Ω 2 (Y ; iR) the purely imaginary curvature form of the unitary connection ∇ L . The order of vanishing of R L at a point y ∈ Y is now defined 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. (1.2) r := max y∈Y r y < ∞.
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. One first result is now the following. (1.4) ψ k 0 (y) = O k −∞ ; y ∈ Y ≤r−1 . The proof of the above result is based on pointwise, diagonal heat kernel asymptotics for the rescaled Bochner Laplacian (see Theorem 16). The leading constant in (1.3) can be identified in terms of the bottom of the spectrum of certain model Laplacians ∆ g T Y y ,j r−2 R L y defined on the tangent space, see Section A.
Without further hypotheses, the structure of the locus Y r may be quite general (locally any closed subset of a hypersurface 3.0.1). To obtain further information on the small eigenvalues we introduce additional assumptions. First we assume Y r = N j=1 Y r,j to be a union of embedded submanifolds with dimensions d j := dim (Y r,j ). Set d max j := max {d j } N j=1 and let NY r,j := T Y ⊥ r,j ⊂ T Y denote the normal bundle of each Y r,j . Note that there is a natural density on each NY r,j coming from the metric. At points y ∈ Y r , the first non-vanishing jet of the curvature j r−2 y R L ∈ S r−2 T * y Y ⊗ Λ 2 T * y Y may be thought of as an element of the product with the r − 2 symmetric power. We say that the curvature R L vanishes non-degenerately along Y r if (1.6) i s v j r−2 y R L = 0, ∀s ≤ r − 2 =⇒ v ∈ T y Y r , where i s above denotes the s-fold contraction of the symmetric part of j r−2 y R L . Denote by N c 1 k 2/r , c 2 k 2/r the Weyl counting function for the number of eigenvalues of ∆ k and by χ [c 1 ,c 2 ] the characteristic function for the given intervals. In Section 3.2 we show that under the nondegeneracy hypothesis (1.6), the Schwartz kernel of the model Laplacian on the tangent space is rapidly decaying in the normal directions. One next result is now as follows.
Theorem 2. Assuming Y r ⊂ Y to be a union of embedded submanifolds along which the curvature vanishes non-degenerately (1.6), the counting function satisfies the asymptotics If further Y r is a union of points, the smallest eigenvalue has a complete asymptotic expansion λ 0 (k) = k 2/r N j=0 λ 0,j k −j/r + O k −(2N +1)/r .
Next, we consider the case when Y n−1 , h T Y is a complex Hermitian manifold, of even dimension n − 1. The bundles L, h L , F, h F are then assumed to be holomorphic with L of rank one. Taking ∇ L , ∇ F to be the Chern connections one also has the associated Kodaira Laplacian q k : Ω 0,q Y ; F ⊗ L k → Ω 0,q Y ; F ⊗ L k , 0 ≤ q ≤ m, acting on tensor powers. The first eigenvalue of the above is typically 0 with ker q k = H q X; F ⊗ L k being cohomological and corresponding 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 Π q k (y, y) = N q k j=1 |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 and thus controls pointwise norms of sections in ker q k in the spirit of (1.4). To obtain the asymptotics for Π q k (y, y), we specialize to the case of Riemann surface (n − 1 = 2). Furthermore in addition to vanishing at finite order (1.2), the curvature is assumed to be semi-positive: R L (w,w) ≥ 0, for all w ∈ T 1,0 Y . Under these assumptions one has H 1 X; F ⊗ L k = 0 for k ≫ 0 with the asymptotics of the Bergman kernel Π k := Π 0 k being given by the following. Theorem 3. Let Y be a compact Riemann surface and (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 another Hermitian holomorphic vector bundle. Then for every N ∈ N the Bergman kernel has the pointwise asymptotic expansion on diagonal, where 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).
Note that at points where R L is positive one has r y = 2 and the above expansion recovers the usual Bergman kernel expansion at these points. The presence of fractional exponents, at points where the curvature vanishes, given in terms of the order of vanishing represents a new feature. It would be desirable to have a more explicit formula for the leading term c 0 at vanishing points for the curvature. As an example 27 computes the leading term explicitly in the case of semi-positive line bundles obtained from branched coverings. Finally, we note that unlike (1.4) the Bergman kernel expansion (1.7) does not exhibit any concentration phenomenon.
The result of Theorem 1 was shown by Montgomery [50] in the case when Y is a Riemann surface with R L vanishing to second order along a curve. It has since been actively explored in increasing generality and refinement [23,30,31,52]. We remark that our non-degeneracy assumption (1.6) is less restrictive than in these earlier works. As was the motivation in [50], the proof here uses the relation of the Bochner Laplacian with the sub-Riemannian (sR) Laplacian on the unit circle bundle of L. This is a manifestation of the semiclassical/microlocal correspondence in this context. In particular the proof of Theorem 16 exploits a standard heat kernel expansion for the sR Laplacian [7,16,41,58] on the circle bundle. A proof of the sR heat kernel expansion based on sR methods is the topic of recent research [3,19].
The Bergman kernel expansion for positive line bundles on a compact complex manifold was first proved in [17,61], motivated by [60], and subsequently in [20,45] by a method similar to the one here. We refer to [44] for a detailed account of this technique and its applications. For semi-positive line bundles the expansion was previously only known on the positive part in some cases. In [8] the expansion is proved on the positive part, and furthermore away from the augmented base locus, assuming the line bundle to be ample. In [35] the expansion is proved on the positive part when one twists by the canonical bundle (i.e. F = K Y ). In [32] a similar expansion, on the positive part when twisted by the canonical bundle, is proved assuming h L to be only bounded from above by a semi-positive metric. On a related note, [2] proves a weighted estimate for the Bergman kernels of a positive but singular Hermitian line bundle over a Riemann surface under the assumption that the curvature has singularities of Poincaré type at a finite set.
The paper is organized as follows. In Section 2 we begin with some standard preliminaries on sub-Riemannian geometry and the sR Laplacian. In particular 2.1.1 gives a proof of the ondiagonal expansion for the sR heat kernel. In Section 3 we specialize to the case of sR structures on unit circle bundles. Here 3.1 proves Theorem 1 based on an analogous heat kernel expansion for the Bochner Laplacian on tensor powers Theorem 16. Further 3.2 and 3.3 prove the Weyl law and expansion of the first value of Theorem 2 respectively. In 4 we come to the case of the Kodaira Laplacian on tensors powers of semi-positive line bundles on a Riemann surface. Here similar rescaling methods prove the Bergman kernel expansion Theorem 3 in 4.1. There are several applications of the Bergman kernel expansion to geometric quantization. In 4.2 we prove a version of Tian's approximation theorem for a semi-positive integral form by induced Fubini-Study metrics. In Section 4.3 we develop the theory of Toeplitz operators. The main tool is the expansion of their kernel, similar to that of the Bergman kernel, from which we deduce the semiclassical behavior of their norm and composition as well as their spectral density limit. In Section 4.4 we show the equidistribution of zeros of random sections to the semi-positive curvature form. In Section 4.5 we prove an asymptotic formula for the holomorphic torsion in the semi-positive case.

