On L p -cohomology and the geometry of metrics on moduli spaces of curves

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1 On L p -cohomology and the geometry of metrics on moduli spaces of curves Lizhen Ji and Steven Zucker October 18, 2010 Abstract Let M g,n be the moduli space of algebraic curves of genus g with n punctures, which is a noncompact orbifold. Let M DM g,n denote its Deligne-Mumford compactification. Then M g,n admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between M g,n (resp., Teichmüller spaces) with these metrics and certain noncompact locally symmetric spaces (resp. symmetric spaces of noncompact type) with their natural metrics. In this chapter, we observe that for all 1 < p <, the L p -cohomology of M g,n with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of M DM g,n, and hence the L p -cohomology is the same for different values of p. This suggests a rank-one nature of the moduli space M g,n from the point of view of L p -cohomology. On the other hand, the L p -cohomology of M g,n with respect to the incomplete Weil-Petersson metric is either the cohomology of M DM g,n or that of M g,n itself, depending on whether p 4 3 or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics. 1 Introduction and the setting Let M g,n be the moduli space of algebraic curves over C of genus g with n punctures. There has been much work devoted to parallels between M g,n and locally symmetric spaces of finite volume, in particular, Hermitian locally symmetric spaces Γ\D, where D is a bounded symmetric domain in Partially Supported by NSF grant DMS Partially Supported by NSF grant DMS

2 some C d, and Γ Aut(D) is a lattice subgroup of the group of holomorphic automorphisms of D. This is our point of departure. Let T g,n be the Teichmüller space that parametrizes the complex structures on a compact orientable surface S g,n of genus g with n punctures, together with a marking on S g,n (i.e., choice of a set of generators of π 1 (S g,n )). Then T g,n is a complex manifold biholomorphic to a bounded domain in C 3g 3+n (when 3g 3 + n > 0). The mapping class group Mod g,n = Diff + (S g,n )/Diff 0 (S g,n ) acts properly and biholomorphically on T g,n, where Diff + (S g,n ) denotes the group of all orientation-preserving diffeomorphisms of S g,n and Diff 0 (S g,n ) is the identity component of Diff + (S g,n ). The quotient Mod g,n \T g,n is the moduli space M g,n. We note that the Mod g,n - quotient nullifies the role of the marking on S g,n, so M g,n parametrizes the complex structures alone. 1 By [19], M g,n has a natural structure of a quasi-projective algebraic variety. The moduli space M g,n = Mod g,n \T g,n is noncompact. It admits the Deligne-Mumford compactification M DM g,n, which parametrizes stable curves [19]. This is a complex projective algebraic variety, in fact a compact complex orbifold. The mapping class group admits normal subgroups Γ of finite index that act freely on T g,n, for which M Γ g,n := Γ\T g,n is a manifold. For such Γ, M g,n is the quotient of M Γ g,n by the finite group G = Γ\Mod g,n, and M DM g,n is the quotient by G of the smooth compactification of Γ\T g,n. For convenience, sometimes we will formulate and prove our assertions for such M Γ g,n in this paper. One can regard T g,n as a sort of analogue of a bounded symmetric domain D, and Mod g,n as an analogue of a lattice subgroup of Aut(D), starting with the following. The space D has a canonical complete Kähler metric that is invariant under the action of Aut(D), and it descends to Γ\D. When Γ is arithmetically-defined and Γ\D is noncompact, 2 the latter admits the rather canonical Baily-Borel Satake compactification Γ\D BB [6] (a normal projective variety), showing that Γ\D is a quasi-projective variety. The analogy extends to the Deligne-Mumford compactification M DM g,n and the Baily-Borel compactification Γ\D BB (though the former is a much nicer space). By construction, the boundary of Γ\D BB is the union of lower dimensional Hermitian locally symmetric spaces. For example, if D is equal to the Siegel upper half space S n of degree n and Γ\D is a Siegel modular variety, then the boundary (Γ\D BB ) is the union of Siegel modular varieties 1 For a good early survey of the above, see [10]. 2 We shall assume this throughout the rest of this chapter. 2

3 of lower degrees. Similarly, the boundary of M DM g,n is the union of moduli spaces of stable curves of smaller genus and with more punctures. Thus, the boundaries in both cases are hereditary in nature. Another analogy between these compactifications Γ\D BB and M DM g,n is the way they are constructed. The Baily-Borel compactification Γ\D BB is the Γ-quotient of a partial compactification D of D that is obtained by adding rational boundary components from the closure of D as a bounded symmetric domain, and giving it a suitable topology. Then the Teichmüller space T g,n also admits a partial compactification called the augmented Teichmüller space T g,n in [1] that is also the completion with respect to the Weil-Petersson metric [85], and the quotient of T g,n by Mod g,n is homeomorphic to the Deligne-Mumford compactification M DM g,n. For any finite index subgroup Γ of Mod g,n, the quotient Γ\ T g,n is also compact. For some special subgroups Γ, such as those given in [60], the compactification Γ\ T g,n is homeomorphic to a smooth projective variety that admits a finite mapping onto M DM g,n. There are numerous natural complete Mod g,n -invariant Riemannian and Finsler metrics on T g,n. These include the Teichmüller (Finsler) metric, the Bergman metric, the Kobayashi metric (which was shown by Royden [71, 72] to be equal to the Teichmüller metric), the Carathéodory metric, the Kähler-Einstein metric, the McMullen metric, the Ricci metric, and the Liu-Sun-Yau metric (a perturbed Ricci metric). It was shown in [58] that all of these are, in fact, quasi-isometric. 3 Looijenga [59] and Saper-Stern [77] proved that the L 2 -cohomology of Γ\D is canonically isomorphic to the (topological) intersection cohomology of Γ\D BB for what is called the middle perversity. 4 This result was conjectured by the second author [93] and is still often called the Zucker conjecture in the literature even though it is now a theorem. It is then a plausible guess that the L 2 -cohomology of M g,n, with respect to any of the above complete metrics, is canonically isomorphic to the intersection cohomology of M DM g,n with coefficients in R. Since M DM g,n is an orbifold, so has only quotient singularities, the latter is equal to the usual cohomology of M DM g,n with Q- coefficients. This is indeed true. With some motivation from a result of [94] saying that the L p -cohomology of Γ\D is equal to the ordinary cohomology 3 The article [90] independently showed the quasi-isometry of some of these metrics. 4 It seems to be accepted that the result for general Γ can be deduced from the case where Γ is neat. However, there is nothing in the literature that gives a proof of it. 3

