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Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung Gegr¨ undet 1996 {1} {4} {3,4} {2} {3} {1,2} {1} {3} {3,4} {4} {2} {1,2} {1,2,3,4} {2} {3,4} {1,2} {1} {3} {4} Band 13 · 2008

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  • Documenta Mathematica

    Journal der

    Deutschen Mathematiker-Vereinigung

    Gegründet 1996

    {1}

    {4}

    {3,4}

    {2}

    {3}

    {1,2}

    {1} {3}

    {3,4}

    {4}{2}

    {1,2}

    {1,2,3,4}

    {2}

    {3,4}{1,2}

    {1} {3} {4}

    Band 13 · 2008

  • Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung,veröffentlicht Forschungsarbeiten aus allen mathematischen Gebieten und wird intraditioneller Weise referiert. Es wird indiziert durch Mathematical Reviews, ScienceCitation Index Expanded, Zentralblatt für Mathematik.

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    Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung,publishes research manuscripts out of all mathematical fields and is refereed in thetraditional manner. It is indexed in Mathematical Reviews, Science Citation IndexExpanded, Zentralblatt für Mathematik.

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  • Documenta MathematicaBand 13, 2008

    V. Gritsenko, K. Hulek and G. K. SankaranHirzebruch-Mumford Proportionality andLocally Symmetric Varieties of OrthogonalType 1–19

    Tom LeinsterThe Euler Characteristic of a Category 21–49

    Sergey MorozovEssential Spectrum of MultiparticleBrown–Ravenhall Operators in External Field 51–79

    A. Vishik, K. ZainoullineMotivic Splitting Lemma 81–96

    Jörg WinkelmannOn Tameness and Growth Conditions 97–101

    V. Bach, J. Hoppe, D. LundholmDynamical Symmetries inSupersymmetric Matrix 103–116

    Jürgen Ritter and Alfred WeissEquivariant Iwasawa Theory: An Example 117–129

    Guido KingsDegeneration of Polylogarithms and Special Valuesof L-Functions for Totally Real Fields 131–159

    Alex Kumjian, David Pask, Aidan SimsC∗-Algebras Associated to Coverings of k-Graphs 161–205

    Alex Postnikov, Victor Reiner, Lauren WilliamsFaces of Generalized Permutohedra 207–273

    Denis PerrotSecondary Invariants for Frechet Algebrasand Quasihomomorphisms 275–363

    Paul S. Muhly and Baruch SolelSchur Class Operator Functionsand Automorphisms of Hardy Algebras 365–411

    Viorel CosteanuOn the 2-Typical De Rham-Witt Complex 413–452

    Ciro Ciliberto, Gerard van der GeerAndreotti–Mayer Loci and the Schottky Problem 453–504

    iii

  • Wies lawa Nizio lK-Theory of Log-Schemes I 505–551

    Eric A. Carlen and Elliott H. LiebBrascamp-Lieb Inequalitiesfor Non-Commutative Integration 553–584

    Gereon QuickProfinite Homotopy Theory 585–612

    F. DégliseAround the Gysin Triangle II. 613–675

    Matthias KünzerComparison of Spectral SequencesInvolving Bifunctors 677–737

    Goro ShimuraArithmetic of Hermitian Forms 739–774

    Goro ShimuraThe Critical Values of Certain Dirichlet Series 775–794

    Christian Ausoni, Bjørn Ian Dundas and John RognesDivisibility of the Dirac Magnetic Monopoleas a Two-Vector Bundle over the Three-Sphere 795–801

    Timothy LogvinenkoNatural G-Constellation Families 803–823

    Eva ViehmannThe Global Structure of Moduli Spacesof Polarized p-Divisible Groups 825–852

    iv

  • Documenta Math. 1

    Hirzebruch-Mumford Proportionality and

    Locally Symmetric Varieties of Orthogonal

    Type

    V. Gritsenko, K. Hulek and G. K. Sankaran

    Received: September 28, 2006

    Revised: February 17, 2008

    Communicated by Thomas Peternell

    Abstract. For many classical moduli spaces of orthogonal type thereare results about the Kodaira dimension. But nothing is known in thecase of dimension greater than 19. In this paper we obtain the firstresults in this direction. In particular the modular variety definedby the orthogonal group of the even unimodular lattice of signature(2, 8m+ 2) is of general type if m ≥ 5.

    2000 Mathematics Subject Classification: 14J15, 11F55Keywords and Phrases: Locally symmetric variety; modular form;Hirzebruch-Mumford proportionality

    1 Modular varieties of orthogonal type

    Let L be an integral indefinite lattice of signature (2, n) and ( , ) the associatedbilinear form. By DL we denote a connected component of the homogeneoustype IV complex domain of dimension n

    DL = {[w] ∈ P(L⊗ C) | (w,w) = 0, (w,w) > 0}+.

    O+(L) is the index 2 subgroup of the integral orthogonal group O(L) thatleaves DL invariant. Any subgroup Γ of O+(L) of finite index determines amodular variety

    FL(Γ) = Γ \ DL.By [BB] this is a quasi-projective variety.