sub-Riemannian geometry
Sub-Riemannian (sR) geometry is the study of (metric-)distributions in smooth manifolds. More precisely, let X n be an n-dimensional, compact, oriented differentiable manifold X. Let E m ⊂ X be a rank m subbundle of the tangent bundle which is assumed to be bracket generating: sections of E generate all sections of T X under the Lie bracket. The subbundle E is further equipped with a metric g E . We refer to the triple X, E, g E as a sub-Riemannian (sR) structure. Riemannian geometry corresponds to E = T X.
The obvious length function l (γ) := 1 0 |γ| dt maybe defined on the set of horizontal paths of Sobolev regularity one connecting the two points x 0 , x 1 ∈ X as This allows for the definition of the sub-Riemannian distance function via A theorem of Chow [51, Thm 1.6.2], shows that this distance is finite (i.e. there exists a horizontal path connecting any two points) giving the manifold the structure of a metric space X, d E . The canonical flag denoting the span of the jth brackets. The dual-canonical flag is defined as , are the annihilators of the canonical flag. The number r (x) is called the step or degree of nonholonomy of the distribution at x and in general depends on the point x ∈ X. Furthermore the ranks of the subspaces E j (x) might also might depend on x ∈ X (E j 's need not be vector bundles). We define m E j (x) : to be the growth vector of the distribution. It shall also be useful to define the weight vector x ∈ X is regular. Hence in the equiregular case each E j is a subbundle of T X with r (x), m E j (x) and Q (x) all being constants independent of x.
In the equiregular case a canonical volume form on X (analogous to the Riemannian volume) can be defined starting from the sR structure. To define this, first note that any surjection π : V → W between two vector spaces allows one to pushforward a metric g V on V to another π * g V on W . This is simply the metric on W induced via the identification W ∼ = (ker π) ⊥ ⊂ V ; with the metric on (ker π) ⊥ being the restriction of g V . Next for each j ≥ 1 define the linear surjection . . e j ) := adẽ 1 adẽ 2 . . . adẽ j−1ẽ j withẽ j ∈ C ∞ (E) denoting local sections extending e j ∈ E. The pushforward metrics g E j := (B j ) * g E ⊗j are now well defined on E j /E j−1 and hence define canonical volume elements along with its dual isomorphism to now gives a canonical smooth volume form known as the Popp volume form. We remark that although the definition makes sense in general it only leads to a smooth form in the equiregular case. An important notion is the that of a privileged coordinate system at x. To define this, fix a set of local orthonormal generating vector fields U 1 , U 2 , . . . U m near x. The E−order ord E,x (f ) of a function f ∈ C ∞ (X) at a point x ∈ X is the maximum integer s ∈ N 0 for which m j=1 s j = s implies that (U s 1 1 . . . U sm m f ) (x) = 0. Similarly the E−order ord E,x (P ) of a differential operator P at the point x ∈ X is the maximum integer for which ord E,x (P f ) ≥ ord E,x (P ) + ord E,x (f ) holds for each function f ∈ C ∞ (X). One then has the obvious relation ord E,x (P Q) ≥ ord E,x (P ) + ord E,x (Q) for any pair of differential operators P, Q. A set of coordinates (x 1 , . . . , x n ) near a point x ∈ X is said to be privileged if: for all j the set ∂ ∂x 1 , ∂ ∂x 2 , . . . , ∂ ∂x m E j forms a basis for E j (x) of the canonical flag (2.2) and moreover each x j has E-order w E j (x) at x. A privileged coordinate system always exists near any point ( [6] pg. 36). Furthermore the coordinate system may be chosen such that each ∂ ∂x j equals the value of some bracket monomial in the generating vector fields at x. The Popp measure (2.5) then equals the Euclidean measure dx at the point x in these coordinates. The E−order of the monomial x α in privileged coordinates is clearly w.α while the defining vector fields U j all have E−order −1. A basic vector field is one of the form x α ∂ x j for some j and has E−order w.α−w j . We may then use a Taylor expansion to write U j = ∞ q=−1Û (q) j with each vector fieldÛ (q) j being a sum of basic vector fields of E-order q. If one defines the rescaling/dilation δ ε x = (ε w 1 x 1 , . . . , ε wn x n ) in privileged coordinates, the vector fieldsÛ (q) j are those appearing in the corresponding expansion (δ ε ) * U j = ∞ q=−1 ε qÛ (q) j for the defining vector fields. A differential operator P on R n is said to be E−homogeneous of ord E (P ) iff (δ ε ) * P = ε ord E (P ) P . It is clear that the product of two such homogeneous differential operators P 1 , P 2 is homogeneous of ord E (P 1 P 2 ) = ord E (P 1 ) + ord E (P 2 ). The nilpotentization of the sR structure at an arbitrary x ∈ X is the sR manifold given viâ with the metricĝ E corresponding to the identification The nilpotentizationμ of a smooth measure µ at x is also defined as the lead- (q) . These nilpotentizations can be shown to be independent of the choice of privileged coordinates up to sR isometry ([6] Ch. 5). An invariant definition of the nilpotentization can be given at regular points x ∈ X. First define the nilpotent Lie algebra with the Lie bracket [., .] : g x ⊗g x → g x given by the brackets of corresponding vector field extensions. The algebra is clearly graded with the j graded component (g x ) j := (E j /E j−1 ) x with the bracket preserving the grading (g x ) i , (g x ) j ⊂ (g x ) i+j . Associated to the nilpotent Lie algebra g x is a unique simply connected Lie group G with the exponential map giving a diffeomorphism exp : g x → G. The nilpotentization of the sR structure X ,Ê,ĝ E at x can then be identified withX := G and the metric distributionÊ,ĝ E obtained via left translation of ( the nilpotentizationμ of the measure µ onX. One may also profitably view sR geometry as a limit of a Riemannian geometry. Namely choose a complement E ′ ⊂ T X, E ⊕ E ′ = T X , with a metric g E ′ giving rise to a family of Riemannian metrics → g E as ǫ → 0. We call the above a family of Riemannian metrics extending/taming g E .
We prove a first proposition in this regard.
Proposition 4. Let d ǫ denote the distance function for the Riemannian metrics (2.6). Then for each x 1 , x 2 ∈ X one has converges to the sR distance (2.1) as ǫ → 0.