4 of its reductive Borel-Serre compactification for sufficiently large and finite p, we prove, without so much effort, the following more general result. Theorem 1.1 For 1 < p <, the L p -cohomology of M g,n, with respect to the canonical complete metrics above, 5 is isomorphic to the cohomology of M DM g,n. In particular, the L p -cohomology of M g,n does not depend on the value of p. 6 Since the reductive Borel-Serre compactification of Γ\D is generally different from the Baily-Borel compactification when the rank of any irreducible factor of the bounded symmetric domain D is greater than one, or more generally when the covering symmetric space of some irreducible factor of Γ\D has rank greater than 1, 7 the L p -cohomology of Γ\D usually depends on the value of p. Thus, the above result says that the moduli space M g,n behaves in this regard like rank-one locally symmetric spaces or their products. (See the paper [28] for results on rank-1 phenomena of the mapping class groups.) The Teichmüller space T g,n also admits a canonical incomplete Kähler metric, the Weil-Petersson metric. In [76], it was proved that the L 2 - cohomology group of M g with respect to the Weil-Petersson metric is canonically isomorphic to the cohomology of M DM g. Though the theorem in [76] is stated only for the case of M g,0, the same method works for general M g,n ; the L 2 -cohomology of M g,n with respect to the Weil-Petersson metric is isomorphic to the cohomology of M g,n DM. In contrast with Theorem 1.1, we have: Theorem 1.2 For 4 3 p <, the Lp -cohomology of M g,n with respect to the Weil-Petersson metric is isomorphic to the cohomology of M DM g,n, whereas for 1 p < 4 3, the Lp -cohomology of M g,n with respect to the Weil-Petersson metric is isomorphic the cohomology of M g,n itself. Both Theorem 1.1 and Theorem 1.2 are decided locally on M DM g,n, as described in Section 5. 5 Since it is complicated to define norms of differential forms and the volume form for a Finsler metric and the L p -cohomology groups of Riemannian manifolds only depend on the quasi-isometry class of the Riemannian metrics, when we say the L p -cohomology of a Finsler metric, we meant the L p -cohomology of a Riemannian metric that is quasi-isometri to it. 6 The case of p = 2 appeared already in [89]. 7 The Baily-Borel compactification of Γ\D is always a quotient space of the reductive Borel-Serre compactification [97]. 4

5 We wish to remind the reader that the mapping class groups, the topology of M g,n, and its Weil-Petersson metric have been studied extensively by many people, too numerous to list here. On the other hand, the complete Riemannian metrics on M g,n mentioned above have only recently been studied systematically in [58], where the aforementioned quasi-isometries were proved. Some related results had also been proved in [90]. The partial similarity also comes from the spectrum of their Laplace operators. It is known that Γ\D can have nonempty continuous spectrum. 8 On the other hand, while it is proved in [48] that the Laplace operator of M g,n with the Weil-Petersson metric has a discrete spectrum, the spectrum of the Laplace operator for any of the above complete Riemannian metrics is expected to contain rays of the form [a, + ), and their generalized eigenfunctions should be analogous to the Eisenstein series for locally symmetric spaces. Acknowledgment. We would like to thank Mark Goresky for helpful correspondence. 2 L p -cohomology L 2 -cohomology is defined from square-integrable differential forms on a Riemannian manifold. Because L 2 defines a Hilbert space, which is self-dual, it attracted special attention. L p -cohomology is defined analogously for p 2. For compact manifolds, it only recovers the de Rham theorem. We intend to give here an account of the main thrust of the research in this direction of mathematics, we hope without any glaring omissions. We start with the case p = 2, for it was the one studied first. In essence, L 2 -cohomology is the cohomological embodiment of two fundamental notions: L 2 harmonic forms on a non-compact Riemannian manifold M, and cohomology classes in H (M, R) that are represented by L 2 differential forms (via the de Rham theorem). While both of these are pertinent considerations, the idea of defining the L 2 -cohomology, denoted H (2) (M, R), as the hypercohomology of a complex of sheaves on suitable topological compactifications of M represents a modern point of view, which we take in this chapter. The two pioneering works, coming at the end of the 1970 s, were by Cheeger [15] and Zucker [92]. 9 8 The spectral decomposition of locally symmetric spaces has played a fundamental role in the Langlands program (see [56, 34]). 9 An anecdote: There was an important conference Analyse et topologie sur les espaces 5

6 In terms of the latter, it is often the case that H (2) (M, R) is represented by a space of L 2 harmonic forms (e.g., for any metric when M is compact), and the reason why this can fail is easy enough to grasp. Also, the space of cohomology classes in H (M, R) that are represented by L 2 differential forms is just the image of the mapping H (2) (M, R) H (M, R) that is induced by the inclusion of the L 2 de Rham complex in the full de Rham complex. (As such, one might say L 2 de Rham cohomology.) When M is (say) a complete complex Kähler manifold, the space of L 2 C-valued harmonic forms has a decomposition according to complex bidegree (see (3.10) below). This gives rise to the very important Hodge decomposition of H (M, C) in algebraic geometry when M is a smooth projective variety over C. We will treat these matters in greater detail below. Cohomology is defined for cochain complexes C (first, of vector spaces over R). More specifically, when the complex has terms only in non-negative degrees, we mean 0 C 0 d 0 C 1 d 1 C 2 d 2..., (2.1) where the differentials d i satisfy d i d i 1 = 0, so im d i 1 ker d i. The i-th cohomology of C is defined as the real vector space H i (C ) (ker d i )/im d i 1. (2.2) A morphism of complexes Ψ : B C, i.e., a set of linear mappings ψ i : B i C i that are compatible with the differentials (d i C ψi = ψ i+1 d i B ), induces mappings H i (B ) H i (C ). (2.3) Ψ is called a quasi-isomorphism (of complexes) when (2.2) is an isomorphism for all i. The R-valued C forms on a smooth manifold M comprise a cochain complex A (M), with A i (M) being the i-forms and each d i given by the exterior derivative d. The de Rham Theorem identifies H i (A (M)) H i (M, R); (2.4) here the cohomology on the right-hand side is what is defined in algebraic topology, using, e.g., singular or Cech cochains with real coefficients. Moresinguliers at Luminy in July, 1981 (proceedings [81]). The second author recalls that at one point, Cheeger, being a differential geometer, said to MacPherson in his presence something like, I was expecting/hoping there would be more L 2 -cohomology. MacPherson s reply, There are two experts in the world on L 2 -cohomology, and both of them are here. 6