    Documenta Mathematica 13 (2008) 1–19

  • 2 V. Gritsenko, K. Hulek and G. K. Sankaran

    For some special lattices L and subgroups Γ < O+(L) one obtains in thisway the moduli spaces of polarised abelian or Kummer surfaces (n = 3, see[GH]), the moduli space of Enriques surfaces (n = 10, see [BHPV]), and themoduli spaces of polarised or lattice-polarised K3 surfaces (0 < n ≤ 19, see[Nik1, Dol]). Other interesting modular varieties of orthogonal type includethe period domains of irreducible symplectic manifolds: see [GHS3].It is natural to ask about the birational type of FL(Γ). For many classicalmoduli spaces of orthogonal type there are results about the Kodaira dimension,but nothing is known in the case of dimension greater than 19. In this paper weobtain the first results in this direction. We determine the Kodaira dimensionof many quasi-projective varieties associated with two series of even lattices.To explain what these varieties are, we first introduce the stable orthogonalgroup Õ(L) of a nondegenerate even lattice L. This is defined (see [Nik2] formore details) to be the subgroup of O(L) which acts trivially on the discrim-inant group AL = L

    ∨/L, where L∨ is the dual lattice. If Γ < O(L) then wewrite Γ̃ = Γ ∩ Õ(L). Note that if L is unimodular then Õ(L) = O(L).The first series of varieties we want to study, which we call the modular varietiesof unimodular type, is

    F (m)II = O+(II2,8m+2)\DII2,8m+2 . (1)

    F (m)II is of dimension 8m + 2 and arises from the even unimodular lattice ofsignature (2, 8m+ 2)

    II2,8m+2 = 2U ⊕mE8(−1),

    where U denotes the hyperbolic plane and E8(−1) is the negative definitelattice associated to the root system E8. The variety F (2)II is the moduli spaceof elliptically fibred K3 surfaces with a section (see e.g. [CM, Section 2]). Thecase m = 3 is of particular interest: it arises in the context of the fake MonsterLie algebra [B1].The second series, which we call the modular varieties of K3 type, is

    F (m)2d = Õ+

    (L(m)2d )\DL(m)

    2d

    . (2)

    F (m)2d is of dimension 8m+ 3 and arises from the lattice

    L(m)2d = 2U ⊕mE8(−1)⊕ 〈−2d〉,

    where 〈−2d〉 denotes a lattice generated by a vector of square −2d.The first three members of the series F (m)2d have interpretations as modulispaces. F (2)2d is the moduli space of polarised K3 surfaces of degree 2d. Form = 1 the 11-dimensional variety F (1)2d is the moduli space of lattice-polarisedK3 surfaces, where the polarisation is defined by the hyperbolic lattice 〈2d〉 ⊕E8(−1) (see [Nik1, Dol]). For m = 0 and d prime the 3-fold F (0)2d is the modulispace of polarised Kummer surfaces (see [GH]).

    Documenta Mathematica 13 (2008) 1–19

  • Hirzebruch-Mumford Proportionality and . . . 3

    Theorem 1.1 The modular varieties of unimodular and K3 type are varietiesof general type if m and d are sufficiently large. More precisely:

    (i) If m ≥ 5 then the modular varieties F (m)II and F(m)2d (for any d ≥ 1) are

    of general type.

    (ii) For m = 4 the varieties F (4)2d are of general type if d ≥ 3 and d 6= 4.

    (iii) For m = 3 the varieties F (3)2d are of general type if d ≥ 1346.

    (iv) For m = 1 the varieties F (1)2d are of general type if d ≥ 1537488.Remark. The methods of this paper are also applicable if m = 2. Using them,

    one can show that the moduli space F (2)2d of polarised K3 surfaces of degree 2dis of general type if d ≥ 231000. This case was studied in [GHS2], where, usinga different method involving special pull-backs of the Borcherds automorphic

    form Φ12 on the domain DII2,26 , we proved that F (2)2d is of general type if d > 61or d = 46, 50, 54, 57, 58, 60.The methods of [GHS2] do not appear to be applicable in the other casesstudied here. Instead, the proof of Theorem 1.1 depends on the existence of agood toroidal compactification of FL(Γ), which was proved in [GHS2], and onthe exact formula for the Hirzebruch-Mumford volume of the orthogonal groupfound in [GHS1].We shall construct pluricanonical forms on a suitable compactification of themodular variety FL(Γ) by means of modular forms. Let Γ < O+(L) be asubgroup of finite index, which naturally acts on the affine cone D•L over DL.In what follows we assume that dimDL ≥ 3.

    Definition 1.2 A modular form of weight k and character χ : Γ → C∗ withrespect to the group Γ is a holomorphic function

    F : D•L → C

    which has the two properties

    F (tz) = t−kF (z) ∀ t ∈ C∗,F (g(z)) = χ(g)F (z) ∀ g ∈ Γ.

    The space of modular forms is denoted by Mk(Γ, χ). The space of cusp forms,i.e. modular forms vanishing on the boundary of the Baily–Borel compactifi-cation of Γ\DL, is denoted by Sk(Γ, χ). We can reformulate the definition ofmodular forms in geometric terms. Let F ∈ Mkn(Γ, detk) be a modular form,where n is the dimension of DL. Then

    F (dZ)k ∈ H0(FL(Γ)◦,Ω⊗k),

    where dZ is a holomorphic volume form on DL, Ω is the sheaf of germs ofcanonical n-forms on FL(Γ) and FL(Γ)◦ is the open smooth part of FL(Γ)such that the projection π : DL → Γ\DL is unramified over FL(Γ)◦.

    Documenta Mathematica 13 (2008) 1–19

  • 4 V. Gritsenko, K. Hulek and G. K. Sankaran

    The main question in the proof of Theorem 1.1 is how to extend the formF (dZ)k to FL(Γ) and to a suitable toroidal compactification FL(Γ)tor. Thereare three possible kinds of obstruction to this, which we call (as in [GHS2])elliptic, reflective and cusp obstructions. Elliptic obstructions arise if FL(Γ)torhas non-canonical singularities arising from fixed loci of the action of the groupΓ. Reflective obstructions arise because the projection π is branched alongdivisors whose general point is smooth in FL(Γ). Cusp obstructions arise whenwe extend the form from FL(Γ) to FL(Γ)tor.The problem of elliptic obstructions was solved for n ≥ 9 in [GHS2].

    Theorem 1.3 ([GHS2, Theorem 2.1]) Let L be a lattice of signature (2, n)with n ≥ 9, and let Γ < O+(L) be a subgroup of finite index. Then there existsa toroidal compactification FL(Γ)tor of FL(Γ) = Γ\DL such that FL(Γ)tor hascanonical singularities and there are no branch divisors in the boundary. Thebranch divisors in FL(Γ) arise from the fixed divisors of reflections.

