BERGMAN KERNELS RELATED TO HERMITIAN LINE BUNDLES OVER COMPACT COMPLEX MANIFOLDS.

Transcription

1 BERGMAN KERNELS RELATED TO HERMITIAN LINE BUNDLES OVER COMPACT COMPLE MANIFOLDS. BO BERNDTSSON ABSTRACT. We discuss some recent estimates for Bergman kernels of spaces of holomorphic sections or harmonic forms taking values in high powers of a given line bundle. The estimates are applied to the asymptotic study of eigenvalues of Toeplitz operators, sampling sequences and Morse inequalities. This is a survey of result from [16], [7] and [1] 1. INTRODUCTION The object of this paper is to review some work related to the study of Bergman kernels associated to holomorphic hermitian line bundles over compact complex manifolds. The first three sections concern the kernels associated to the spaces of holomorphic sections of the bundle. A lot of relatively recent work treat these questions, see e g [23],[4], [24], [10],[9] and [17]. Our particular angle is the application to sampling sequences on the manifold. A sampling sequence is roughly a discrete set with the property that any holomorphic section to the bundle is determined by its values on the sequence. On a compact manifold the space of holomorphic sections is of course of finite dimension and sampling sequences abund, but the point is to get asymptotic information on sampling sequences for powers L k of a line bundle L as k goes to infinity. This part of the paper is a rewrite of the thesis of N Lindholm, [16], to the setting of compact manifolds. This makes the problems technically simpler, but the main points of the arguments are the same. In section 2 we give an asymptotic upper estimate on the diagonal of the Bergman kernel for bundles of the form L k E as k tends to infinity, and an asymptotic formula in the case when L is positive. These results are rather easy consequences of the submeanvalue inequality for holomorphic functions and Hörmander s L 2 -estimates for the -equation, and they are much less precise than the results quoted above. We then study the full Bergman kernel on the product of with itself, and show that if the bundle is positive, then the kernel decays off the diagonal so rapidly that the norm of the Date: April 25, Mathematics Subject Classification. 46 F 20, 32 A 26. Key words and phrases. Line bundles, Cohomology, Toeplitz operators, sampling sequences, harmonic forms. 1

2 2 kernel tends (in the weak -toplogy) to a measure concentrated on the diagonal. In section 3 these estimates are applied to give a rather simple proof of an eigenvalue estimate for a certain Toeplitz operator, due to Boutet de Monvel and Guillemin, [5]. The proof here uses an idea of H Landau [15]. Landau used the eigenvalue estimate in the study of sampling sequences for Paley-Wiener spaces, and in section 4 we adapt this method to the sampling problem for sections of a bundle. In section 5 we study Bergman kernels associated to harmonic forms with values in a line bundle of the same form as above. The main result is an asymptotic inequality (Theorem 5.1) for the kernels on the diagonal as k goes to infinity, which is then applied to give a rather simple proof of D ly s weak holomorphic Morse inequalities, [11]. Presumably, the same estimate of the Bergman kernel follows also from the heat kernel method of D ly and Bismut [3] (via some uniformity in the time variable), but the main point here is the simplicity of the method. With some more work, the same method also gives the strong Morse inequalities. This part of the paper is a report on the work of R Berman [7]. In section 6 we give a very short account of some results from [1] that give sharper estimate for the asymptotic decay of the Bergman kernels of harmonic forms in the semipositive case. The main idea is to prove a submeanvalue inequality for harmonic forms using the notion of a subharmonic form introduced by Skoda, [21]. Integrating the estimates we obtain essentially optimal asymptotic estimates of the dimensions of the Dolbeault cohomology groups with values in L k E when L is semipositive. I would like to thank Niklas Lindholm and Robert Berman for many discussions on these topics. Thanks are also due to Christophe Mourougane for first emphasising the analogies between sampling problems and Morse inequalities. 2. BERGMAN KERNELS FOR SPACES OF HOLOMORPHIC SECTIONS. Throughout this paper will be a compact hermitian complex manifold, L and E are hermitian holomorphic line bundles over, and F k = L k E, with the understanding that F k is given the natural metric induced by the metrics on L and E. If f and g are sections to a line bundle we let < f, g > denote the pointwise scalar product and (f, g) = < f, g > the integrated product. In the same way f and f denote the pointwise and integrated norms. The Kähler form of the metric on is denoted ω, and the volume form on is ω n = ω n /n!. In this section we consider the space H 0 (F ) of holomorphic sections to F. Let (f j ) be an orthonormal basis of H 0. The Bergman kernel of H 0 is defined by K(x, y) = K y (x) = f j (x) f j (y).