Proof. Let γ j be a sequence of (constant speed) geodesics connecting x 1 , x 2 for a sequence of metrics g T X ǫ j ; ǫ j → 0. Since any g E -geodesic also has g T X ǫ j -length d E (x 1 , x 2 ) we have ∀j. For these constant speed geodesics their g T X ǫ j -energy is bounded and hence so is their g T X ǫ=1 -energy. Thus by the Banach-Alauglou theorem a subsequence of γ jq ⇀ H 1 g T X ǫ=1 γ * converges weakly. Furthermore from (2.7), (2.8) this limit is horizontal. By weak convergence and Cauchy-Schwartz lim inf l ǫ jq γ jq ≥ lim inf l ǫ=1 γ jq ≥ lim inf l (γ * ) ≥ d E (x 1 , x 2 ). From this and (2.7) the proposition follows.
2.1. sR Laplacian. We shall be interested in the sub-Riemannian (sR) Laplacian. It shall be useful to define it as acting on sections of an auxiliary complex Hermitian vector bundle with connection F, h F , ∇ F of rank l. To define this first define the sR-gradient ∇ g E ,F s ∈ C ∞ (X; E ⊗ F ) of a section s ∈ C ∞ (X; F ) by the equation where h E,F := g E ⊗ h F . Next one defines the divergence (adjoint) of this gradient. In the sR context, the lack of canonical volume form presents a difficulty in doing so; hence we shall choose an auxiliary non-vanishing volume form µ. The divergence ∇ E,F * µ ω ∈ C ∞ (X; F ) of a section ω ∈ C ∞ (X; E ⊗ F ) is now defined to be the formal adjoint satisfying The sR-Laplacian acting on sections of F is defined by the equation In terms of a local orthonormal frame {U j } m j=1 for E, we have the expression µ being the divergence of the vector field U j with respect to µ. Remark 5. To remark on how the choice of the auxiliary form µ affects the Laplacian, let µ ′ = hµ denote another non-vanishing volume form where h is a positive smooth function on X. From (2.11), it now follows easily that one has the relation with ∆ g E ,µ denoting the sR Laplacian on functions (i.e. with F = C). Thus the two corresponding Laplacians are conjugate modulo a zeroth-order term.
The sR Laplacian ∆ g E ,F,µ is non-negative and self adjoint with respect to the obvious inner product s, s ′ = X h F (s, s ′ ) µ, s, s ′ ∈ C ∞ (X; F ). Its principal symbol is easily computed to be the Hamiltonian while its sub-principal symbol is zero. The characteristic variety is is the annihilator of E. From the local expression (2.11) and the bracket generating condition on E, the Laplacian ∆ g E ,F,µ is seen to be a sum of squares operator of Hörmander type [33]. It is then known to be hypoelliptic and satisfies the optimal sub-elliptic estimate [54] with a gain of 1 r derivatives (2.14) where r := sup x∈X r (x) is the total step of the distribution. For X non-compact, one also has the local subelliptic estimate (2.15) ψs 2 H 1/r ≤ C ∆ g E ,F,µ ϕs, ϕs + ϕs 2 L 2 , ∀s ∈ C ∞ (X; F ) and ϕ, ψ ∈ C ∞ c (X) with ϕ = 1 on spt (ψ). Thus the sR Laplacian has a compact resolvent, a discrete spectrum of non-negative eigenvalues 0 ≤ λ 0 ≤ λ 1 ≤ . . . approaching infinity and a well-defined heat kernel e −t∆ g E ,F,µ .
Remark 6. For each point p ∈ E ⊥ on the characteristic variety (2.13) the Hessian ∇ 2 σ of the symbol as well as the fundamental matrix F p : T p M → T p M , ∇ 2 σ (., .) := ω (., F p .) are invariantly defined (here ω denotes the symplectic form on the cotangent space). Under the condition (2.16) tr ∀p ∈ E ⊥ , the Laplacian ∆ g E ,F,µ is known to satisfy a better subelliptic estimate with loss of 1 derivative [34]. Furthermore, the heat kernel and trace asymptotics under this assumption were shown in [47,48].
2.1.1. sR heat kernel. We are interested in the asymptotics of the heat kernel e −t∆ g E ,F,µ . As a first step we show the finite propagation speed for the corresponding wave equation.
Lemma 7 (Finite propagation speed). Let f (x; t) be the unique solution to the initial value problem Proof. The result maybe restated in terms of the Schwartz kernel K t (x, y) of e it √ ∆ g E ,F,µ as We choose a family of metrics g T X ǫ (2.6) extending g E . The Riemannian Laplacian ∆ g T X ǫ ,F,µ (still coupled to the form µ) is written F,µ is the sR Laplacian on the complementary distribution E ′ . Via the min-max principle for small eigenvalues this leads to the L 2 convergence Π of the corresponding spectral projectors onto any given interval [0, L]. Following this one has the weak convergence K ǫ t ⇀ K t with K ǫ t (x, y) denoting the Schwartz kernel of e it √ ∆ g T X ǫ ,F,µ . The proposition now follows from the finite propagation speed of ∆ g T X ǫ ,F,µ along with d ǫ → d E as ǫ → 0 by Proposition 4 .
Next we show how the above finite propagation result leads to a localization for the heat kernel. To state this, fix a Riemannian metric g T X and a privileged coordinate ball B g T X ̺ (x) centered at a point x of radius ̺ (depending on x). Let U 1 , . . . , U m be generating vector fields on this ball. Let χ ∈ C ∞ c ([−1, 1] ; [0, 1]) with χ = 1 on − 1 2 , 1 2 . Define the modified measure and vector fields viã on R n in terms of the nilpotentization at x given by these privileged coordinates. These modified vector fields can be seen to be bracket generating for ̺ sufficiently small. The connection on F takes the form , A (0) = 0, in terms of an orthonormal trivialization for F over the ball. A modified connection on R n is now defined as∇ A formula similar to (2.11) now gives an sR Laplacian on R n viã Being semi-bounded from below, it is essentially self-adjoint and has a well defined heat kernel on R n using functional calculus. Furthermore its resolvent maps ∆ g E ,F,µ − z uniformly ∀x, x ′ ∈ X and t ≤ 1. Further there exists ̺ 1 > 0 (depending on x) such that Proof. Both claims are standard applications of finite propagation Lemma 7. With the cutoff χ as before, write the heat kernel in terms of the wave operator , now gives the bound , which combines with the sub-elliptic estimate (2.14) to give (2.18). For (2.19), note that the second summand of (2.21) is exponentially decaying O exp(− We now give the on diagonal expansion for the sR heat kernel. Theorem 9. There exist smooth sections A j ∈ C ∞ (X; End (F ))such that The leading term A 0 = e −∆ĝ ,μ μ (0, 0) is a multiple of the identity given in terms of the scalar heat kernel on the nilpotent approximation.