    Reflective obstructions, that is branch divisors, are a special problem relatedto the orthogonal group. They do not appear in the case of moduli spacesof polarised abelian varieties of dimension greater than 2, where the modulargroup is the symplectic group. There are no quasi-reflections in the symplecticgroup even for g = 3.The branch divisor is defined by special reflective vectors in the lattice L. Thisdescription is given in §2. To estimate the reflective obstructions we use theHirzebruch-Mumford proportionality principle and the exact formula for theHirzebruch-Mumford volume of the orthogonal group found in [GHS1]. We dothe numerical estimation in §4.We treat the cusp obstructions in §3, using special cusp forms of low weight(the lifting of Jacobi forms) constructed in [G2] and the low-weight cusp formtrick (see [G2] and [GHS2]).

    2 The branch divisors

    To estimate the obstruction to extending pluricanonical forms to a smooth

    projective model of FL(Õ+

    (L)) we have to determine the branch divisors ofthe projection

    π : DL → FL(Õ+

    (L)) = Õ+

    (L) \ DL. (3)According to [GHS2, Corollary 2.13] these divisors are defined by reflectionsσr ∈ O+(L), where

    σr(l) = l −2(l, r)

    (r, r)r,

    coming from vectors r ∈ L with r2 < 0 that are stably reflective: by this wemean that r is primitive and σr or −σr is in Õ

    +(L). By a (k)-vector for k ∈ Z

    we mean a primitive vector r with r2 = k.

    Documenta Mathematica 13 (2008) 1–19

  • Hirzebruch-Mumford Proportionality and . . . 5

    Let D be the exponent of the finite abelian group AL and let the divisordiv(r) of r ∈ L be the positive generator of the ideal (l, L). We note thatr∗ = r/ div(r) is a primitive vector in L∨. In [GHS2, Propositions 3.1–3.2] weproved the following.

    Lemma 2.1 Let L be an even integral lattice of signature (2, n). If σr ∈ Õ+

    (L)

    then r2 = −2. If −σr ∈ Õ+

    (L), then r2 = −2D and div(r) = D ≡ 1 mod 2 orr2 = −D and div(r) = D or D/2.

    We need also the following well-known property of the stable orthogonal group.

    Lemma 2.2 For any sublattice M of an even lattice L the group Õ(M) can be

    considered as a subgroup of Õ(L).

    Proof. Let M⊥ be the orthogonal complement of M in L. We have as usual

    M ⊕M⊥ ⊂ L ⊂ L∨ ⊂M∨ ⊕ (M⊥)∨.

    We can extend g ∈ Õ(M) to M ⊕M⊥ by putting g|M⊥ ≡ id. It is clear thatg ∈ Õ(M ⊕ M⊥). For any l∨ ∈ L∨ we have g(l∨) ∈ l∨ + (M ⊕ M⊥). Inparticular, g(l) ∈ L for any l ∈ L and g ∈ Õ(L). 2

    We can describe the components of the branch locus in terms of homogeneousdomains. For r a stably reflective vector in L we put

    Hr = {[w] ∈ P(L⊗ C) | (w, r) = 0},

    and let N be the union of all hyperplane sections Hr ∩ DL over all stablyreflective vectors r.

    Proposition 2.3 Let r ∈ L be a stably reflective vector: suppose that r andL do not satisfy D = 4, r2 = −4, div(r) = 2. Let Kr be the orthogonalcomplement of r in L. Then the associated component π(Hr ∩ DL) of thebranch locus N is of the form Õ+(Kr)\DKr .

    Proof. We have Hr ∩ DL = P(Kr) ∩DL = DKr . Let

    ΓKr = {ϕ ∈ Õ+

    (L) | ϕ(Kr) = Kr}. (4)

    ΓKr maps to a subgroup of O+(Kr). The inclusion of Õ(Kr) in Õ(L)

    (Lemma 2.2) preserves the spinor norm (see [GHS1, §3.1]), because Kr hassignature (2, n− 1) and so Õ+(Kr) becomes a subgroup of Õ

    +(L).

    Therefore the image of ΓKr contains Õ+

    (Kr) for any r. Now we prove that

    this image coincides with Õ+

    (Kr) for all r, except perhaps if D = 4, r2 = −4

    and div(r) = 2.

    Documenta Mathematica 13 (2008) 1–19

  • 6 V. Gritsenko, K. Hulek and G. K. Sankaran

    Let us consider the inclusions

    〈r〉 ⊕Kr ⊂ L ⊂ L∨ ⊂ 〈r〉∨ ⊕K∨r .

    By standard arguments (see [GHS2, Proposition 3.6]) we see that

    | detKr| =| detL| · |r2|

    div(r)2and [L : 〈r〉 ⊕Kr] =

    |r2|div(r)

    = 1 or 2.

    If the index is 1, then it is clear that the image of ΓKr is Õ+

    (Kr). Let usassume that the index is equal to 2. In this case the lattice 〈r〉∨ is generatedby r∨ = −r/(r, r) = r∗/2, where r∗ = r/ div(r) is a primitive vector in L∨. Inparticular r∨ represents a non-trivial class in 〈r〉∨ ⊕ K∨r modulo L∨. Let ustake k∨ ∈ K∨r such that k∨ 6∈ L∨. Then k∨ + r∨ ∈ L∨ and

    ϕ(k∨)− k∨ ≡ r∨ − ϕ(r∨) mod L.