7 over, cup product on cohomology is induced by the exterior product of differential forms. The de Rham Theorem is best understood by introducing the sheaves of germs of differential forms on M [83]. Let A M be the complex (cf. (2.1)) of sheaves of vector spaces over R on M given by the data: for U open in M, U A (U). Sheaves allow for a standard way to deduce global results from local ones. Because each sheaf A i M admits partitions of unity, induced by arbitrarily fine partitions of unity on functions (viz. A 0 M ), so these are called fine sheaves. One deduces the de Rham Theorem as the global consequence of the local calculation (Poincaré lemma) { H i (A R if i = 0, (U)) = (2.5) 0 otherwise whenever U is a contractible open subset of M. The rather trivial complex of sheaves R[0], consisting of R in degree zero, and 0 in all other degrees, maps to A (M) with image the locally constant functions on M. The de Rham theorem follows from the fact that the morphism R[0] A M is a quasi-isomorphism in the category of complexes of sheaves on M, which is just (2.5). Next, we assume that M has a specified Riemannian metric g. This imparts a notion of length to tangent vectors to M, and thereby a length to all tensors on M. Thus, if ϕ A (M), one gets a non-negative function ϕ, the (pointwise) length of ϕ, on M. We also get from g a volume form dv M on M, and corresponding L p seminorms: ( ϕ (p) = M ϕ p dv M ) 1/p (2.6) for any p with 1 p <. The definition of the L seminorm ϕ ( ) goes in the usual way. We say that ϕ A (M) is L p when ϕ (p) <. The simplest way to proceed is: Definition 2.1 (i) The L p de Rham complex of M is the largest subcomplex of A (M) contained in the L p forms. Its elements are {ϕ A (M) : ϕ and dϕ are L p }, and it is denoted A (p) (M). (ii) The L p -cohomology of M is the cohomology of the L p de Rham complex of M, i.e., H(p) i (M) = Hi (A (p)(m)). (2.7) 7

8 It is clear that the L p -cohomology of M depends on g only to the extent that the L p de Rham complex does; quasi-isometric metrics on M yield the same L p de Rham complex, hence the same L p -cohomology. (Metrics g 1 and g 2 are said to be quasi-isometric when there are positive real numbers c 1 c 2 such that g 1 c 1 g 2 and g 1 c 2 g 2.) We return to the case p = 2. The starting point of the role of L 2 - cohomology is the Hodge theorem: Theorem 2.2 [44]. Let M be a compact Riemannan manifold. Then in every (de Rham) cohomology class in H (M, R), there is one and only one harmonic differential form. One notes that there is no mention of L 2 in the above statement since smooth differential forms on compact manifolds are automatically square integrable. The proof in [44] relies on the theory of integral equations. The reworking of the theorem in terms of Hilbert spaces is due to Gaffney [32, 33], based on an earlier treatment by Kodaira [54]. Before we explain this, we point out direct consequences of the Hodge theorem. The basic reason is that cohomology groups can be studied using the unique canonical representatives instead of cohomology classes. 1. It gives an analytic proof that the de Rham cohomology groups of a compact Riemannian manifold are finite dimensional. The reason is that the spectrum of the Laplacian (i.e., Hodge-Laplacian, or de Rham Laplace operator) on differential forms of a compact Riemannian has a discrete spectrum (or more to the point that 0 is an isolated point in the spectrum with finite multiplicity) [82, p. 226]. 2. The de Rham cohomology groups of a compact oriented Riemannian manifold satisfies the Poincaré duality, since for each harmonic differential form, it is easy to find another harmonic form which pairs nontrivially with it (see (2.20) below or [82, p. 226]). 3. On a compact orientable Riemannian manifold with nonnegative Ricci curvature, every harmonic 1-form is parallel. Furthermore, if the Ricci curvature is positive definite at some point, then every harmonic 1- form is zero and the first cohomology group is zero. In the proof, the Weitzenböck formula was applied to harmonic 1-forms, and the positivity of the Ricci curvature forces vanishing of the harmonic form. This result of Bochner has been developed into a powerful method called Bochner technique. One important result proved by this technique is the Kodaira vanishing theorem. See the book [88] and the 8

9 survey [9] for many variants of the Bochner vanishing theorem on harmonic 1-forms. 4. The Hodge theorem on the existence and uniqueness of harmonic representatives of cohomology classes is a crucial step in the Hodge theory (or decomposition) which will be explained in the next section. See also the discussion in [16, 1.2] about three steps to obtain the Kähler package of a Kähler manifold. 5. When M is a compact locally symmetric space Γ\D, the harmonic representatives of cohomology classes are automorphic forms, and the Hodge theorem establishes a connection between cohomology groups of arithmetic groups and automorphic forms. See the book [12] and references there. Let M be an oriented Riemannian manifold of dimension m, and let : A i (M) : A m i (M) be the 0-th order operator for which we have at each point of M ϕ, ψ dv M = ϕ ψ whenever ϕ, ψ A i (M). (2.8) The formal adjoint of d is the first-order operator δ : A i+1 (M) A i (M) for which the formula dϕ, ψ = ϕ, δψ whenever ϕ A i (M) and ψ A i+1 (M) (2.9) holds locally on M. It is easy to check that δ = 1 d = ± d ; the sign in the right-hand side is 1 for all i when m is even. If one of ϕ, ψ has compact support, Stokes theorem implies that (2.9) holds for the global L 2 inner product on A (M); to distinguish the latter from the pointwise inner product, we introduce the notation ϕ, ψ (2). Unfortunately, the forms in A i (p)(m) (p is temporarily arbitrary again) are merely dense in the Banach space of L p i-forms, so one must pass to the completion to draw conclusions by functional analysis. Once that is done, d is only a densely-defined unbounded operator. One must then consider the various closures of d. At the extremes, we have the maximal, or weak, closure d max, whose domain is the set of L p i-forms ϕ whose distributional exterior derivative dϕ is given by an L p (i + 1)-form, and the minimal closure d min determined by starting instead with C forms of compact support. The extreme closures of δ are defined analogously. It is important to recognize the strict, as opposed to just formal, adjoint of a given closure of, say d, and we present it for d max. It is clear that (2.9) 9