3 Thus K is a section to the bundle F x F y over, and we think of K y as a section to F with values in F. Note that F x F y has a natural metric inheritetd from F s so we can talk about K(x, y). The definition of K is made so that if f is any element of H 0 then (2.1) (f, K y ) = f(y). The restriction of K to the diagonal is a section to F F, and we let B(x) = K(x, x) be its pointwise norm. We shall refer to B as the Bergman function of F. Thus A basic property of B is that B(x) = f j (x) 2. (2.2) B(x) = sup f(x) 2 / f 2 where the supremum is taken over all non-zero sections to H 0. It is also clear that (2.3) B(x) = H 0 (F ), where H 0 stands for the dimension of H 0, since each term in the formula for B contributes a 1 to the integral. We shall now give some estimates for B. Much more precise results are known (at least in the strictly positive case), see [23],[4], [24] and [10], but we shall see that already very elementary estimates have some interesting applications. We fix a point x in and choose local coordinates z near x and a local trivialization of our bundles L and E (and hence of F = F k ). The local coordinates are chosen so that z(x) = 0 and such that the metric of is Euclidean w r t z at z = 0. We can also choose the local trivializations of L and E so that the metrics are given by φ = φ 0 +o( z 2 ) and ψ = ψ 0 +o( z 2 ) respectively, where φ 0 and ψ 0 are quadratic forms. The metric on F k is then given by kφ + ψ. To estimate B(x) from above we now estimate the value at x of a section to F of norm 1. Let f be given by the holomorphic function h with respect to the local trivialization. First note that (2.4) f(x) 2 = h(0) 2 z <c k / k h 2 e kφ 0. z <c k / k e kφ 0 since h 2 is plurisubharmonic. If we choose c k to be a sequence tending to infinity sufficiently slowly then the numerator of (2.4) can be estimated from above by (1 + ɛ k ) f 2 = 1 + ɛ k, where ɛ k tends to zero as k grows. As for the denominator there are two different cases depending on the signs of the eigenvalues of the quadratic 3

4 4 form φ 0, λ 1,...λ n. If one of the eigenvalues is smaller than or equal to zero, then k z <c n e kφ 0 = k /k 1/2 z <c k e φ 0 tends to infinity as k tends to infinity. If on the other hand all of the eigenvalues are strictly positive then lim z <c kn e kφ 0 = π n /λ k / 1..λ n. k Since we can choose the section f to be a normalized extremal for (2.2) we have proved the next proposition. Theorem 2.1. Let F k = L k E be a hermitian holomorphic line bundle over a compact hermitian manifold, and let B k be its Bergman function. Let + be the set of points in where the curvature form of L has only positive eigenvalues, and let χ 0 be the characteristic function of +. Then lim sup B k /k n χ 0 (x)λ 1...λ n /π n where the λ i : s are the eigenvalues of L s curvature form at x with resepct to the metric of. It can rather easily be verified that on a compact manifold we can choose the coordinate patches and the local trivializations above in such a way that the rough estimate (2.5) B k (x) Ck n is uniform in x. By Fatou s lemma we can then integrate the inequality in Theorem 2.1 and use (2.3) to get an estimate for the dimension of H 0 (F k ). Theorem 2.2. With the same notation and assumptions as in Theorem 2.1 we have that lim sup H 0 (F k ) /k n 1/π n c(l) n /n! + where c(l) is the curvature form of L. Proof. If ω n is the volume form of then λ 1...λ n ω n = c(l) n /n!. Now assume that the curvature form of the bundle L is strictly positive everywhere ( i e the bundle is positive). We shall see that then the limsup of Theorem 2.1 is actually a limit and equality holds. Again we fix a point x and consider the quadratic form φ 0 which is the second order term in the Taylor expansion of the metric of L in the coordinates z. Let h be the holomorphic function which is identically equal to one in C n. The norm of h in L 2 (e φ 0 ) equals (π n /λ 1...λ n ) 1/2. Let χ k be a cut-off function in C n that equals one when ζ < c k and vanishes when ζ > 2c k, and put h (k) (z) = hχ k (z k). Since h (k) is for large k supported in a small neighbourhood of