    We note that if ϕ ∈ ΓKr then ϕ(r) = ±r. Hence

    ϕ(k∨)− k∨ ≡{

    0 mod L if ϕ(r) = r

    r∗ mod L if ϕ(r) = −r

    Since ϕ(r∗) ≡ r∗ mod L, we cannot have ϕ(r) = −r unless div(r) = 1 or 2.Therefore we have proved that ϕ(k∨) ≡ k∨ mod Kr (Kr = K∨r ∩ L), exceptpossibly if D = 4, r2 = −4, div(r) = 2. 2

    The group Õ+

    (L) acts on N . We need to estimate the number of componentsof Õ

    +(L) \ N . This will enable us to estimate the reflective obstructions to

    extending pluricanonical forms which arise from these branch loci.For the even unimodular lattice II2,8m+2 any primitive vector r has div(r) = 1.Consequently r is stably reflective if and only if r2 = −2.For L

    (m)2d the reflections and the corresponding branch divisors arise in two

    different ways, according to Lemma 2.1. We shall classify the orbits of suchvectors.

    Proposition 2.4 Suppose d is a positive integer.

    (i) Any two (−2)-vectors in the lattice II2,8m+2 are equivalent moduloO+(II2,8m+2), and the orthogonal complement of a (−2)-vector r is iso-metric to

    K(m)II = U ⊕mE8(−1)⊕ 〈2〉.

    (ii) There is one Õ+

    (L(m)2d )-orbit of (−2)-vectors r in L

    (m)2d with div(r) = 1. If

    d ≡ 1 mod 4 then there is a second orbit of (−2)-vectors, with div(r) = 2.The orthogonal complement of a (−2)-vector r in L(m)2d is isometric to

    K(m)2d = U ⊕mE8(−1)⊕ 〈2〉 ⊕ 〈−2d〉,

    Documenta Mathematica 13 (2008) 1–19

  • Hirzebruch-Mumford Proportionality and . . . 7

    if div(r) = 1, and to

    N(m)2d = U ⊕mE8(−1)⊕

    (1 2

    1−d2 1

    ),

    if div(r) = 2.

    (iii) The orthogonal complement of a (−2d)-vector r in L(m)2d is isometric to

    II2,8m+2 = 2U ⊕mE8(−1)

    if div(r) = 2d, and to

    K(m)2 = U⊕mE8(−1)⊕〈2〉⊕〈−2〉 or T2,8m+2 = U⊕U(2)⊕mE8(−1)

    if div(r) = d.

    (iv) Suppose d > 1. The number of Õ(L(m)2d )-orbits of (−2d)-vectors with

    div(r) = 2d is 2ρ(d). The number of Õ(L(m)2d )-orbits of (−2d)-vectors with

    div(r) = d is

    2ρ(d) if d is odd or d ≡ 4 mod 8;2ρ(d)+1 if d ≡ 0 mod 8;2ρ(d)−1 if d ≡ 2 mod 4.

    Here ρ(d) is the number of prime divisors of d.

    Proof. If the lattice L contains two hyperbolic planes then according to the

    well-known result of Eichler (see [E, §10]) the Õ+(L)-orbit of a primitive vectorl ∈ L is completely defined by two invariants: by its length (l, l) and by itsimage l∗ + L in the discriminant group AL, where l∗ = l/ div(l).i) If u is a primitive vector of an even unimodular lattice II2,8m+2 then div(u) =1 and there is only one O(II2,8m+2)-orbit of (−2)-vectors. Therefore we cantake r to be a (−2)-vector in U , and the form of the orthogonal complement isobvious.ii) In the lattice L

    (m)2d we fix a generator h of its 〈−2d〉-part. Then for any

    r ∈ L(m)2d we can write r = u + xh, where u ∈ II2,8m+2 and x ∈ Z. It is clearthat div(r) divides r2. If f | div(r), where f = 2, d or 2d, then the vector uis also divisible by f . Therefore the (−2)-vectors form two possible orbits ofvectors with divisor equal to 1 or 2. If r2 = −2 and div(r) = 2 then u = 2u0with u0 ∈ 2U⊕mE8(−1) and we see that in this case d ≡ 1 mod 4. This givesus two different orbits for such d. In both cases we can find a (−2)-vector rin the sublattice U ⊕ 〈−2d〉. Elementary calculation gives us the orthogonalcomplement of r.iii) This was proved in [GHS2, Proposition 3.6] for m = 2. For general m theproof is the same.

    Documenta Mathematica 13 (2008) 1–19

  • 8 V. Gritsenko, K. Hulek and G. K. Sankaran

    iv) To find the number of orbits of (−2d)-vectors we have to consider two cases.a) Let div(r) = 2d. Then r = 2du+ xh and r∗ ≡ (x/2d)h mod L, where u ∈II2,8m+2 and x is modulo 2d. Moreover (r, r) = 4d

    2(u, u)− x22d = −2d. Thusx2 ≡ 1 mod 4d. This congruence has 2ρ(d) solutions modulo 2d. For any suchx mod 2d we can find a vector u in 2U ⊕mE8(−1) with (u, u) = (x2 − 1)/2d.Then r = 2du+ xh is primitive (because u is not divisible by any divisor of x)and (r, r) = −2d.b) Let div(r) = d. Then r = du+xh, where u is primitive, r∗ ≡ (x/d)h mod Land x is modulo d. We have (r∗, r∗) ≡ −2x2/d mod 2Z and x2 ≡ 1 mod d.For any solution modulo d we can find as above u ∈ 2U ⊕mE8(−1) such thatr = du+ xh is primitive and (r, r) = −2d. It is easy to see that the number ofsolutions {x mod d |x2 ≡ 1 mod d} is as stated. 2Remark. To calculate the number of the branch divisors arising from vectorsr with r2 = −2d one has to divide the corresponding number of orbits foundin Proposition 2.4(iv) by 2 if d > 2. This is because ±r determine differentorbits but the same branch divisor. For d = 2 the proof shows that there isone divisor for each orbit given in Proposition 2.4(iv).