Proof. By Lemma 8, it suffices to demonstrate the expansion for the localized heat kernel e −t∆ g E ,F,µ (0, 0) at the point x. Next, the heat kernel of the rescaled sR-Laplacian Rearranging and setting x = x ′ = 0, t = 1; gives and it suffices to compute the expansion of the left hand side above as the dilation ε → 0. To this end, first note that the rescaled Laplacian has an expansion under the privileged coordinate dilation Here each∆ is an ε-dependent second order differential operators on R n of E-order at least N − 1. The coefficient functions of∆ are uniformly (in ε) C ∞ -bounded. The first term is a scalar operator given in terms of the nilpotent approximation at the point x. This expansion (2.25) along with the subelliptic estimates now gives ∀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 subelliptic estimate we have For M ≫ 0 sufficiently large, Sobolev's inequality gives an expansion for the corresponding Schwartz kernels of (2.28) in C 0 (R n × R n ). The heat kernel expansion now follows by plugging the resolvent expansion into the Helffer-Sjöstrand formula. Finally, to see that the expansion only involves only even powers of ε (or that (2.22) has no half-integer powers of t), note that the operators∆ We note that one may similarly prove an on diagonal expansion ∀ϕ ∈ S (R), by plugging the resolvent expansion (2.28) into the Helffer-Sjöstrand formula for ϕ, in the final step above. However both the above and the expansion Theorem 9 hold only pointwise along the diagonal. In particular the leading order Q (x) is in general a function of the point x on the diagonal. This hence does not immediately give heat trace or spectral function asymptotics for the sR Laplacian as the expansion might not be uniform or integrable in x. However in the equiregular case, where Q (x) = Q is constant, a uniform set of privileged coordinates (privileged at each point in a neighborhood of x) maybe chosen in the proof. This gives the uniformity of the expansion in x and one obtains the following.
Theorem 10. In the equiregular case there is a trace expansion with leading term given by Thus, the Weyl counting functions satisfies The above two theorems are by now well known cf. [49,58,7,59,41,16]. The sR proof based on privileged coordinate dilations is from [3], [19].

Bochner Laplacian on tensor powers
A natural place where sub-Riemannian structures arise is on unit circle bundles. To precise, let us consider X, E, g E to be a corank 1 sR structure on an n-dimensional manifold X (i.e. rankE = n − 1). We assume that there is a free S 1 action on X with respect to which the sR structure is invariant and transversal (i.e. the generator e ∈ C ∞ (T X) of the action and E are transversal at each point). The quotient Y := X/S 1 is a then a manifold with a Riemannian metric g T Y induced from g E . Equivalently, the natural projection π : X → Y is a principal S 1 bundle with connection given by the horizontal distribution E. Let L := X × ρ S 1 → Y be the Hermitian line associated to the standard one dimensional representation ρ of S 1 with induced connection ∇ L and curvature R L . It is possible to restate the bracket generating condition in terms of the curvature R L . First note that since the distribution is of corank 1, the growth vector at x is simply a function of the step r (x) and given by Equivalently, the canonical flag (2.2) is given by .
Also note that the weight vector at x is while the Hausdorff dimension is given by Q (x) = n−1+r (x). On account of the S 1 invariance, each of m E (x) ,r (x) and Q (x) descend to functions on the base manifold Y . The degree of nonholonomy r (x) at x is now characterized in terms of the order of vanishing of the curvature R L .
Proposition 11. The degree of nonholonomy where j l π(x) R L denotes the l-th jet of the curvature R L .
Proof. In terms of local coordinates on Y and a local orthonormal section l for L, we may write , while E = ker dθ + a L with θ being the induced coordinate on each fiber of X. The proposition now follows on noting Thus we see that the bracket generating condition is equivalent to the curvature R L having a finite order of vanishing at each point of Y . Horizontal curves on X can also be characterized in terms of the base Y . Namely, a horizontal curveγ on X corresponds to a pair (γ, s) of a curve γ on the base Y along with an parallel orthonormal section s of L along it |s| = 1,∇ L γ s = 0, with the correspondence simply being given by projection/lift. The length ofγ is simply the length of γ on the base. It is clear that the horizontal lift of a Riemannian geodesic on Y is a sR geodesic on X. The sR distance function on X is now also characterized in terms of the holonomy distance on the base manifold Y d L hol ((y 1 , l y 1 ) ; (y 2 , l y 2 )) = inf γ∈Ω L(( y 1 ,ly 1 );(y2,ly 2 )) l (γ) 3.0.1. Structure of Y r . As noted before, the function y → r y is upper semi-continuous and gives a decomposition of the manifold open. We next address the structure of Y r , the locus of highest vanishing order for the curvature.
Proposition 12. The subset Y r ⊂ Y is locally the closed subset of a hypersurface.
Proof. First express the curvature R L = R L ij dy i ∧ dy j in some coordinates centered at y ∈ Y r . By definition one has with Y r being described by the above equations near y. The second equation (3.3) implies that one of the functions ∂ α y R L ij , |α| = r − 3, has a non-zero differential and cuts out a hypersurface.
The following examples show that this is the most that can be said about Y r in general.
Example 13. Let 0 ∈ S ⊂ R y 2 be any arbitrary (non-empty) closed subset. By the Whitney y 2 ,y 3 be any arbitrary (non-empty) closed subset. By the Whitney extension theorem, there exists f ∈ C ∞ R 2 y 2 ,y 3 such that S = f −1 (0). The (closed) two form R L = y 1 dy 1 dy 2 + f (y 2 , y 3 ) dy 2 dy 3 on R 3 y is the curvature form of some connection on trivial line bundle on R 3 y . We clearly have r = 3 with Y 3 = {y|y 1 = f (y 2 , y 3 ) = 0} = {0} × S ⊂ R 3 y 1 ,y 2 ,y 3 in this example.
3.1. Smallest eigenvalue. The unit circle bundle of L being X, the pullback C ∼ = π * L → X is canonically trivial with 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 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 µ X an invariant density on X (e.g. the pullback of the Riemannian density) inducing a density µ Y on Y .
Pick an auxiliary complex Hermitian vector bundle with connection F, h F , ∇ F on Y and we denote by the same notation its pullback to X. 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 , preserves the decomposition (3.4) and acts via Bochner Laplacian (1.1) on the tensor powers F ⊗ L k , with adjoint being taken with respect to µ Y . As a first result we show how the subelliptic estimate (2.14) immediately gives a general spectral gap property for the Bochner Laplacian.
Proof. The subelliptic estimate (2.14) on the circle bundle is . Letting s = π * s ′ be the pullback of an orthonormal eigenfunction s ′ of ∆ k with eigenvalue λ on the base gives k 2/r ≤ C (λ + 1)as required.
The spectral gap above was earlier shown for symplectic curvature [28,43] (where r = 2) and for surfaces with curvature vanishing to first order along a curve [50] (where r = 3).