5 the origin it naturally defines a section, f (k), to F k over. Then f (k) (x) = 1 and (2.6) lim k n f (k) 2 = π n /λ 1...λ n Let α (k) = f (k) ; a (0, 1)-form with values in F k. Then k n α (k) 2 tends to zero. Since the curvature of F k is for large k strictly greater than the curvature of the anticanonical bundle it follows from Hörmander s L 2 - estimates that we can solve the equation u (k) = f (k) with k n u (k) 2 also tending to zero. Moreover, α (k) vanishes for z < 1/k 1/2, so u (k) is holomorphic there, and a submeanvalue estimate implies that u (k) (x) also tends to zero. Let g (k) = f (k) u (k). Then g (k) is a holomorphic section to F k, g (k) (x) tends to 1, and k n g (k) 2 tends to π n /λ 1...λ n. By (2.2) we get that lim inf B k /k n λ 1...λ n /π n We have therefore proved Theorem 2.3. Assume L is strictly positive. Then uniformly on, and lim B k/k n = λ 1...λ n /π n lim H0 (F k ) /k n = c(l) n /n! We now turn to estimates of the Bergman kernel off the diagonal. The next result says that if the bundle is positive, the Bergman kernel decays sufficiently rapidly off the diagonal so that the norm of the kernel tends to a measure concentrated on the diagonal. Theorem 2.4. Let L be a strictly positive line bundle over and let K k be the Bergman kernel of F k = L k E. Define a sequence of measures µ k on by µ k = k n K k (x, y) 2 ω n (x)ω n (y), where ω n is the volume form on. Then µ k tends to π n c(l) n /n! [ ] in the weak*-topology as k goes to infinity. ( Here [ ] denotes the current of integration on the diagonal in, and c(l) is the Chernform of the pullback of L to under either one of the projection maps.) In [16], Theorem 2.4 is deduced from the explicit estimate k n K(x, y) Ce ɛk1/2 d(x,y), where d is some distance function on and ɛ is some positive number. This estimate in turn is proved using a method going back to Kerzman, [14], that involves a careful estimate of the L 2 -minimal solution to the equation. Here we will use a simpler method, proposed by R Berman [8], that although it does not give the explicit deacy estimate, gives the weak convergence result quite easily. 5

6 6 Theorem 2.4 means explicitly that if g is a continuous function on then lim k n g(x, y) K k (x, y) 2 ω n (x)ω n (y) = π n g(x, x)c(l) n /n!. This follows from Lebesgue s theorem on dominated convergence if we can prove two things. The first is that for any y in (2.7) lim k n g(x, y) K k (x, y) 2 ω n (x) = π n g(y, y)d(y), where D(y) = λ 1...λ n is the product of the eigenvalues of the form c(l) with respect to the metric ω at the point y, and the second fact we need is that (2.8) k n g(x, y) K k (x, y) 2 ω n (x) C for a fixed constant C, so that Lebesgue s theorem can be applied. Note first that by the reproducing property of the Bergman kernel (2.9) k n K k (x, y) 2 ω n (x) = k n K k (y, y) π n D(y), by Theorem 2.3, and that this expression is uniformly bounded in y by (2.5). It therefore suffices to prove that, for any y in, (2.10) k n K(x, y) 2 ω n (x) z >ɛ tends to zero for any ɛ > 0, where z is a local coordinate chosen as before with z = 0 corresponding to the point x = y. Just as before we also suppose the coordinates are chosen so that the metric ω is euclidean when z = 0, and we also choose local trivializations of the bundles L and E with respect to which the metrics are given by functions φ = φ 0 + o( z 2 ) and ψ = ψ 0 + o( z 2 ) respectively. Now, with respect to the local trivializations the kernel K(x, y) is given by a holomorphic function, h(z) (considering y as fixed). Then K k (y, y) 2 = h(y) 2 h(z) 2 e kφ 0 dλ(z)/ e kφ 0 dλ(z). z <ɛ k z <ɛ k Since by Theorem 2.3, K k (y, y) 2 is asymptotic to (k n π n D(y)) 2, and since z <ɛ k e kφ 0 dλ(z) is asymptotic to k n π n D(y) 1 if k 1/2 ɛ k tends to infinity, it follows that (2.11) h(z) 2 e kφ 0 dλ(z) k n π n D(y) + o(k n ). z <ɛ k But, if ɛ k is sufficiently small (but still satisfies k 1/2 ɛ k ) h(z) 2 e kφ 0 dλ(z) (1 + δ k ) K k (x, y) 2 ω n (x), z <ɛ k z <ɛ k