    3 Modular forms of low weight

    In this section we let L = 2U ⊕ L0 be an even lattice of signature (2, n) withtwo hyperbolic planes. We choose a primitive isotropic vector c1 in L. Thisvector determines a 0-dimensional cusp and a tube realisation of the domainDL. The tube domain (see the definition of H(L1) below) is a complexificationof the positive cone of the hyperbolic lattice L1 = c

    ⊥1 /c1. If div(c1) = 1 we call

    this cusp standard (as above, by [E] there is only one standard cusp). In thiscase L1 = U ⊕ L0. In [GHS2, §4] we proved that any 1-dimensional boundarycomponent of Õ

    +(L) \ DL contains the standard 0-dimensional cusp if every

    isotropic (with respect to the discriminant form: see [Nik2, §1.3]) subgroup ofAL is cyclic.Let us fix a 1-dimensional cusp by choosing two copies of U in L. (One hasto add to c1 a primitive isotropic vector c2 ∈ L1 with div(c2) = 1). ThenL = U ⊕ L1 = U ⊕ (U ⊕ L0) and the construction of the tube domain may bewritten down simply in coordinates. We have

    H(L1) = Hn = {Z = (zn, . . . , z1) ∈ H1 × Cn−2 ×H1; (ImZ, ImZ)L1 > 0},where Z ∈ L1 ⊗ C and (zn−1, . . . , z2) ∈ L0 ⊗ C. (We represent Z as a columnvector.) An isomorphism between Hn and DL is given by

    p : Hn −→ DL (5)

    Z = (zn, . . . , z1) 7−→(− 1

    2(Z,Z)L1 : zn : · · · : z1 : 1

    ).

    The action of O+(L⊗R) on Hn is given by the usual fractional linear transfor-mations. A calculation shows that the Jacobian of the transformation of Hn

    Documenta Mathematica 13 (2008) 1–19

  • Hirzebruch-Mumford Proportionality and . . . 9

    defined by g ∈ O+(L⊗R) is equal to det(g)j(g, Z)−n, where j(g, Z) is the last((n+2)-nd) coordinate of g

    (p(Z)

    )∈ DL. Using this we define the automorphic

    factor

    J : O+(L ⊗ R)×Hn+2 → C∗(g, Z) 7→ (det g)−1 · j(g, Z)n.

    The connection with pluricanonical forms is the following. Consider the form

    dZ = dz1 ∧ · · · ∧ dzn ∈ Ωn(Hn).

    F (dZ)k is a Γ-invariant k-fold pluricanonical form on Hn, for Γ a subgroupof finite index of O+(L), if F

    (g(Z)

    )= J(g, Z)kF (Z) for any g ∈ Γ; in other

    words if F ∈Mnk(Γ, detk) (see Definition 1.2). To prove Theorem 1.1 we needcusp forms of weight smaller than the dimension of the corresponding modularvariety.

    Proposition 3.1 For unimodular type, cusp forms of weight 12 + 4m exist:that is

    dimS12+4m(O+(II2,8m+2)) > 0.

    For K3 type we have the bounds

    dimS11+4m(Õ+

    (L(m)2d )) > 0 if d > 1;

    dimS10+4m(Õ+

    (L(m)2d )) > 0 if d ≥ 1;

    dimS7+4m(Õ+

    (L(m)2d )) > 0 if d ≥ 4;

    dimS6+4m(Õ+

    (L(m)2d )) > 0 if d = 3 or d ≥ 5;

    dimS5+4m(Õ+

    (L(m)2d )) > 0 if d = 5 or d ≥ 7;

    dimS2+4m(Õ+

    (L(m)2d )) > 0 if d > 180.

    Proof. For any F (Z) ∈ Mk(Õ+

    (L)) we can consider its Fourier-Jacobi expan-sion at the 1-dimensional cusp fixed above

    F (Z) = f0(z1) +∑

    m≥1fm(z1; z2, . . . zn−1) exp(2πimzn).

    A lifting construction of modular forms F (Z) ∈Mk(Õ+

    (L)) with trivial char-acter by means of the first Fourier–Jacobi coefficient is given in [G1], [G2].We note that f1(z1; z2, . . . , zn−1) ∈ Jk,1(L0), where Jk,1(L0) is the space ofthe Jacobi forms of weight k and index 1. A more general construction of theadditive lifting was given in [B2] but for our purpose the construction of [G2]is sufficient.

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  • 10 V. Gritsenko, K. Hulek and G. K. Sankaran

    The dimension of Jk,1(L0) depends only on the discriminant form and the rankof L0 (see [G2, Lemma 2.4]). In particular, for the special cases of L = II2,8m+2

    and L = L(m)2d we have

    Jcuspk+4m,1(mE8(−1)) ∼= Sk(SL2(Z))

    andJcuspk+4m,1(mE8(−1)⊕ 〈−2d〉) ∼= Jcuspk,d ,

    where Jcuspk,d is the space of the usual Jacobi cusp forms in two variables ofweight k and index d (see [EZ]) and Sk(SL2(Z)) is the space of weight k cuspforms for SL2(Z).The lifting of a Jacobi cusp form of index one is a cusp form of the same weight

    with respect to O+(II2,8m+2) or Õ+

    (L(m)2d ) with trivial character. The fact

    that we get a cusp form was proved in [G2] for maximal lattices, i.e., if d issquare-free. In [GHS2, §4] we extended this to all lattices L for which theisotropic subgroups of the discriminant AL are all cyclic, which is true in allcases considered here.To prove the unimodular type case of Proposition 3.1 we can take the Jacobiform corresponding to the cusp form ∆12(τ). Using the Jacobi lifting construc-tion we obtain a cusp form of weight 12 + 4m with respect to O+(II2,8m+2).For the K3 type case we need the dimension formula for the space of Jacobicusp forms Jcuspk,d (see [EZ]). For a positive integer l one sets

    {l}12 =

    ⌊ l12⌋ if l 6≡ 2 mod 12

    ⌊ l12⌋ − 1 if l ≡ 2 mod 12.Then if k > 2 is even

    dimJ cuspk,d =

    d∑

    j=0

    ({k + 2j}12 −

    ⌊j2

    4d

    ⌋),

    and if k is odd

    dimJ cuspk,d =

    d−1∑

    j=1

    ({k − 1 + 2j}12 −

    ⌊j2

    4d

    ⌋).