3.1.1. Heat kernels. Next we would like to investigate the heat kernel of the Bochner Laplacian ∆ k . From (3.5), we have the relation between the heat kernels with l y 1 , l y 2 denoting two unit elements in the fibers of L above y 1 , y 2 respectively. We again note that the kernels are computed with respect to the densities µ X , µ Y chosen before. The above relation together with (2.18) first gives If we choose a coordinate system centered at a point y ∈ Y and a trivialization l of L that is parallel along coordinate rays starting at the origin, the connection form can be expressed in terms of the curvature as ∇ L = d + ia L , It is now easy to see that the induced coordinate system on the unit circle bundle X is privileged at each point on the fiber above y.
Next, using (3.6) with T = ε 2 t and y 1 = y 2 belonging to this coordinate chart one has where δ ε denotes the privileged coordinate dilation as before. Now setting y 1 = εy = δ ε y, the equations (2.24), (2.25) as in the proof of Theorem 9 give that the right hand side above has an expansion uniformly in k ∈ N, t ≤ 1 and y ∈ B R (0), ∀R > 0. A slight difference above being that the coefficients a j (y, θ ′ ; t) above are computed with respect to the nilpotent sR Laplacian∆ g T Y y ,j r−2 R L y (A.5) on the product S 1 θ ×R n−1 rather than (2.26) on Euclidean space. In particular the leading term is a 0 (y, θ ′ ; t) = e −t∆ g T Y ,j r−2 R L (y, 0; y, θ ′ ). Now let ε = k − 1 r , and set r 1 (y) := 1 − r(y) r to get following a stationary phase expansion in θ ′ . Above we again note that the remainders are uni- as in (2.27), for some set of second-order differential operators △ j , j = 1, 2, . . . , (see also (3.9) below). Furtherρ denotes an almost analytic continuation of ρ satisfying ρ (x) = e −tx , x ≥ 0. Finally, setting y = y 1 = y in (3.10), we have arrived at the following.
Theorem 16. The heat kernel of the Bochner Laplacian ∆ k has the pointwise expansion We note that the above heat kernel expansion like Theorem 9 is in general again not uniform in the point y on the diagonal and does not immediately give heat trace asymptotics. We shall explore the heat trace asymptotics in the next subsection 3.2.
We end this subsection by showing how the heat kernel expansion of Theorem 16 immediately gives the estimates on the first positive eigenvalue/eigenfunction of the Bochner Laplacian of Theorem 1. First for any 0 < t 1 < t 2 , y ∈ Y r and R > 0 one has the following estimate at leading order This already gives an upper bound on the first eigenvalue. To identify the constant (1.5) one takes the limit as t 1 → t 2 to obtain For the lower bound on λ 0 (k), first note that as in (2.29) one may prove an on diagonal expansion ∀ϕ ∈ S (R), and where the coefficient a ϕ j has the form (3.11) withρ replaced with an analytic continuation of ϕ. Next note that each of the terms C z k 2/r ∆ k (y, y) on using (3.12) and (3.13).
3.2. Weyl law. We now prove the first part of Theorem 2 on the asymptotics of the Weyl counting function N c 1 k 2/r , c 2 k 2/r ; under the assumption that Y r = N j=1 Y r,j is a union of embedded submanifolds, of dimensions d j := dim (Y r,j ), along which the curvature R L vanishes non-degenerately (1.6). By a standard Tauberian argument, this shall follow from the following heat trace expansion.
Theorem 17. Given f ∈ C ∞ (Y ), the heat trace of the Bochner Laplacian satisfies the asymptotics Moreover, the leading terms above are given by in terms of the pullback to normal bundle of f | Y r,j .
Proof. By Theorem 16 it suffices to consider f supported in a sufficiently small neighborhood of a given point y ∈ Y r,j . We then choose a coordinate system is given by the vanishing of the last n − 1 − d j of these coordinates. Further we may assume ∂ y j n−1 j=1 to be orthonormal at y. A trivialization for L is chosen which is parallel with respect to coordinate rays starting at the origin. The curvature may then be written The non-degeneracy condition (1.6) is now equivalent to i.e. the (r − 2)-order vanishing locus is locally the same for R L and its leading part R L 0 . The model operator (A.4) on the tangent space is given in terms of this leading part of the curvature. We may also similarly define to be the model k-Bochner Laplacian for each k > 0. uniformly for k −1/r ≥ ε and y ∈ B 1 (0). Furthermore, substituting t = 1 ε 2 k 2/r in (3.9) we obtain uniformly for k ∈ N, k −1/r ≤ ε and y ∈ B 1 (0). The leading term above is identified with the heat kernel of the given form so that ord y R L 0 < r − 2 by (3.17). Then follows by a stationary phase type argument as in Theorem 16. A similar argument applied to the subsequent terms in (3.20), which are given by convolution integrals with the leading part, shows that dθ ′ e −ikε r θ ′ a 2j y, θ ′ ; 1 ε 2 k 2/r = O (k −∞ ), ∀j. In particular the terms of (3.20) are integrable in ε for fixed k. Finally (3.19), (3.20), (3.21) and a Taylor expansion for f near y = 0 combine to give (3.16).

3.3.
Expansion for the ground state. Next we show the expansion for the first eigenvalue under the non-degeneracy assumption (1.6) and when Y r is a union of points. The technique below borrows from [10] Ch. 9. Firstly we now choose the trivialization for L to be parallel with respect to geodesics starting at the origin in some geodesic ball B 2̺ (y) centered at any y ∈ Y r . The curvature may then be written in these coordinates/trivialization with R pq;α being identified with the components of the nonvanishing jet j r−2 R L . The non-degeneracy condition (1.6) is now equivalent to ∂ β R L 0 (y) = 0, ∀ |β| < r − 2 =⇒ y = 0.
We shall begin by constructing quasimodes localized at any y ∈ Y r . The Bochner Laplacian ∆ k = ∇ F ⊗L k * ∇ F ⊗L k can be written in this local frame/coordinates where as in (3.8) and with a F being the Christoffel symbol for F in some orthonormal trivialization. Further, the adjoint is taken with respect to the metric g T Y pq = δ pq + O (y 2 ) when expressed in geodesic coordinates. With χ ∈ C ∞ c ([−1, 1] ; [0, 1]) with χ = 1 on − 1 2 , 1 2 , we define the modified connections on R n−1 viã Further we choose a modified metricg T Y which is Euclidean outside B 2̺ (y) and agrees with g T Y on B ̺ (y). This defines the modified Bochner Laplacian (3.24)∆ k := ∇ F ⊗L k * ∇ F ⊗L k agreeing with ∆ k =∆ k on the geodesic ball B ̺ (y).
Using a Taylor expansion and (3.23), (3.24), the rescaled Bochner Laplacian 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 n−1 . Furthermore the functions appearing in (3.27) 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 (3.28) are uniformly (in k) C ∞ bounded. The leading term of (4.23) is computed in terms of the the model Bochner Laplacian on the tangent space T Y (A.3). We note that these operators (3.27) are the same as those appearing in (3.11). We choose 0 < ̺ ≪ 1 such that the B ̺ (y)'s, y ∈ Y r , are disjoint. Next with E 0,y := ker ∆ g T Y y ,j r−2 R L y − λ 0,y the smallest eigenspace of △ 0 , any normalizedψ ∈ E 0,y defines a quasimode We then defineẼ 0,y to be the span of the quasimodes corresponding to an orthonormal basis of E 0,y . SetẼ 0 := ⊕ y∈ȲrẼ0,y ⊂ C ∞ Y ; F ⊗ L k withẼ ⊥ 0 being its L 2 orthogonal complement. We now have the following proposition.