7 7 where δ k tends to zero. Comparing with (2.9) we see that z >ɛ k K(x, y) 2 ω n (x) = o(k n ) which completes the proof of Theorem EIGENVALUES OF TOEPLITZ OPERATORS In this section we assume that L is strictly positive. Let χ be a bounded measurable function on and consider the symmetric form on H 0 (F k ) defined by (f, f) χ = χ f 2. By linear algebra, there is a selfadjoint linear operator on H 0, T χ, such that (f, g) χ = (T χ f, g). It is clear that T χ f is precisely the orthogonal projection of χf on the space of holomorphic sections, so T χ f(y) = < χf, K y >. We are interested in the eigenvalue distribution of T χ for k large. We will have use for the special case when χ is the characteristic function of a domain in when we discuss sampling problems in the next section and it also turns out to be convenient to treat this case first. The idea to use Toeplitz operators to study sampling is taken from H Landau [15] who worked in the setting of Paley-Wiener spaces. The method we use for the estimates also follows closely the ideas of Landau. So, we now let χ be the characteristic function of a domain Ω and write T Ω for T χ. By the spectral theorem, T Ω has an orthonormal basis of eigenfunctions (or eigensections ), and we may of course use this basis to define the Bergman kernel. The trace of T Ω is easily computed by t(ω, k) = tr(t Ω ) = f j 2 = B k. Ω Ω By Theorem 2.3 it follows that lim k n t(ω, k) = π Ω n c(l) n /n!. In a similar way we can compute the Hilbert-Schmidt norm, tr(tω 2 ) and find t 2 (Ω, k) = tr(tω) 2 = K(x, y) 2. By Theorem 2.4 we conclude that Ω Ω too. lim k n t 2 (Ω, k) = π Ω n c(l) n /n!

8 8 Fix a large value of k and let the eigenvalues of T Ω be τ 1,...τ N. We have then seen that k n τ j = k n τ 2 j + o(1). Fix a number γ < 1 but close to one. Since k n τ j (1 τ j ) = o(1) and each term in this sum is positive (the eigenvalues are obviously smaller than 1) we get that lim k n τ j γ τ j = 0. Therefore lim k n τ j = π n c(l) τ Ω n /n! j>γ for any such γ. Now denote by N(T Ω, γ) the number of eigenvalues τ j that are greater than γ. Since the two limits above hold for an arbitrary γ the next theorem easily follows. Theorem 3.1. Assume the line bundle L is strictly positive. Then, for any 0 < γ < 1 lim k n N(T Ω, γ) = π Ω n c(l) n /n!. From this is not hard to deduce, by comparison, the asymptotic behaviour of the eigenvalue distribution of T χ for a general continuous function χ. We just state the result and refer the reader to [16] for the small argument that is missing. Theorem 3.2. Let χ be a continuous real-valued function on. Let τ j be the eigenc values of T χ and put dξ k = k n δ τj (so that dξ is a sum of Dirac measures on the line). Then dξ k tends, in the weak* -topology to the push forward of the measure c(l) n /n! under the map χ. 4. SAMPLING We still assume that L is positive and consider the space of sections to F k, H 0 (F k ). Let, for k > 0, D k be a discrete set of points in, that we assume to be separated so that any two different points have a distance bounded from below by a positive constant times k 1/2. We shall say that the sequence of sets (D k ) is sampling for the sequence of bundles F k if there is a uniform constant A such that A 1 k n D k f(x) 2 f 2 Ak n D k f(x) 2.

9 9 Let dν k = k n D k δ x. If D k satisfies the separation condition above then the measures dν k have total mass bounded by a constant independent of k. Hence any sequence of dν k s has a subsequence that is weak -convergent. We shall prove: Theorem 4.1. Assume that (D k ) is sampling for F k. If a subsequence of the dν k s converges weakly to the measure dν, then dν π n c(l) n /n!. For the proof we first need a lemma. Lemma 4.2. Let Ω be a domain in and let Ω δ be the set of points with distance smaller than δ to Ω. Assume (D k ) is sampling for F k and let f be a section to F k that vanishes on D k Ω k 1/2. Then there is a number γ < 1, only depending on A, such that f 2 γ f 2. Proof. By the assumptions we have that Ω f 2 Ak n D k f(x) 2 and the sum is only taken over points in the complement of Ω k 1/2. Since f is holomorphic it can be verified that for any point x f(x) 2 Ck n f 2 U k (x) where U k (x) is a suitable neighbourhood of x of diameter of size k 1/2. Since the points in D k are separated by a distance of this order we can sum over the points in D k Ω c and then get k 1/2 f 2 CA f 2 = CA( f 2 f 2 ). Ω c Ω Hence f 2 (CA 1)/CA f 2 which proves the lemma Ω We can now prove the theorem in the following way. Let N k be the number of points in D k Ω k 1/2. We claim that N k N(T Ω, γ), where γ is the number given in the lemma, and T Ω is the Toeplitz operator from the previous section. Indeed, if this inequality does not hold, we can find a section f to F k which is a linear combination of eigensections to T Ω