    This gives the bounds claimed. For k = 2, using the results of [SZ] one canalso calculate dim Jcusp2,d : there is an extra term, ⌈σ0(d)/2⌉, where σ0(d) denotesthe number of divisors of d. This gives dimJcusp2,d > 0 if d > 180 and for somesmaller values of d. 2

    4 Kodaira dimension results

    In this section we prove Theorem 1.1. We first explain the geometric back-ground. Let FL(Γ)tor be a toroidal compactification as in Theorem 1.3. In

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  • Hirzebruch-Mumford Proportionality and . . . 11

    particular all singularities are canonical and there is no ramification divisorwhich is contained in the boundary. Then the canonical divisor (as a Q-divisor)is given by KFL(Γ)tor = nM − V −D where M is the line bundle of modularforms of weight 1, n is the dimension of FL(Γ), V is the branch locus (which isgiven by reflections) and D is the boundary. Hence in order to construct k-foldpluricanonical forms we must find modular forms of weight kn which vanishof order k along the branch divisor and the boundary. This also suffices sinceFL(Γ)tor has canonical singularities.Our strategy is the following. For Γ ⊆ Õ+(L) we choose a cusp form Fa ∈ Sa(Γ)of low weight a, i.e. a strictly less than the dimension. Then we consider ele-ments F ∈ F kaMk(n−a)(Γ, detk): for simplicity we assume that k is even. Suchan F vanishes to order at least k on the boundary of any toroidal compactifi-cation. Hence if dZ is the volume element on DL defined in §3 it follows thatF (dZ)k extends as a k-fold pluricanonical form to the general point of everyboundary component of FL(Γ)tor. Since we have chosen the toroidal compact-ification so that all singularities are canonical and that there is no ramificationdivisor which is contained in the boundary the only obstructions to extendingF (dZ)k to a smooth projective model are the reflective obstructions, comingfrom the ramification divisor of the quotient map π : DL → FL(Γ) studied in§2.Let DK be an irreducible component of this ramification divisor. Recall fromProposition 2.3 that DK = P(K ⊗ C) ∩ DL where K = Kr is the orthogonalcomplement of a stably reflective vector r. For the lattices chosen in The-orem 1.1 all irreducible components of the ramification divisor are given inProposition 2.4.

    Proposition 4.1 We assume that k is even and that the dimension n ≥ 9.For Γ ⊆ Õ+(L), the obstruction to extending forms F (dZ)k where F ∈F kaMk(n−a)(Γ) to FL(Γ)tor lies in the space

    B =⊕

    K

    B(K) =⊕

    K

    k/2−1⊕

    ν=0

    Mk(n−a)+2ν(Γ ∩ Õ+

    (K)),

    where the direct sum is taken over all irreducible components DK of the rami-fication divisor of the quotient map π : DL → F(Γ).

    Proof. Let σ ∈ Γ be plus or minus a reflection whose fixed point locus is DK .We can extend the differential form provided that F vanishes of order k alongevery irreducible component DK of the ramification divisor.If Fa vanishes along DK then K gives no restriction on the second factor of themodular form F .Now let {w = 0} be a local equation for DK . Then σ∗(w) = −w (this isindependent of whether σ or −σ is the reflection). For every modular formF ∈ Mk(Γ) of even weight we have F

    (σ(z)

    )= F (z). This implies that if

    F (z) ≡ 0 on DK , then F vanishes to even order on DK .

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    We denote by M2b(Γ)(−νDK) the space of modular forms of weight 2b whichvanish of order at least ν along DK . Since the weight is even we haveM2b(Γ)(−DK) = M2b(Γ)(−2DK). For F ∈ M2b(Γ)(−2νDK) we consider(F/w2ν) as a function on DK . From the definition of modular form (Def-inition 1.2) it follows that this function is holomorphic, Γ ∩ ΓK-invariant(see equation (4)) and homogeneous of degree 2b + 2ν. Thus (F/w2ν)|DK ∈M2(b+ν)(Γ∩ ΓK). In Proposition 2.3 we saw that, ΓK contains Õ

    +(K) as sub-

    group of Õ+

    (L)(with equality in almost all cases), so we may replace Γ ∩ ΓKby Γ ∩ Õ+(K). In this way we obtain an exact sequence

    0→M2b(Γ)(−(2 + 2ν)DK)→M2b(Γ)(−2νDK)→M2(b+ν)(Γ ∩ Õ+

    (K)),

    where the last map is given by F 7→ F/w2ν . This gives the result. 2

    Now we proceed with the proof of Theorem 1.1.

    Let L be a lattice of signature (2, n) and Γ < Õ+

    (L): recall that k is even.According to Proposition 4.1 we can find pluricanonical differential forms onFL(Γ)tor if

    CB(Γ) = dimMk(n−a)(Γ)−∑

    K

    dimB(K) > 0, (6)

    where summation is taken over all irreducible components of the ramificationdivisor (see the remark at the end of §2). It now remains to estimate thedimension of B(K) for each of the finitely many components of the ramificationlocus in the cases we are interested in, namely Γ = O+(II2,8m+2) and Γ =

    Õ+

    (L(m)2d ).

    According to the Hirzebruch-Mumford proportionality principle

    dimMk(Γ) =2

    n!volHM (Γ)k

    n +O(kn−1).