Proof.
For two quasimodes ψ k , ψ ′ k localized at different points ofȲ r one has ψ k , following a similar off-diagonal decay for the kernel of ϕ 1 k 2/r ∆ k as (3.7). We now consider ψ k , ψ ′ k ∈Ẽ 0,y of the form (3.30) localized at some y ∈Ȳ r . To this end (and as in the proof of Theorem 9) note first that by a finite propagation argument we have Next, the expansion (3.26) 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 we have For M ≫ 0 sufficiently large, Sobolev's inequality gives an expansion for the corresponding Schwartz kernels in (3.39) in C 0 (R n−1 × R n−1 ). Plugging the resulting expansion into the Helffer-Sjöstrand formula then gives ∀N ∈ N and for some (k-independent) C ϕ j ∈ C 0 (R n−1 × R n−1 ), j = 0, 1, . . . . Finally, the last equation with (3.30), (3.38) gives as required.
Following this we note from (3.36) that the low lying eigenvalues of ∆ k are given by Spec ∆ k | E 0 = Spec ∆ k | P 0Ẽ0 for k ≫ 0. However, since the matrix coefficients of ∆ k | P 0Ẽ0 were just shown to have an expansion, the expansion for the smallest eigenvalue now follows from an application of standard perturbation theory for self-adjoint matrices (see [39] Ch. 2).

Kodaira Laplacian on tensor powers
We now specialize to the case when Y is a complex Hermitian manifold with integrable complex structure J. For the arguments of this section, we shall further need to restrict to the two dimensional case, i.e. Y is a Riemann surface (see Remark 22). The metric g T Y is induced from the Hermitian metric on the complex tangent space T C Y = T 1,0 Y . Further (L, h L ), (F, h F ) are chosen to be a Hermitian, holomorphic bundles where L is of rank one. We denote by ∇ L , ∇ F the corresponding Chern connections. The curvature R L of ∇ L is a (1, 1) form which is further assumed to be semi-positive We also assume as before that the curvature R L vanishes at finite order at any point of Y . 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. Denote by Ω 0, * X; F ⊗ L k ;∂ k the Dolbeault complex and define the Kodaira Laplace and Dirac operators acting on Ω 0, * X; F ⊗ L k k := Clearly, D k interchanges while k preserves Ω 0,0/1 . We denote D ± k = D k | Ω 0,0/1 and 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 (4.3) 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 Λ 0, * ω R F := R F (w,w)wiw in terms of an orthonormal section w of T 1,0 Y . The Lichnerowicz formula for the above Dirac operator ([44] Thm 1.4.7) simplifies for a Riemann surface and is given by Proof. Writing s = |s|w ∈ Ω 0,1 Y ; F ⊗ L k in terms of a local orthonormal sectionw gives (4.4). This gives from Proposition 15, (4.5) and (4.6).
We now derive as a corollary a spectral gap property for Kodaira Dirac/Laplace operators D k , k corresponding to Proposition 15.
Proof. From Proposition 20, it is clear that Spec 1 k ⊂ c 1 k 2/r − c 2 , ∞ for some c 1 , c 2 > 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.
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 H 1 (Y ; E) = 0 whenever deg E > 2g − 2. 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 22 below.
The vanishing H 1 Y ; F ⊗ L k = 0 for k sufficiently large gives 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 22. The Corollary 21 also gives a corresponding spectral gap for the 'renormalized Laplacian' ( [28,43]), ∆ # k := ∆ k − kτ L = D 2 k + τ F , acting on functions : there exist c 0 , c 1 , c 2 > 0 such that for any k, with the number of eigenvalues in the interval [−c 0 , c 0 ] again being the holomorphic Euler characteristic (4.8). The argument for Proposition 20 breaks down in higher dimensions since there are more components to 2ω R L − τ L in the Lichnerowicz formula (4.5) which semipositivity 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 [24].

Bergman kernel expansion.
We now investigate the asymptotics of the Bergman kernel on the Riemann surface Y . This is the Schwartz kernel Π k (y 1 , y 2 ) of the projector onto the nullspace of k (4.9) with respect to the L 2 inner product given by the metrics g T Y , h F and h L . Alternately, if s 1 , s 2 , . . . , s N k denotes an orthonormal basis of eigensections of H 0 X; F ⊗ L k then (4.10) Π k (y 1 , y 2 ) = N k j=1 s j (y 1 ) ⊗ s j (y 2 ) * .
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 similar to (3.8). We now define a modified frame {ẽ 1 ,ẽ 2 } 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 metricg T Y and almost complex structureJ 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 in (3.22) with iR L 0 (e 1 , e 2 ) ≥ 0. Further we may as before define the modified connections∇ F ,∇ L (3.23) 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 byR L (3.23) and is semi-positive by (4.12). Equation (3.23) also givesR L = R L 0 +O (̺ ry−1 ) and that the (r y − 2)-th derivative/jet ofR L is non-vanishing at all points on R 2 for (4.13) 0 < ̺ < c j ry−2 R L (y) .
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 formulã the adjoint being taken with respect to the metricg T Y and corresponding volume form. Also the endomorphismsR F ,τ F ,τ L and ω (κ) are the obvious modifications of (4.4) defined using the curvatures of∇ F ,∇ L andg T Y respectively. The above (4.15) again agrees with k on B ̺ (y) while the endomorphismsR F ,τ F , ω (κ) all vanish outside B ̺ (y). Being semi-bounded below (4.15) is essentially self-adjoint. A similar argument as Corollary 21 gives a spectral gap (4.17) Spec 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 sub-elliptic estimate (2.14), the projectorΠ k 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 ofg T Y .
We are now ready to prove the Bergman kernel expansion Theorem 3.
Proof of Theorem 3. First choose ϕ ∈ S (R s ) even satisfyingφ ∈ C c − ̺ 2 , ̺ 2 and ϕ (0) = 1. For c > 0, set ϕ 1 (s) = 1 [c,∞) (s) ϕ (s). On account of the spectral gap Corollary 21, and as ϕ 1 decays at infinity, we have for a ∈ N. Combining the above with semiclassical Sobolev and elliptic estimates gives  It thus suffices to consider the Bergman kernel of the model Kodaira Laplacian (4.15) on R 2 .