10 holds globally for an L 2 differential form ψ if and only if ϕ dϕ, ψ (2) (2.10) is a bounded functional of ϕ in the domain of d max, or equivalently, on smooth L 2 forms ϕ. One then writes (d max ) (ψ) for the L 2 form representing that functional. It is clear that (d max ) is a closure of δ, and indeed (d max ) = δ min. (2.11) In particular, if one knows that d max = d min, all closures of d coincide, likewise for δ, and also vice versa. Remark 2.3 One can find the following assertions in [95, 1], for instance. (i) One can replace A (p) (M) by the larger complex L (p)(m) given by the domain of d max. The inclusion A (p) (M) L (p)(m) is a quasi-isomorphism. In other words, the two complexes define the same notion of L p -cohomology. 10 We thereby get the reformulation of (2.1) as H i (p) (M) = ker di max/im d i 1 max. (2.12) (ii) When M is the interior of a compact Riemannian manifold-with-corners (i.e., where the metric does not degenerate at the boundary), d max coincides with the closure of d on the complex of C differential forms that are smooth at the boundary. The resulting complex is a quasi-isomorphic subcomplex of A (M). Likewise, d max is the closure of δ on the subcomplex of such forms that vanish on the boundary. It is now necessary to be careful about what we mean by an L 2 harmonic differential form. Harmonic forms are always smooth. At the formal level, they are given by the solutions of the equation ϕ = 0, where is the second-order Laplacian operator dδ + δd that takes A i (M) to itself. Already when i = 0, the solutions of this equation can be infinite-dimensional. This operator has various closed extensions. One can define max and min as before, but the most useful closure is str, the one given by the strict termby-term closure. Writing d from now on instead of d max, str has as domain the space of L 2 forms ϕ in the domain of d and d, such that dϕ is in the domain of d and d ϕ is in the domain of d. Under these conditions, one can expand str ϕ, ϕ (2), making use of the global version of (2.9) (compare (2.16) below), str ϕ, ϕ (2) = d ϕ, d ϕ (2) + dϕ, dϕ (2). (2.13) 10 Thus, both d max and d min can be described in terms of closures of graphs. 10

11 From this we see that str ϕ = 0 if and only if dϕ = 0 and d ϕ = 0. A differential i-form ϕ will be said to be strictly harmonic when these equalities hold, and we write ϕ H i (M). In general ker d i is a closed subspace of the L 2 i-forms, but its subspace im d i 1 need not be. One has then that ( ker d i = im d i 1 ker d i (im ker d i 1 ) ) = ker d i ker(d i ) = H(2) i (M) From (2.12) with p = 2, we obtain H i (2) (M) = Hi (2) (M) ( im d i 1 /im d i 1 ) (2.14) (2.15) (algebraic direct sum). Some elementary functional analysis yields that the second summand is either 0 or of infinite dimension over R. The latter case occurs when M = R. Remark 2.4 (i) For a compact manifold M, (2.15) is basically what Gaffney proved in [33], where d i 1 is shown to have closed range. In [32], he proved that on a complete Riemannian manifold, d max = d min, so (2.9) holds in the L 2 -complex, viz., dϕ, ψ (2) = ϕ, δψ (2), (2.16) and then str = max. (ii) If dim H i (2) (M) <, e.g., when M is compact, then Hi (2) (M) H(2) i (M). By (i), the right-hand side is just the space of all harmonic i- forms. This proves Theorem 2.2. (iii) It is possible that H i (2) (M) be infinite-dimensional because Hi (2) (M) is infinite-dimensional. (iv) For any p, the reduced L p -cohomology can be defined as the following alteration of (2.12): ( ) ( ) H i (p)(m) = ker d i max / im d i 1 max. (2.17) With p = 2, H i (2)(M) is always isomorphic to H(2) i (M), but the cohomological flavor is gone. On M, there is a natural pairing A i (2)(M, R) Am i (2) (M, R) A m (1)(M, R). (2.18) 11

12 When M is compact, one can express the Poincaré duality (a perfect pairing) in terms of closed differential forms, P : H i (M, R) H m i (M, R) H m (M, R) R, P (ϕ, ψ) = ϕ ψ. (2.19) In particular, this can be given at the level of harmonic forms; indeed, as commutes with, one has an isomorphism : H i (M, R) H m i (M, R), P (ϕ, η) = ϕ, η (2). (2.20) The above carries over to complete manifolds M. The pairing on L 2 harmonic forms is still a perfect pairing, so (2.20) induces a Poincaré duality P : H(2) i (M, R) Hm i (2) (M, R) R. (2.21) With a little care, one defines and makes use of complexes of L p sheaves (as promised). One notes first that the assignment, for U open in M, U A i (p) (U) gives rise only to the sheaf of i-forms that are locally Lp, i.e., A M itself. To remember the global Lp condition, one must first specify a compactification M of M as a Hausdorff topological space. The L p sheaf of i-forms on M is the assignment, for U open in M, U A i (p)(u M). For all M, the resulting complex of sheaves A (p) (M) has A (p)(m) as its complex of global sections. For this fact to be useful for understanding the L p - cohomology of M, one needs that the metric is fine on M, so that A (p) (M) is a complex of fine sheaves (see [93, 4]); there must exist a partition of unity on M whose cut-off functions at the boundary of M have bounded differential with respect to g, for multiplication only by such functions preserves the two L p conditions in the definition of A i (p)(u M). Whether such functions exist is something that depends on the choice of M. Remark 2.5 The analogue of Remark 2.3(i) holds at the level of sheaves. There is a weak L p complex of sheaves L (p)(m) for any compactification M of M, containing A (p)(m) as a quasi-isomorphic subcomplex, for which (M) is its complex of global sections. L (p) There are useful variants of L p -cohomology. Let M be a Riemannian manifold and w a positive, say smooth, function on M. The L p seminorm with weight w is ( ) 1/p ϕ (p),w = ϕ p w dv M. (2.22) M 12 M