    The exact formula for the Hirzebruch-Mumford volume volHM for any indef-inite orthogonal group was obtained in [GHS1]. It depends mainly on thedeterminant and on the local densities of the lattice L. Here we simply quotethe estimates of the dimensions of certain spaces of cusp forms.The case of II2,8m+2 is easier because the branch divisor has only one irre-ducible component defined by any (−2)-vector r. According to Proposition 2.4the orthogonal complement Kr is K

    (m)II . This lattice differs from the lattice

    L(m)2 , whose Hirzebruch-Mumford volume was calculated in [GHS1, §3.5], only

    by one copy of the hyperbolic plane. Therefore

    volHM Õ+

    (L(m)2 ) = (B8m+4/(8m+ 4)) volHM Õ

    +(K

    (m)II ),

    and hence, for even k,

    dimMk(Õ+

    (K(m)II )) =

    21−4m

    (8m+ 1)!· B2 . . . B8m+2

    (8m+ 2)!!k8m+1 +O(k8m),

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    where the Bi are Bernoulli numbers. Assume that m ≥ 3. Let us take a cuspform

    F ∈ S4m+12(O+(II2,8m+2))from Proposition 3.1. In this case the dimension of the obstruction space B ofProposition 4.1 for the pluricanonical forms of order k = 2k1 is given by

    k1−1∑

    ν=0

    dimM(4m−10)k+2ν(Õ+

    (K(m)II )) =

    24m+2

    (8m+ 2)!· B2 . . . B8m+2

    (8m+ 2)!!

    ((1 +

    1

    4m− 10)8m+2 − 1

    )((4m− 10)k1)8m+2

    +O(k8m+1)

    In [GHS1, §3.3] we computed the leading term of the dimension of the spaceof modular forms for O+(II2,8m+2). Comparing these two we see that theconstant CB(O

    +(II2,8m+2)) in the obstruction inequality (6) is positive if andonly if

    B4m+24m+ 2

    >(1 +

    1

    4m− 10)8m+2 − 1. (7)

    Moreover F (m)II is of general type if CB(O+(II2,8m+2)) > 0. From Stirling’sformula

    5√πn( nπe

    )2n > |B2n| > 4√πn( nπe

    )2n. (8)

    Using this estimate we easily obtain that (7) holds if m ≥ 5. Therefore we haveproved Theorem 1.1 for the lattice II2,8m+2.

    Next we consider the lattice L(m)2d of K3 type. For this lattice the branch divisor

    of F (m)2d is calculated in Proposition 2.4. It contains one or two (if d ≡ 1 mod 4)components defined by (−2)-vectors and some number of components definedby (−2d)-vectors. To estimate the obstruction constant CB(Γ) in (6) we use thedimension formulae for the space of modular forms with respect to the group

    Õ+

    (M), where M is one of the following lattices from Proposition 2.4: L(m)2d

    (the main group); K(m)2d and N

    (m)2d (the (−2)-obstruction); M2,8m+2, K

    (m)2 and

    T2,8m+2 (the (−2d)-obstruction). The corresponding dimension formulae werefound in [GHS1] (see §§3.5, 3.6.1–3.6.2, 3.3 and 3.4). The branch divisor of(−2d)-type appears only if d > 1. We note that

    volHM (Õ+

    (T2,8m+2)) > volHM (Õ+

    (K(m)2 )). (9)

    Therefore in order to estimate CB(Γ) we can assume that all (−2d)-divisors de-fined by stably reflective (−2d)-vectors r with div(r) = d (see Proposition 2.4)are of the type T2,8m+2.We put k = 2k1, w = n − a and n = 8m+ 3. For the obstruction constant in(6) we obtain

    CB(Õ+

    (L(m)2d )) > dimM2k1w(Õ

    +(L

    (m)2d ))−B(−2) −B(−2d) (10)

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  • 14 V. Gritsenko, K. Hulek and G. K. Sankaran

    where

    B(−2) = dimB(K(m)2d ) + dimB(N

    (m)2d ),

    B(−2d) = 2ρ(d)−1(dimB(M2,8m+2) + 2

    hd dimB(T2,8m+2))

    and B(K) is the obstruction space from Proposition 4.1. By hd we denote thesum δ0,d(8) − δ2,d(4), where d(n) is d mod n and δ is the Kronecker delta (seeProposition 2.4 and the remark following it).For any lattice considered above

    B(K) =

    k1−1∑

    ν=0

    dimM2(k1w+ν)(Õ+

    (K))

    =28m+3

    (8m+ 3)!Ew(8m+ 3)volHM (Õ

    +(K))(k1w)

    8m+3 +O(k8m+21 )

    where Ew(8m+ 3) = (1 +1w )

    8m+3.All terms in (10) contain a common factor. First

    dimM2k1w(Õ+

    (L(m)2d )) = C

    k1,wm,d

    ∣∣∣∣B8m+4B4m+2

    ∣∣∣∣√d+O(k8m+21 ), (11)

    where

    Ck1,wm,d =

    24m+1+δ1,d

    (8m+ 3)!

    |B2 . . . B8m+2|(8m+ 2)!!

    |B4m+2|4m+ 2

    d4m+32

    p|d(1− p−(4m+2))(k1w)8m+3.

    We note that 24m+1B4m+24m+2 = π−(4m+2)Γ(4m+ 2)ζ(4m+ 2).

    From [GHS1, (16)] it follows that

    volHM (Õ+

    (K(m)2d )) =

    2δ1,d−δ4,d(8)B2 . . . B8m+2

    (8m+ 2)!!d4m+

    32 π−(4m+2)Γ(4m+ 2)L(4m+ 2,

    (4d

    )).

    We can use the formula for the volume of N(m)2d in the following form:

    volHM (Õ+

    (N(m)2d )) =

    21+δ1,d−(8m+4)d4m+32B2 . . . B8m+2

    (8m+ 2)!!π−(4m+2)Γ(4m+ 2)L(4m+ 2,

    (d

    ))

    (see [GHS1, 3.6.2]). It follows that

    B(−2) = Ck1,wm,d Ew(8m+3)(2

    8m+3−δ4,d(8)PK(4m+2)+PN(4m+2))+O(k8m+21 )

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    where

    PK(n) = (1− 2−n)δ0,d(2)L(n,

    (4d∗))

    L(n, χ0,4d)

    p|d

    1− p−n1 + p−n

    and

    PN (n) =L(n,

    (d∗))

    L(n, χ0,d)

    p|d

    1− p−n1 + p−n

    .