10 10 corresponding to eigenvalues greater than γ, which vanishes on D k Ω k n (fewer equations than unknowns!). For such a section f 2 > γ f 2, Ω contradicting the lemma. Combining this with the estimate of N(T Ω, γ) from Theorem3.1, we immediately get that dν c(l) n /n!. Ω Ω This easily implies the theorem. 5. BERGMAN KERNELS FOR SPACES OF HARMONIC FORMS. We now start to consider the higher order Dolbeault cohomology groups, H q (F k ). By Hodge s theorem each cohomology class has a unique harmonic representative, or in other words H q is isomorphic to the space of harmonic forms that we denote by H q. By definition, a form α is harmonic if α = 0, where = +. We shall study the asymptotic behaviour of the Bergman kernels of H q, but only treat the kernels on the diagonal. Let α j be an orthonormal basis for the space H q of harmonic (0, q)-forms with values in F k. As before α j denotes the pointwise norm of α j which now depends both on the metric on the bundle and the metric on. We define the Bergman function of H q by B q k (x) = α j 2. A bit more information can be obtained by looking at a closely related form of bidegree (q, q), B q k that we call the Bergman form of Hq. To define B q k, choose a local trivialization of F and represent an F -valued form α by a scalar form v in terms of the trivialization. If the metric on F is given by e τ we put [α ᾱ] = c q v ve τ, where c q is a number of modulus 1 chosen to make the form non-negative. Clearly this expression does not depend on the choice of trivialization. Now define B q k (x) = [α j ᾱ j ]. One verifies that the definition is independent of the choice of basis. B q k is a positive form in the sense that the associated quadratic form on the space of (0, n q)-forms at x (θ, θ)ω n = c n q θ θ B q k is positive. The trace of B q k defined by tr(b q k ) ω n = B q k ω n q

11 equals B q k. As in section 2 we fix a point x in, choose local coordinates so that z(x) = 0 and the metric form ω = β = i/2 dz j d z j at z = 0. We also choose trvializations of L and E ( and hence of F = F k ) so that the metrics on L and E are respectively given by φ = φ 0 + o( z 2 ) ψ = ψ 0 + o( z 2 ), where φ 0 and ψ 0 are quadratic forms. Consider the space of harmonic forms in L 2 (C n, e φ 0 ), and let B q (C n, φ 0 ) and B q (C n, φ 0 ) be the Bergman function and form of this space. The definitions are in complete analogy with the corresponding definitions on, even though they now involve infinite sums. ( Since point evaluations are bounded operators on the space of harmonic forms, it is classical that the series converge). We have now come to the principal result of this section (see [7]). Theorem 5.1. With notation and assumptions as above we have lim sup k n B q k (x) Bq (C n, φ 0 )(0), and lim sup k n B q k (x) Bq (C n, φ 0 )(0) Here the limsup in the second part of the statement means that for any (n q, 0)-form θ at z = 0 lim sup B q k (x) c n qθ θ B q (C n, φ 0 )(0) c n q θ θ. It is clearly enough to prove the second statement, about the Bergman forms, since the first one follows by taking traces. We will have use for the following lemma. Lemma 5.2. If θ is an (n q, 0)-form at x, then B q k (x) c n qθ θ = sup[α(x) ᾱ(x)] c n q θ θ =: S k,θ (x), where the supremum is taken over all harmonic (0, q)-forms of norm smaller than 1. Proof. The somewhat cumbersome formulation in the lemma is dictated by our thinking of B q k as a differential form. Unwinding the definitions we see that the statement kan be refomulated as follows. Let v be a (0, q) form at x with values in F. Then < αj (x), v > 2 = sup < α(x), v > 2 =: σ, 11 the supremum again being over all harmonic forms of norm smaller than 1. If α = c j α j and α is of norm 1, then c j 2 = 1 and < α(x), v > 2 = c j < α j, v > 2 < α j (x), v > 2.

12 12 For the reverse inequality, note that < αj (x), v > 2 =< γ(x), v > if Since we get which completes the proof. γ = < α j (x), v >α j. γ 2 = < α j (x), v > 2 < γ(x), v > ( < α j (x), v > 2 ) 1/2 σ, A similar statement of course holds for the Bergman form in C n, and in this case we denote the extremal function at the origin S C n,θ(0). The proof of Theorem 5.1 follows closely the arguments in section 2. Fix x, a form θ at z = 0, and let for each k α (k) be the element in H q which maximizes S k,θ (x). Express α (k) in terms of the local coordinates z and local trivialization, chosen as before, and define a scalar form in C n by a (k) (ζ) = k n/2 α (k) (k 1/2 ζ) if ζ < c k, and a (k) (ζ) = 0 if ζ c k. The c k s will be chosen as a sequence that slowly tends to infinity, and by α (k) (k 1/2 ζ) we understand that we have scaled the coefficients of α (k). (We could have scaled the differentials as well, i e considered the pullback of α (k) under the map ζ k 1/2 ζ.) The norm of a (k) in L 2 (C n, φ 0 ) is k ζ <c n α (k) (k 1/2 ζ) 2 e φ 0 = α (k) 2 e kφ 0, k z <c k k 1/2 where for a brief moment we have let the norms denote the Euclidean norms. If c k tends to infinity sufficiently slowly this expression can be estimated from above by (1 + ɛ k ) α (k) 2 ω n = 1 + ɛ k where ɛ k tends to zero. Any subsequence of k s therefore has a subsequence (k l ) such that a (kl) converges weakly in L 2 (C n, φ 0 ) to a form a ( ) of norm not exceeding 1. We claim that the convergence is also uniform on the unit ball (in fact, on any compact) and that the limit is harmonic in L 2 (C n, φ 0 ). Accepting this, we see that a ( ) is a contender for the extremal for S C n,θ(0). Since each α (k) was an extremal for S k,θ and the sequence is also pointwise convergent it follows that lim l k n l S k,θ (x) S C n,θ(0).