13 When w = 1, this is just (2.6). Replacing (2.6) by (2.22) in Definition 2.1, one defines the weighted L p -complex A (p)(m; w) and then the weighted L p -cohomology of M: H(p) i (M; w) = Hi (A (p)(m; w)). (2.23) The notion of weighted L p sheaves on a compactification M of M, denoted A (p)(m; w), is correspondingly defined, and likewise the analogue of Remark 2.5. A second variant lies in differential forms with coefficients in a flat (complex) vector bundle. Let M denote the universal covering space of M. Then M = Γ\ M, where Γ is the fundamental group of M. Assume that Γ acts freely on a finite-dimensional vector space V/C, i.e., V is a representation space for Γ. The projection of M V onto M induces, via the quotient by Γ, with Γ taken to be acting diagonally on the former, a vector bundle V over M. Any fixed v V is taken to define a locally constant section of V; let V denote the sheaf of locally constant sections of V. The complex of sheaves A (M, R) V, with differential given locally as d 1, whose complex of global sections is denoted A (M, V), is quasi-isomorphic to V. If one puts a metric on V as well as M, one can define A (p) (M, V) and Lp -cohomology groups H (p) (M, V) parallel to Definition 2.1, and sheaves A (p)(m, C) V.11 The question that we are pursuing in this chapter is: Problem 2.6 If M is a given non-compact Riemannian manifold, is there a compactification M of M for which the metric is fine on M, and H (p) (M) (is finite-dimensional and) admits a topological interpretation on M? When p = 2, the topological interpretation will have to satisfy Poincaré duality, because of (2.21). In the cases where Problem 2.6 has been solved, information drawn from the explicit asymptotics of the metric, as read on M, are used. Indeed, Theorems 1.1 and 1.2 are proved by examining A (p) (MDM g,n ) for the respective metrics. Remark 2.7 One can generalize the above definition of L p -cohomology to orbifolds. Differential forms, partitions of unity, L p -sheaves, etc. can be defined on orbifolds. (See [78] and [23]). In this paper, we concentrate on the case of smooth manifolds, with the sense that one can deduce the general case from that. 11 Actually, weighted L p -cohomology can be viewed as a case of the preceding where the bundle V is trivial of rank one, metrized by w. 13

14 Remark 2.8 We list here some interesting topics of a similar-sounding nature that do not pass through Problem 2.6, and we have elected to exclude. (i) The investigation of reduced L 2 -cohomology, which is by Remark 2.4(iv) focused on the L 2 harmonic forms: [21, 22, 39]. (ii) The L 2 -cohomology of a covering group (see [3, 20]). (iii) The image of H (p) (M) H (M), i.e., the (de Rham) cohomology classes represented by an L p differential form (with p = 2, [2, 11, 45]). (iv) Bounded cohomology in the sense of Gromov [40]. (v) The cohomology of infinite chains with l p -coefficients (see [20]). 3 Hodge decomposition We retain the notation from the previous section. Suppose that our Riemannian manifold M is a complex manifold of (complex) dimension n (so m = 2n). We assume that the metric g is hermitian, i.e., that the almostcomplex structure tensor J of M is an isometry of g. The C-valued differential forms decompose according to bidegree (or type) A j (M, C) = A p,q (M) with p, q 0 and A p,q (M) = A q,p (M), p+q=j (3.1) with the bar denoting complex conjugation. An analogous decomposition holds for the sheaves A j (M, C). At issue is whether the decomposition (3.1) passes to H j (M, C). One extends the Laplacian C-linearly to A j (M, C). For a j-form decomposed according to (3.1): ϕ = ϕ p,q, (3.2) ϕ = ϕ p,q, of course. However, it is not true in general that ϕ p,q is a form of type (p, q). When the Hermitian metric is Kählerian one then says that M is a compact Kähler manifold meaning that the alternating bilinear form ω, given pointwise by ω(x, Y ) = g(jx, Y ), (3.3) is a closed differential 2-form (of type (1, 1)), 12 the following is true: 12 It is easy to see that ω n /n! is the volume form of M, and that ω = ω n 1 /n!. 14

15 Theorem 3.1 (Hodge decomposition). Let M be a compact Kähler manifold. Then: (i) (A p,q (M)) A p,q (M). Thus ϕ is harmonic if and only if each ϕ p,q (from (3.1)) is harmonic. (ii) H j (M, C) = p+q=j Hp,q (M), where H p,q (M) denotes the space of cohomology classes represented under (3.2) by harmonic forms of type (p, q). Moreover, H p,q (M) = H q,p (M) (iii) The decomposition in (ii) is independent of the Kähler metric. Remark 3.2 We give some indication of the proof of Theorem 3.1. (i) The first assertion of Theorem 3.1 is usually proved by showing that when M is a Kähler manifold, can be expressed as an operator that, for obvious reasons, preserves each A p,q (M). Using the decomposition d = +, with the type (1, 0) component of d that takes A p,q (M) to A p+1,q (M), and its complex conjugate, one has the analogues of (expressed at the formal level): = +, = +. (3.4) Both of the above operators preserve A p,q (M). One says that a form ϕ of type (p, q) is -harmonic (resp. -harmonic) when ϕ = 0 (resp. = 0). Then one computes that = + + [cross-terms]. (3.5) When the metric is Kählerian, the cross-terms vanish and =, so = 2. Furthermore, (3.1) induces on the space of C-valued harmonic j-forms H j (M, C), H j (M, C) = H p,q (M). (3.6) p+q=j Inserting this into Theorem 2.2, yields (ii). The abstract version of the such a decomposition is called a (real) Hodge structure of weight j. Thus, one has constructed a real Hodge structure of weight j on H j (M, R). (ii) The study of is fundamental in complex function theory in more than one variable, and it plays a role as asserted in item (i) above. Indeed, there are analogues of (2.4) and (2.5) for and O M (V), the sheaf of holomorphic sections of a holomorphic vector bundle V on M (the Dolbeault lemma), here with O M (V) = Ω p M, the sheaf of holomorphic p-forms on M. By the Dolbeault lemma (see [38]), the cohomology of the -complex, with 15