Next with the rescaling/dilation δ k −1/r y = k −1/r y 1 , . . . , k −1/r y n−1 as in 3.3, the rescaled Kodaira Laplacian (4.21) ⊡ := k −2/ry (δ k −1/r ) * ˜ k satisfies ϕ ˜ k k 2/ry (y, y ′ ) = k 2/ry ϕ (⊡) yk 1/ry , y ′ k 1/ry (4.22) for ϕ ∈ S (R). Using a Taylor expansion via (3.23), (4.14) the rescaled Dirac operator has an expansion Here each is a (k-independent) self-adjoint, second-order differential operator while each y α a α j;pq (y; k) ∂ yp ∂ yq + b α j;p (y; k) ∂ yp + c α j (y; k) is a k-dependent self-adjoint, second-order differential operator on R 2 . Furthermore the functions appearing in (4.24) 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 (4.25) are uniformly (in k) C ∞ bounded. Using It is now clear from (4.21) that for ϕ supported and equal to one near 0. In light of the spectral gap (4.17), the equation (4.22) specializes to (4.27)Π k (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 (4.23) 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 (4.28) Then by repeated applications of the local elliptic estimate using (4.23) we have For M ≫ 0 sufficiently large, Sobolev's inequality gives an expansion for the corresponding Schwartz kernels in (4.30) in C l (R 2 × R 2 ), ∀l ∈ N 0 . Next, plugging the above resolvent expansion into the Helffer-Sjöstrand formula as before gives As ϕ was chosen supported near 0, the spectral gap properties (4.17), 38 give The expansion is now a consequence of (4. 19), (4.20) and (4.27). Finally, in order to show that there are no odd powers of k −j/ry , one again notes that the operators ⊡ j (4.24) change sign by (−1) j under δ −1 x := −x. Thus the integral expression (4.28) 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).  Proof. The proof is a modification of the previous. First note that a similar localization to (4.20) is valid in C l , ∀l ∈ N 0 , and for y in a uniform neighborhood of y. Next differentiating (4.27) with y = y ′ gives (4.34) ∂ α yΠ k (y, y) = k (2+|α|)/ry ∂ α y Π ⊡ yk 1/ry , yk 1/ry , ∀α ∈ N 2 0 . Finally, the expansion (4.31) being valid in C l , ∀l ∈ N 0 , maybe differentiated and plugged into the above with y = 0 to give the theorem.
Remark 24. The expansion (1.7) 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. [44,Theorem 4.1.1]. In particular the first two coefficients for y ∈ Y 2 are given by 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 As before with Theorem 9 and Theorem 16 the expansions of Theorem 3 and Theorem 23 are not uniform in the point on the diagonal. We next give uniform estimates on the Bergman kernel useful in Section 4.4. Below we set C r 1 : with the o (1) terms being uniform in y ∈ Y .
Proof. Note that theorem Theorem 3 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 (4.13); which in turn affects the C ∞ -norms of the coefficients of (4.25) via (3.23). 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 (4.37) j ry−2 R L (y) ≥ (1 − ε) j ry−2 R L (y) , We begin by rewriting the model Kodaira Laplacian˜ k (4.15) near y in terms of geodesic coordinates centered at y. In the region a rescaling of˜ k by δ k −1/2 , now centered at y, shows as in (4.36). Now, in the region a rescaling of˜ k by δ k −1/3 centered at y similarly shows Next, in the region Continuing in this fashion, we are finally left with the region . . , C ry−3 j ry−3 R L (y) /j ry−4 R L (y) k 2/(ry −1) .

In this region we have
following (4.37) 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 (4.35). 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 26. 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 (4.37) holds for each y ∈ Y and y ∈ B cε|j ry−2 R L | (y). Then rewrite the model Kodaira Laplacian˜ k (4.15) near y in terms of geodesic coordinates centered at y. In the region a rescaling of˜ k by δ k −1/2 , now centered at y, shows following 24 as r y = 2. Diving the above by (4.38) gives Next, in the region a rescaling of˜ k by δ k −1/3 centered at y similarly shows as in Theorem 23. Dividing this by (4.40) gives Continuing in this fashion as before eventually gives ∀y ∈ Y , y ∈ B cε|j ry −2 R L | (y), ∀α ∈ N 2 0 . By compactness one again finds a uniform c 1,ε such that ∀y ∈ Y , proving the lemma.
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 ). The leading term of (1.7) is given by the model Bergman kernel Π ⊡ 0 (0, 0) of the operator ⊡ 0 = bb † , b † = 2∂z + a, a = r 4 z (zz) r/2−1 R L 0 f (y) . An orthonormal basis for ker (⊡ 0 ) is then given by . This gives the first term of the expansion c 0 (y) = Π ⊡ 0 (0, 0) = 1 2π at the vanishing/branch point y in this example.

Induced Fubini-Study metrics.
A theorem of Tian [60], with improvements in [17,61] (see also [44, 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 [44, 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 (4.48)) are embeddings for k ≫ 0. Moreover, the Kodaira map induces a canonical isomorphism This implies immediately (see e.g. [44, (5 ∂ 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 are the global potentials of the metrics h and Θ * k h F S with respect to Then (4.42) can be written as We obtain by (1.7) that 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, and we get the same bound O k −1 for higher derivatives, obtaining again (4.44). 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 (4.44) and (4.47). 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 25 is that the base locus Bl F ⊗ L k := y ∈ Y |s (y) = 0, s ∈ H 0 Y ; F ⊗ L k = ∅ is empty for k ≫ 0. This shows that the subspace is a hyperplane for each y ∈ Y . One may identify the Grassmanian G d k − 1; H 0 Y ; F ⊗ L k , d k := dimH 0 Y ; F ⊗ L k , 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 28. 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 (4.46). Moreover, uniformly on Y . On compact sets of Y 2 the estimates (4.44) and (4.47) hold.
Proof. The proof follows from (4.43), (4.45) and the uniform estimate of Lemma 26 on the derivatives of the Bergman kernel.
As we noted before, the bundle L satisfying the hypotheses of Theorem 28 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 (4.51) is slower than in the positive case (4.44).
One can easily prove a generalization of Theorem 28 for vector bundles (F, h F ) of arbitrary rank (see [44, S 5.1.4] for the case of a positive bundle (L, h L )). We have then Kodaira maps uniformly on Y .

Toeplitz operators.
A generalization of the projector (4.9) and Bergman kernel (4.10) 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 3. For positive line bundles the analogous result was proved in [18,Theorem 2] for compact Kähler manifolds and F = C and in [44,Lemma 7.2.4 and (7.4.6)], [46,Lemma 4.6], in the symplectic case.
Theorem 29. 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 (4.53) 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 (4.53) and the localization/rescaling properties (4.20), (4.27) one has Next as in Section A, ϕ (⊡) (., 0) ∈ S (V ) for ϕ ∈ S (R) in the Schwartz class. Thus plugging (4.31) and a Taylor expansion We now show that the Toeplitz operators (4.53) can be composed up to highest order generalizing the results of [14] in the Kähler case and F = C and [44,, [46,Theorems 1.1 and 4.19] in the symplectic case.