13 Therefore lim sup S k,θ (x) S C n,θ(0), which, combined with lemma 5.2 gives Theorem 5.1. It remains to show that a (kl) converges uniformly on compacts and that the limit form is harmonic in L 2 (C n, φ 0 ). For this, note that since α (k) is harmonic on, a (k) is harmonic for ζ < c k for the Laplace operator, (k), on forms in C n determined by the weightfactor e τ (k) where τ (k) (ζ) = (kφ + ψ)(k 1/2 ζ) and the metric form ω (k) = kω(k 1/2 ζ). As k tends to infinity τ (k) tends to φ 0 and ω (k) tends to the Euclidean metric form. One verifies that it follows from this that the Laplace operators (k) converge to the Laplace operator in L 2 (C n, φ 0 ), in the sense that the coefficents of these second degree elliptic operators converge uniformly on compacts. In particular, (k) form a family of second order elliptic operators that is uniformly bounded and uniformly elliptic on compacts. From this it follows from Garding s inequality and the Sobolev inequality that on any relatively compact domain in C n, derivatives of any order of a harmonic form can be uniformly bounded (independently of k) by the norm of the form in L 2 on a slightly larger domain. Hence any bounded sequence of such forms has a subsequence that converges uniformly with all its derivatives on compacts, and if the sequence is already weakly convergent, the whole sequence converges uniformly on compacts. Finally, the limit form is clearly harmonic in L 2 (C n, φ 0 ). A weak consequence of the above is a (k) (0) can be bounded by a constant times the norm of a (k) over a ball of radius 1, which in turn can be bounded from above by Ck n/2 α (k). Hence S k,θ (x) Ck n for any θ of norm 1, and by the lemma B q k (x) Ckn. On a compact manifold one can verify that the constant here can be chosen independently of the point, so sup B q k Ckn. To get concrete information from Theorem 5.1 we need to compute the Bergman kernel of the space of harmonic forms in L 2 (C n, φ 0 ) at the origin, B q (C n, φ 0 )(0). Theorem 5.3. Let λ 1...λ n be the eigenvalues of the quadratic form φ 0, and let D(φ 0 ) be the product of the eigenvalues. Assume first that exactly q of the eigenvalues are negative, and let I be the multindex of length q consisting of the indices corresponding to negative eigenvalues. Then In particular B q (C n, φ 0 ) = π n ( 1) q D(φ 0 )c q dz I d z I. B q (C n, φ 0 ) = π n ( 1) q D(φ 0 ). If it does not hold that exactly q of the eigenvalues are negative, then B q (C n, φ 0 ) = 0 13

14 14 Proof. Let α I be a a (0, q)-form of the form α I = ad z I where I is a multiindex of length q and dz I = dz i1...dz iq. Let be the formal adjoint of with respect to the scalar product in L 2 (C n, φ 0 ) and let δ i be the formal adjoint of j =: / z i. We may assume that φ 0 = λ j z j 2 is diagonalized. Then a simple and elegant computation, using that j and δ i commute for i j ([7]) shows that the Laplacian of α I is given by α I = ( i I c δ i i a + i I i δ i a)d z I. In particular the standard coordinate representation of a (0, q)-form diagonalizes, so if a form is harmonic, each component is also harmonic. Integrating by parts we find that 0 = ( α I, α) = i a 2 + δ i a 2. i I c I Hence, if α I is harmonic then its coefficient a is holomorphic with respect to z i for any i outside I. If α I is square integrable against the weight factor e φ 0 and different from 0 it follows that all the eigenvalues λ i are positive for i outside of I. On the other hand δ i a also vanishes for i in I. This is easily seen to mean that e λ i z i 2 ā is holomorphic with respect to z i, which in turn implies that all λ i must be negative if i is in I. In particular the form φ 0 has exactly q negative eigenvalues if there are nontrivial harmonic forms, so we have proved the last part of the theorem. Assume this is the case and that α I is harmonic. Then we have seen that h(z) = e i I λ i z i 2 a( z), where z means that we have conjugated the variable z i for i in I is holomorphic and square integrable with respect to e φ 0 where φ 0 = λ i z i 2. Hence the computation of the Bergman form is reduced to the computation of the Bergman function for a space of holomorphic functions, in which case the answer is well known ( e g by a method like the one used in section 2). This completes the proof. Now let (q) be the set of points in where the curvature form of L has exactly q negative eigenvalues, and let D[x] = D(φ 0 ) if φ 0 is the quadratic form associated to the point x as above. Combining Theorems 5.1 and 5.3 we get that (5.1) lim sup k n B q k (x) π n ( 1) q D[x]. This easily implies the following important result of D ly, see [11], [12].