16 differentials : A p,q (M) A p,q+1 (M), can be identified with sheaf cohomology: H q (M, Ω p M ) Hp,q (M) H p,q (M). (3.7) (iii) Assertion (iii) of Theorem 3.1 may come as a surprise. It is proved by showing that H p,q (M) can be described by a sufficiently algebraic statement that is independent of the Kähler metric. Specifically, if one writes F p H j (M) for the sum of the cohomology classes of closed forms of types (r, s), with r p, then H p,q (M) = F p H j (M) F q H j (M); (3.8) This filtration F is the restriction of one on A (M, C), and that is induced from the corresponding filtration (also denoted F ) of Ω M ; the latter complex and its - (Dolbeault) resolution are quasi-isomorphic to C M. There are topological conditions imposed by the existence of a Kähler metric on M. For one, H 2 (M, R) 0. Also, we see directly from Theorem 3.1(ii) that when j is odd, dim H j (M) must be even. One can go further. Let L denote the 0-th order operator (of type (1, 1)) given by wedge product with [ω], viz., L : A j (M, R) A j+2 (M, R). Then L actually takes H j (M) to H j+2 (M) (as and L commute) as well as the canonical cup product with [ω] from H j (M) to H j (M). One has L n j : H j (M, R) H 2n j (M, R) (for j < n), (3.9) which is an isomorphism. 13 There is additional structure on the cohomology of M, but we will not need that for our purposes. (See [84] for that, as well as for (3.9).) Examples. We present a good supply of Kähler manifolds. There are three standard ones: C (with the Euclidean metric), CP n (with the P GL n+1 - invariant Fubini-Study metric), and a hermitian symmetric space D of noncompact type (with a metric invariant under the isometry group of D). Two basic constructions provide many more. Any complex submanifold N of a Kähler manifold M is Kählerian (with the metric of M restricted to N). In particular, every smooth algebraic subvariety of CP n is Kähler; in [55] Kodaira characterized smooth projective varieties as Kähler manifolds for which some Kähler metric satisfies [ω] H 2 (M, Q). The other construction is taking a quotient of M by a discrete group acting on M without fixed 13 There is some relation between (3.9) and that makes use of the primitive decomposition. 16

17 points. When Γ is an arithmetically-defined subgroup of the isometries of D, the quotient Γ\D is a locally symmetric variety. If M is a complete (non-compact) Kähler manifold with finite-dimensional L 2 -cohomology, one still has the decomposition of L 2 harmonic forms ((3.6) in the compact case) H j (2)(M, C) = p+q=j H p,q (2)(M), (3.10) as the notions of formal and strict harmonic forms coincide by Remark 2.4(i). This induces Hodge decompositions, via (2.15): H j (2)(M, C) = p+q=j H p,q (2)(M). (3.11) Problem 2.6 is asking for a topologically-defined cohomology group on some compactification M, to which (3.11) imparts Hodge structures of weight j in degree j. 4 Intersection (co)homology We will give a treatment of intersection cohomology with middle perversity, as it occurs for complex projective algebraic varieties. Leading to [35], Goresky and MacPherson were thinking: for a compact, orientable, stratified space X of dimension m (with singularities), one knows that Poincaré duality, i.e., the perfect pairing in the manifold case (even valid with Z-coefficients): H m i (X, Z) H i (X, Z) Z (4.1) (cf. (4.1), with coefficients in R, (2.21)) generally fails to hold. Is there a construction of something else for which Poincaré duality always holds, and which recovers the usual duality (4.1) when X is a manifold? They made a construction involving simplicial chains (as used by Lefschetz) with a triangulation of X, together with a parameter, the perversity, which specifies how much one relaxed the condition of general position for chains and their boundaries (compare Definition 2.1). For instance, if one insists on general position with respect to the strata of X (the 0 perversity), one gets the cohomology groups H m i (X) from i-cycles, and with the vacuous condition 17

18 one gets the homology groups H m i (X) from (m i)-cycles. The intersection pairing (at the level of simplicial chains) induces the usual dual pairing H m i (X, Q) H m i (X, Q) Q, (4.2) defined even with Z-coefficients. In general, there is a notion of a dual pair of perversities (see [35]). There is the analogue of (4.2) giving the duality between the corresponding notions of intersection cohomology groups. It is a fact of life that intersection cohomology, beyond usual homology and cohomology, is not a cohomology theory in the sense of algebraic topology, as it is not a homotopy invariant. For the purpose of this exposition, we assume that X is a compact algebraic subvariety of some CP l, with complex dimension n. One knows (see [37]) that X admits a stratification by subvarieties. By this, we mean that one has a descending chain of projective subvarieties X X 0 X 1... X n, with codim X (X j ) j; then X j = X j X j+1 is the stratum of complex codimension j, a space without singularities, for which the Whitney conditions (see [37]) hold. This ensures that transverse slices along each stratum are locally constant, specifically are homeomorphic X j. Note that Λ j is necessarily of odd to the (real) cone on the link Λ j of real dimension, namely 2j 1, and comes with an induced stratification with all strata of even real codimension. The middle perversity, m, for a space with all strata of even codimension, is the specification that an i-chain is permitted to intersect X j in a chain of dimension at most i j. In particular, the intersection must be empty whenever j > i. The homology groups that one gets are IH (X) m ; we will drop the symbol m from the notation. Then intersection of chains defines a perfect pairing with coefficients in Q [35]: IH 2n i (X, Q) IH i (X, Q) Q. (4.3) By using the dual numbering, we can rewrite (4.3) as cohomology: IH i (X, Q) IH 2n i (X, Q) Q. (4.4) There is an analogue of (2.5) that characterizes intersection cohomology. Let U be a contractible open set about a point of X j of the form W Cone(Λ j ), with W a contractible open set in X j. { IH i IH i (Λ j ) if i < j, (U) = (4.5) 0 otherwise. 18

19 There is one instance in which intersection cohomology is expressible as ordinary cohomology, namely when X has only isolated singularities. In this case, X 1 = X n and the global consequence of (4.5) is H i ( X 0 ) if i < n, IH i (X) H i (X) if i > n, im {H n (X) H n ( X 0 )} if i = n. (4.6) (The Hodge structure in this case is known without invoking L 2 -cohomology by [18].) There are other, softer characterizations of IH (X) m given in [36]. Another class of simple cases comes directly from application of these: Proposition 4.1 For a complex orbifold, its intersection cohomology is canonically isomorphic to its ordinary cohomology group. The idea that self-dual L 2 -cohomology would be the intersection cohomology of some compactification arose inevitably. As was mentioned in Section 1, the Zucker conjecture from [94] was confirmed by [59, 77], by completely different methods: 14 Theorem 4.2 Let Γ\D be a locally symmetric variety, with a metric induced by an invariant metric on D. Then, 15 H (2) (Γ\D, V) IH (Γ\D BB, V), (4.7) whenever V is induced by a trivial flat bundle V = D V on D, with Aut(D) hence Γ acting diagonally on the two factors, and is metrized by an admissible inner product. Here, the Kähler metric is complete, and we are using the variant above Problem 2.6, which makes sense for intersection cohomology as well. 16 The proofs of Theorem 4.2 assume that Γ is a neat arithmetic subgroup. Every arithmetic group contains a normal neat subgroup of finite index. It seems to 14 For an exposition of both proofs, see [98]. 15 for local reasons on Γ\D BB. More precisely, there is a complex of sheaves of intersection cochains, and it is shown to be quasi-isomorphic to the complex of L 2 -sheaves on Γ\D BB, by use of the characterization in [36]. 16 A precursor of Theorem 4.2 was given in [92], though its relation with intersection cohomology was not sensed until later (see [16]). The main theorem of [92] was generalized independently, and as intersection cohomology, in [14] and [51]. 19