Remark 31. Similar to the previous remark 24, 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 T k : Then this class is closed under composition and one may define a formal star product on (cf. [14,18,46]). Furthermore ∀f, g ∈ C ∞ c (Y 2 ; End (F )), with {·, ·} 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 (4.53), called Szegő-type limit formulas [15,27]. The spectral measure of T f,k is defined via We now have the following asymptotic formula.
Theorem 32. The spectral measure (4.59) satisfies Proof. Since Spec (T f,k ) ⊂ − f ∞ , f ∞ by (4.55), the equation (4.60) is equivalent to We first prove that the trace of a Toeplitz operator (4.53) satisfies the asymptotics (4.61) tr To this end first note that the expansion of Theorem 29 is uniform on compact subsets K ⊂ Y 2 while |T f,k (y, y)| = O (k) uniformly in y ∈ Y as in Lemma 25. Further as with Proposition 12 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 we have K j ⊂ K j+1 , ∩ ∞ j=1 K j = Y ≥3 . One may then breakup the trace integral from which (4.61) follows on knowing 1 2π

Following this one has
tr T l f,k = tr T f l ,k + O f k 1−1/r ∀l ∈ N from (4.8), (4.56). A polynomial approximation of the compactly supported function by (4.61) as required.
The analogous result for projective manifolds endowed with the restriction of the hyperplane bundle was originally proved in [15,Theorem 13.13], [27] and for arbitrary positive line bundles in [9], see also [42]. In [36, Theorem 1.6] the asymptotics (4.61) are proved for a semi-classical spectral function of the Kodaira Laplacian on an arbitrary manifold.

Random sections.
In this section we generalize the results of [57] 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 of finite dimension χ Y ; F ⊗ L k − 1 (4.8). 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. Proof. The proof follows [44] following (4.51) and it thus suffices to show Y ϕ (s k ) → 0, µ-almost surely with being the given random variable. But (4.62) gives The above result may be alternatively obtained using L 2 estimates for the∂-equation of a modified positive metric as in [22, S 4].
Example 34. (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 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.
The above theorem interpolates between the case of SU (2)/elliptic polynomials (r = 2) [13] and the case of Kac polynomials (r = ∞) [29,38,56]. For recent results on the distribution of zeroes of more general classes of random polynomials we refer to [5,12,37]. 4.5. Holomorphic torsion. In this section we give an asymptotic result for the holomorphic torsion of the semi-positive line bundle L generalizing that of [11] (see also [44, 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 k := exp − 1 2 ζ ′ k (0) . Next, with τ L , ω R L as in (4.4) and t > 0, set 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.
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).
Theorem 37. The holomorphic torsion satisfies the asymptotics as k → ∞. 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 (4.70), (4.71), (4.72), (4.73) and (4.74).

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 (A.1) S r−s−2 V * ⊗ Λ 2 V * ∋ i s v R V = 0, ∀s ≤ r − 2 =⇒ T y Y ∋ v = 0. 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 ∂ yp + i r y q y α R pq,α 2 .
As in (3.5), the above may now be related to the (nilpotent) sR Laplacian on the the product S 1 θ × V given by (A.5)∆ g V ,R V := − n q=1 ∂ yp + i r y q y α R pq,α ∂ θ

2
, 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 (3.6), 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 Dirac and Laplace operators, intrinsically associated to the tuple V, g V , R V , J V , via We may choose complex orthonormal basis {w j } m j=1 of V 1,0 that diagonalizes the tensor R V : R V (w i ,w j ) = δ ij R jj ; R ij ∈ S r−2 V * . This gives complex coordinates on V in which (A.8) may be written as with each R jj (z), 1 ≤ j ≤ dimV /2, being a real homogeneous function of order r − 2.
Being symmetric with respect to the standard Euclidean density and semi-bounded below, both ∆ g V ,R V and V are essentially self-adjoint on L 2 . The domains of their unique self-adjoint extensions are Dom ∆ g V ,R V = ψ ∈ L 2 |∆ g V ,R V ψ ∈ L 2 , respectively. We shall need the following information regarding their spectrum.
Proof. The proof is similar to those of Proposition 15 and Corollary 21. Introduce the deformed Laplacian ∆ k := ∆ g V ,kR V obtained by rescaling the tensor R V . From (A.4) ∆ k = k 2/r R∆ g V ,R V R −1 are conjugate under the rescaling R : C ∞ (V ; E) → C ∞ (V ; E), (Ru) (x) := u yk 1/r implying Spec (∆ k ) = k 2/r Spec ∆ g V ,R V EssSpec (∆ k ) = k 2/r EssSpec ∆ g V ,R V (A.11) By an argument similar to Proposition 15, one has Spec (∆ k ) ⊂ c 1 k 2/r − c 2 , ∞ for some c 1 , c 2 > 0 for R V = 0. From here Spec ∆ g V ,R V ⊂ [c, ∞) follows. Next, under the nondegeneracy condition, the order of vanishing of the curvature homogeneous curvature R V (of the homogeneous connection a R V (A.2)) is seen to be maximal at the origin: ord y R V < r − 2 for y = 0. Following a similar sub-elliptic estimate (2.15) on V × S 1 θ as in Proposition 15, we have k 2/(r−1) u 2 ≤ C ∆ k u, u + u 2 L 2 , ∀u ∈ C ∞ c (V \ B 1 (0)) , holds on the complement of the unit ball centered at the origin. Combining the above with Persson's characterization of the essential spectrum (cf. [53,1] Ch. 3) we have EssSpec (∆ k ) ⊂ c 1 k 2/(r−1) − c 2 , ∞ . From here and using (A.11), EssSpec ∆ g V ,R V = ∅ follows. The proof of the final part is similar following k 2/r Spec g V ,R V ,J V = Spec g V ,kR V ,J V = Spec ( k ) ⊂ {0} ∪ c 1 k 2/r − c 2 , ∞ , k := g V ,kR V ,J V , by an argument similar to Corollary 21.
Next, the heat e −t∆ g V ,R V , e −t g V ,R V ,J V and wave e it √ ∆ g V ,R V , e it √ g V ,R V ,J V operators being well-defined by functional calculus, a finite propagation type argument as in Lemma 8 gives ϕ ∆ g V ,R V (., 0) ∈ S (V ), ϕ g V ,R V ,J V (., 0) ∈ S (V ) are in the Schwartz class for ϕ ∈ S (R). Further, when EssSpec ∆ g V ,R V = ∅ any eigenfunction of ∆ g V ,R V also lies in S (V ). Finally, on choosing ϕ supported close to the origin, the Schwartz kernel Π g V ,R V ,J V (., 0) ∈ S (V ) of the projector Π g V ,R V ,J V onto the kernel of g V ,R V ,J V is also of Schwartz class.