15 Theorem 5.4. (D ly s weak holomorphic Morse inequalities.) lim sup k n H q (F k ) ( 1) q c(l) n /n! (q) Proof. By the remarks preceeding Theorem 5.3 we have that k n B q k C. The theorem therefore follows from (2.3) and (5.1) by Fatou s lemma, since D[x]ω n = c(l) n /n!. D ly also proved Strong holomorphic Morse inequalities for the dimensions of the cohomology groups H q (F k ). These state that for any 0 m n, (5.2) q m( 1) m q H q (F k ) ( 1) m q I(q), q m where I(q) = ( 1) q (q) c(l) n /n!. The strong Morse inequalities are obtained from estimates of the dimensions of spaces of forms spanned by eigenforms corresponding to small eigenvalues of the Laplacian, just as in [13], which in turn uses an idea of Witten [22]. Let H q µ(f ) be the linear span of forms satisfying α = να where ν µ. The Laplace operator is a selfadjoint operator from H q µ(f ) to itself, so we can decompose the space as the orthogonal sum of the kernel and the range. Since = + we get H q µ(f ) = H q + H q = H q + A q + B q where A q is the range of and B q is the range of. The -operator is an isomorphism between B q and A q+1, so if lower case letters denote the dimensions of the corresponding spaces we get m m m m ( 1) m q h q µ = ( 1) m q h q + ( 1) m q a q + ( 1) m q a q+1 = 0 = 0 0 m ( 1) m q h q + ( 1) m a 0 + a m+1. 0 As a 0 = 0 and a m+1 is non-negative, it follows that m m (5.3) ( 1) m q h q µ ( 1) m q h q, 0 so to prove the strong Morse inequalities it suffices to have good information on h q µ for some appropriately chosen µ. This can be obtained from the following estimate of Bergman kernels

16 16 Theorem 5.5. Let B k,µk be the Bergman function for the spaces H q µ k (F k ). If (µ k ) is an appropriately chosen sequence such that µ k tends to infinity and µ k /k tends to zero, then (5.4) lim k n B k,µk (x) = π n ( 1) q D[x] if x lies in (q). Outside of (q) the limit is 0. We refer to [7] for the proof of Theorem5.5. Integrating (5.4) we find that m m lim k n h q µ k (F k ) = I(q) 0 which, together with (5.3) gives the strong Morse inequalities. 6. A SHARPER ESTIMATE IN THE SEMIPOSITIVE CASE An immediate consequence of Theorem 5.4 is that if L is semipositive, i e if c(l) has only non-negative eigenvalues everywhere, then (6.1) H q (F k ) = o(k n ). This estimate was first obtained by Siu, see[18], [19], and was the main step in his proof of the Grauert-Riemenschneider conjecture. The next result (see [1]) is a strengthening of (6.1), which gives the optimal order of vanishing of the dimensions in the semipositive case. Theorem 6.1. Assume c(l) 0 on. Then there is a constant such that if µ k. In particular H q µ(f k ) C(µ + 1) q k n q, H q (F k ) Ck n q. To have an example where Theorem 6.1 is sharp, it suffices to consider manifolds of the form = M N where M and N are manifolds of dimension q and n q respectively.for the line bundle L we take the pullback under the projection map of a positive line bundle, L, on N. Then any form α on of type α = π Mγπ Nf, where γ is a harmonic (0, q)-form on M and f is a holomorphic section to L k is a harmonic (0, q)-form on with values in L k. Since, by Theorem 2.3 we have a space of dimension Ck n q to choose f from it follows that the last part of Theorem 6.1 is sharp if only M has non-trivial cohomology of degree q. An elaboration of this argument shows that the first part is also sharp. The proof of Theorem 6.1 is also based on an estimate of the Bergman function. To simplify, we discuss only the case µ = 0. We thus consider the Bergman function and the Bergman form of H q (F k ), and shall prove that B q k (x) Ckn q 0