20 be expected that the case of general Γ would follow routinely, but there is no account in the literature. It would need a proof of the following assertion, one that makes use of [36], but that would be too technical for this exposition: Proposition 4.3 Let Y be a complex space with a Riemannian metric on the nonsingular part Y 0. Assume that a finite group G acts holomorphically on Y and quasi-isometrically on Y 0. Assume that the intersection complex IC (Y ) and the L p -complex L (p)(y ) are quasi-isomorphic. Then for Z = G\Y then IC (Z) is canonically quasi-isomorphic to L (p)(z). In particular, IH (Z) is isomorphic to H (p) (Z). By Equation (3.11) and Theorem 4.2, a Hodge structure is constructed for IH (Γ\D BB, R). There is another construction, coming via the D- module machinery of polarizable Hodge modules [73], which is understood to be the right Hodge structure. More than twenty years have passed, yet it is only in the simplest cases that the two Hodge structures have been proved to coincide; the two constructions are very different, hard to compare. A longstanding conjecture was made by Cheeger, Goresky and MacPherson, and it appeared in [16]. It is a reversal of Problem 2.6. Let X be a singular projective algebraic variety, and let M be its smooth locus. The projective embedding defines a metric on M by restriction of the Fubini-Study metric. The metric is, up to quasi-isometry, independent of the embedding. Evidently the metric on M is incomplete. The conjecture states: Conjecture 4.4 [16] Let X be as above. Then H (2) (M) IH (X), and moreover, this imparts a Hodge structure to IH (X). Because the metric is incomplete, the last clause is an additional condition. Recall that we were considering complete metrics in Remark 2.4(i), for which (2.15) is talking about the space of all L 2 harmonic forms. The isomorphism of Conjecture 4.4 in the case of varieties with only isolated singularities is proved in [66], deriving it in the limit from results in [75] for certain complete metrics. L 2 -cohomology, a natural consideration given Remark 3.2(ii), for the incomplete metric is studied in [69, 70], and the Hodge decomposition of the L 2 -cohomology is proved. See [46] for the first proof of the first part of the above conjecture when n = 2. 20

21 5 Metrics on M g,n and the proofs of Theorems 1.1 and 1.2 We recall some well-known metrics on M g,n and give the determination of their L p -cohomology. As announced in Section 1, it is convenient to work with the spaces M Γ g,n = Γ\T g,n, for Γ a finite-index normal subgroup of Mod g,n that acts freely on T g,n. We understand to have done so, though the case of Mod g,n and general Γ can be deduced from that by Proposition 4.3. a) The Weil-Petersson metric. There is a very natural Mod g,n -invariant incomplete metric on T g,n, known as the Weil-Petersson metric. Its definition is given in terms of the quadratic differentials on a reference curve, i.e., one corresponding to a choice of a point of T g,n ; quadratic differentials can be identified with tangent vectors to T g,n at that point (see [10] and [87]). We address first the proof of Theorem 1.2. Let S k be a boundary stratum of M DM g,n of codimension k. By [57], each point x S k has a fundamental system of neighborhoods U of the form U k k, with U k {0} giving a contractible neighborhood of x in S k that is a bounded domain. Then we have U := U M g,n U k ( ) k. (5.1) Determining the sheaves A (p) (MDM g ) is done by using the asymptotic formula for the metric in the coordinates (5.1). Indeed, the key point of the proof is to use results of Masur [61] on the Weil-Petersson metric of M g near the boundary divisors of M g DM to obtain a quasi-isometric model of the Weil-Petersson metric. For the case of M g,n, the Weil-Petersson metric has been described precisely in [85] and the corresponding asymptotic behavior can be obtained. It can be found in [76, p.352]: ω wp (U) = ω(u k ) + j ω hj ( j). (5.2) Here, each ω hj is the metric of a horn on a circle, viz., of the form ω hj = dr 2 + r 2c j dθ 2 for some c j > 1; indeed, all c j = 3. Thus, the Weil-Petersson metric is fine on M DM g,n. Saper invokes the L 2 Künneth formula, applying it to (5.2) to yield H (2) (U ) H (2) (U k) j H (2) ( j : h j ). (5.3) 21

22 With Cheeger s calculation [15] of the L 2 -cohomology of a horn, one gets { H(2) i R if i = 0, ( ; h j ) = (5.4) 0 otherwise; informally stated, the L 2 -condition serves to fill in the origin of. same holds for U k, so he obtains that likewise { H(2) i (U R if i = 0, ) = 0 otherwise; The (5.5) from which it follows that the sheaf of L 2 -differential forms is quasi-isomorphic to the constant sheaf, and hence H i (2) (M g,n; ω wp ) H i (M DM g,n ). (5.6) One can carry out an analogous determination of H (p) (M g,n; ω wp ). But first we place Theorem 1.2 in a broader context. By definition, the c-horn H c (N) on a Riemannian manifold N is the product (0, 1] N with metric ds 2 = dr 2 + r 2c ds 2 N for some c 1. It is incomplete. Cheeger s general determination [15, Lemma 3.4] of the L 2 -cohomology of a c-horn H c (N) on a Riemannian manifold N (in terms of the L 2 -cohomology of N) has been generalized to L p - cohomology by Youssin [91]. For our exposition, we extract a special case that is somewhat more general than what we need: Proposition 5.1 Let H c (N) be the c-horn on a compact Riemannian manifold N, whose dimension n is odd. Then { H j (p) (H H j (N) if p < (n + c 1 )/j, c(n)) 0 otherwise. Corollary 5.2 The L p -cohomology of H c (S 1 ) is given by 1. H 0 (p) (H c(s 1 )) R for all p; 2. H 1 (p) (H c(n)) { R if p < c+1 c, 0 otherwise. 3. H 2 (p) (H c(s 1 )) = 0 for all p. 22

23 Because S 1 is an abelian Lie group, acting isometrically on the L p de Rham complex of H c (S 1 ), one can prove Corollary 5.2 directly, avoiding most of the detailed analysis needed for the proof of Proposition 5.1. We give here the argument. Lemma 5.3 Let H c (S 1 ) (0, 1] S 1 denote the c-horn on a circle. Then dr is an L p 1-form for all p, and dθ is an L p 1-form if and only if p < c+1 c. Proof. We note that the pointwise norms dr p = 1, dθ p = r pc and dv = r c dr dθ. Then 2π 1 dθ p (p) = r c pc dr dθ. 0 It follows that dθ is an L p 1-form on the horn if and only if c pc > 1, i.e., p < c+1 c. QED The preceding lemma suggests: Lemma 5.4 Let I = {r : 0 < r 1} be the half-open unit interval. Then the weighted L p -cohomology of I (1 p < ) with weights of the form r α (α R) is given by { 1. H(p) 0 (I; R if α > 1, rα ) = 0 if α 1, 2. H 1 (p) (I; rα ) = 0 for all p. Proof. The statement about H 0 is easy, for it is only a question of whether 1 L p (I; r α ) := { f f(r) p r α dr < }. As for H 1, there is an evident operator B given by B(f(r)dr) = F (r): { r F (r) = 1 f(t)dt if α > 1, r 0 f(t)dt if α 1, (5.7) such that df = f(r)dr in the weak sense. (Indeed, in the first case, B takes smooth 1-forms to smooth functions, i.e., F (r) = f(r) holds in the sense of smooth functions.) By Remark 2.3, it suffices to verify that B is a bounded 23

24 operator on L (p) (I; rα ), which we show only in the second case, the first being similar. Using Fubini s theorem, we estimate 1 B(fdr) p r 1 r (p) = f(t)dt rα dr f(t) p dt r α dr = t r α dr f(t) p dt ψ(t)t α f(t) p dt f p (p) = fdr p (p), where ψ(t) = { t when α < 1, t log t when α = 1. QED We can decompose A (p) (H c(s 1 ) into the direct sum of the S 1 -invariant forms and the forms with integral zero over the S 1 -orbits; both are subcomplexes. The former is isomorphic to A (p) (I; rc ) A (p) (I; rc pc )dθ, (5.8) Applying Lemma 5.4 already produces the right-hand side of (2) in Corollary 5.2. It remains to show that the complementary factor is acyclic (i.e., has trivial cohomology). The main issue lies with H 1. Let ϕ be a smooth 1-form and write ϕ = f(r, θ)dr + g(r, θ)dθ, with f L p (I S 1 ; r c ), g L p (I S 1 ; r c pc ), and the partial derivatives g r and f θ are equal. Let G(r, θ) denote the θ-antiderivative of g(r, θ) given by G(r, θ) = θ θ 0 g(r, η)dη, (5.9) for any θ 0 S 1. This is well-defined whenever g has mean zero. Then G is a smooth function in L p (I S 1 ; r c cp ), by a calculation simpler than the one in (5.8) (5.9); a fortiori it is in L p (I S 1 ; r c ). As g r = f θ, G r (r, θ) = f(r, θ) f(r, θ 0 ). The preceding yields that ϕ = dg + f(r, θ 0 )dr. If we choose θ 0 so that f(r, θ 0 ) L p (I S 1 ; r c ), i.e., outside a set of measure zero in S 1, we have by Lemma 5.4 that f(r, θ 0 )dr, hence also ϕ, is exact in the sense of L p - cohomology. This concludes our proof of Corollary 5.2. Remark 5.5 The way that Proposition 5.1 is proved, in effect, in [15] and [91] is by proving that the weak and minimal closures of d on A (H c (N)) coincide. Once one knows that, one can also adapt the criterion from [93, (2.29)] (see also (2.41) of op.cit.), of which what we did above is a variant. 24

25 We note that when c = 3, the dividing point for the two cases in Corollary 5.2 occurs at p = 4 3. We conclude, finishing the proof of Theorem 1.2: Proposition 5.6 a) For all p, the analogue of (5.3) holds: H (p) (U ) H (p) (U k) j H (p) ( j ). b) When 4 3 p <, the analogue of the identification in (5.5) holds for H(p) i (U ), and H (p) (M g; ω wp ) H (M DM g ). c) When 1 p < 4 3, { (1) H(2) i R if i = 0 or i = 1 ( ; h j ) = 0 otherwise, (so (5.5) must be correspondingly altered); 17 (2) H (p) (M g,n; ω wp ) H (M g,n ). Proof. Statement a) is an application of the L p -Künneth theorem, as the factors have finite-dimensional L p -cohomology. This yields the local and global assertions in b) and c). b) Complete metrics. Because the Weil-Petersson metric is incomplete, its geometry becomes more complicated. The statement of the Hodge theorem for ω wp, which identifies the L 2 -cohomology with a space of L 2 harmonic forms, can impose (idealized) boundary conditions for str (compare the issue in Conjecture 4.4). For a complete metric having finite-dimensional L 2 -cohomology, the latter is isomorphic to the space of all L 2 harmonic forms. There are several constructions and/or use of complete metrics in the literature. Foremost is the Teichmüller metric, which is defined as follows. For any two points in T g,n (representing two marked Riemann surfaces of genus g with n punctures, S and S, there is a unique quasi-conformal mapping S S that has least distortion K 1 and is compatible with the markings. Then the Teichmüller distance d(s, S ) is defined to be log K. Other well-known examples of complete metrics include: the Bergman metric, Kähler-Einstein metric, McMullen metric [64], Ricci metric (the negative of the Ricci curvature of the Weil-Petersson), the perturbed Ricci metric of Liu, Sun and Yau [58], and any Poincaré metric adapted to the boundary of M DM g. Though there are infinitely many quasi-isometry classes of complete 17 Informally, the L p -condition is mild enough that the puncture still counts. 25