17 uniformly on. By Theorem 5.3, this means that α(x) 2 Ck n q α 2 for any α in H q (F k ). The proof proceeds in two steps. First, by a scaling argument, like the one used in section 5, we prove that (6.2) α(x) 2 Ck n z k 1/2 α 2. Here z are the local coordinates used in the previous section. Defining α (k) as in that section we see that (6.2) means that α (k) (0) 2 C α (k) 2, ζ 1 which follows from Gardings inequality and Sobolev estimates just as before. To improve the estimate we now need to show that (6.3) α 2 Ck q α 2. z k 1/2 This means that a harmonic form cannot be too concentrated near a single point. For this we will use an important differential inequality for differential forms. Theorem 6.2. Let α be a harmonic form of bidegree (0, n q) on with values in a line bundle F. Then (6.4) i ([γ γ] ω q 1 ) c(f ) [γ γ] ω q 1 c ω γ 2 ω n. Here c ω is a constant depending only on the metric ω which vanishes if the metric is Kähler. Theorem 6.2 was first proved by Siu [20] in the Kähler case. A proof of the result as stated above can be found in [1]. ( Here we would like to point out a mistake in [1]. There the inequality (6.4) is stated as an inequality for i ([γ γ]) ω q 1, but it is the inequality as stated above that is actually used. In the Kähler case, or when q = 1, the two formulas coincide, and both inequalities hold true in the general case. ) Notice that when the curvature form c(l) is negative, and the metric is Kähler, it follows from (6.4) that the form S = [γ γ] is a subharmonic (n q, n q)-form in the sense that i S ω n q 1 0. It was proved by Skoda, [21], that such forms enjoy certain submeanvalue properties. These submeanvalues properties are the essential ingredient in the proof of (6.3), and we shall now briefly explain how that works. Let α be a harmonic (0, q)-form with values in F k = L k E. We can also think of α as a harmonic (n, q)-form with values in F k K 1. Since anyway we can choose our bundle E at will we see that it suffices to prove (6.3) for harmonic (n, q)-forms with values in F k. Now recall that there is a natural isomorphism between F -valued (n, q)- forms and forms of degree (0, n q) with values in F 1. Expressed in terms 17

18 18 of a local trivialization of F with respect to which the metric is given by e τ the isomorphism is (6.5) α γ =: αe τ, where * is the Hodge *-operator. If we give F 1 the induced metric (e τ in the trivialization) then (6.5) exchanges the operators and, and in particular preserves the Laplace operator and harmonic forms. Given a harmonic (n, q)-form α we now put (6.6) T α = [γ γ], with γ defined by (6.5). It follows from Theorem 6.2 that if F = F k = L k E where c(l) is non-negative everywhere, then i (T α ω q 1 ) C α 2 ω n, with C independent of k. Now we return to our point x and the coordinate system z used above. Put σ Tα (r) = T α ω q. z <r In [21], Skoda proved that if the metric ω is Euclidean and if i T 0, then σ T (r)/r 2q is nondecreasing. By a variant of his argument one can prove that in the above situation we have that σ Tα (r)/r 2q is almost nondecreasing in the sense that there is a constant such that (6.7) σ Tσ (r)/r 2q Cσ Tα (R)/R 2q if r < R ( and R is so small that we remain in the coordinate neighbourhood ). But T α ω q = γ 2 ω n = α 2 ω n. Applying (6.7) with r = k 1/2 and R equal to a fixed small constant we get (6.3). This completes the sketch of the proof of Theorem 6.1 REFERENCES [1] BERNDTSSON, B: :, An eigenvalue estimate for the Laplacian.. Preprint, Chalmers University (2001) (to appear in Journal of Differential Geometry). [2] BERNDTSSON, B AND CHARPENTIER, PH,: A Sobolev mapping property of the Bergman kernel., M.ath. Z. 235 (2000) [3] J-M BISMUT: D ly s asymptotic Morse inequalities: a heat equation proof., J.. Funct. Anal. 72 (1987), no. 2, [4] T BOUCHE: Convergence de la métrique de Fubini-Study d un fibré linéaire positif., A.nn. Inst. Fourier (Grenoble) 40 (1990), no. 1, [5] L BOUTET DE MONVEL AND V GUILLEMIN: The spectral theory of Toeplitz operators., Annals of Mathematics Studies, 99. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, [6] DONNELLY, H. AND FEFFERMAN, C: :, L 2 -cohomology and index theorem for the Bergman metric.. Ann. of Math. (2) 118 (1983), no. 3,

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