    Here χ0,f denotes the principal Dirichlet character modulo f .We note that |PK(n)| < 1 and |PN (n)| < 1 for any d. We conclude that

    B(−2) < Ck1,wm,d Ew(8m+ 3)b(−2)

    where b(−2) = 28m+3 − 1.The (−2d)-contribution is calculated according to [GHS1, 3.3–3.4]. We notethat Õ

    +(T2,8m+2) is a subgroup of Õ

    +(M2,8m+2). We obtain

    B(−2d) ≤ Ck1,wm,d Ew(8m+ 3)b(−2d)

    where for d > 2

    b(−2d) =2ρ(d)

    d

    (4

    d

    )4m+ 124(2hd(1 + 2−(4m+2) − 2−(8m+3)) + 2−(8m+3)).

    As a result we see that that the obstruction constant CB(Õ+

    (L(m)2d )) is positive

    if

    β(w)m,d =

    ∣∣∣∣B4m+2B8m+4

    ∣∣∣∣Ew(8m+ 3)(b(−2) + b(−2d)) <√d.

    Using (8) we get

    ∣∣∣∣B4m+2B8m+4

    ∣∣∣∣ <5

    4√

    2

    (πe

    2m+ 1

    )4m+21

    28m+4.

    For m ≥ 5 we choose a cusp form Fa of weight a = 4m + 10, i.e. we takew = 4m − 7 in Proposition 4.1. Such a cusp form exists for all d ≥ 1 byProposition 3.1. Using the fact that β(−2d) ≤ β(−4) for any d ≥ 2 and the valueb(−4) = 2

    4m+ 52 , we see that

    β(4m−7)m,d < (1 +

    1

    4m− 7)8m+3 5

    8√

    2

    (πe

    2m+ 1

    )4m+228m+3 + 24m+

    52 + 1

    28m+3,

    which is smaller than 1 if m ≥ 5. This proves Theorem 1.1 for m ≥ 5.For m = 4 there exists a cusp form Fa of weight 4m+ 6 if d 6= 1, 2, 4, i.e. wetake w = 4m − 3. To see that β(13)4,d <

    √d we need check this only for d = 3

    because b(−2d) < b(−6) for d > 3. One can do it by direct calculation.

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  • 16 V. Gritsenko, K. Hulek and G. K. Sankaran

    For m ≤ 3 we choose Fa of weight 4m + 2, i.e. we take w = 4m + 1. Sucha cusp form exists if d > 180 according to Proposition 3.1. For such d we see

    that β(−2d) < 1. Then the obstruction constant CB(Õ+

    (L(m)2d )) is positive if

    ∣∣∣∣B4m+2B8m+2

    ∣∣∣∣ (1 +1

    4m+ 1)8m+3(28m+3 + 2) <

    √d.

    This inequality gives us the bound on d in Theorem 1.1.This completes the proof of Theorem 1.1.

    In the proof of Theorem 1.1 above we have seen that the (−2)-part of the branchdivisor forms the most important reflective obstruction to the extension of the

    Õ+

    (L(m)2d )-invariant differential forms to a smooth compact model of F

    (m)2d . Let

    us consider the double covering SF (m)2d of F(m)2d for d > 1 determined by the

    special orthogonal group:

    SF (m)2d = S̃O+

    (L(m)2d ) \DL(m)2d → F

    (m)2d .

    Here the branch divisor does not contain the (−2)-part. Theorem 4.2 belowshows that there are only five exceptional varieties SF (m)2d with m > 0 andd > 1 that are possibly not of general type.

    The variety SF (2)2d can be interpreted as the moduli space of K3 surfaces ofdegree 2d with spin structure: see [GHS2, §5]. The three-fold SF (0)2d is themoduli space of (1, t)-polarised abelian surfaces.

    Theorem 4.2 The variety SF (m)2d is of general type for any d > 1 if m ≥ 3. Ifm = 2 then SF (2)2d is of general type if d ≥ 3. If m = 1 then SF

    (1)2d is of general

    type if d = 5 or d ≥ 7.

    Proof. The case m = 2 is [GHS2, Theorem 5.1], and the result for m ≥ 5 isimmediate from Theorem 1.1. For m = 1, 3 and 4 we can prove more thanwhat follows from Theorem 1.1.The branch divisor of SF (m)2d is defined by the reflections in vectors r ∈ L

    (m)2d

    such that −σr ∈ S̃O+

    (L(m)2d ), because the rank of L

    (m)2d is odd. Therefore

    r2 = −2d, by Proposition 2.1.If F ∈ M2k+1(S̃O

    +(L

    (m)2d )) is a modular form (note that the character det is

    trivial), d > 1 and z ∈ D•L

    (m)2d

    is such that (z, r) = 0, then

    F (z) = F (−σr(z)) = F (−z) = (−1)2k+1F (z).

    Therefore any modular form of odd weight for S̃O+

    (L(m)2d ) vanishes on the

    branch divisor.

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    To apply the low-weight cusp form trick used in the proof of Theorem 1.1

    one needs a cusp form of weight smaller than dimSF (m)2d = 8m + 3. ByProposition 3.1 there exists a cusp form F11+4m ∈ S11+4m(S̃O

    +(L

    (m)2d )). For

    m ≥ 3 we have that 11 + 4m < 8m + 3. Therefore the differential formsF k11+4mF(4m−8)k(dZ)

    k, for arbitrary F(4m−8)k ∈ M(4m−8)k(S̃O+

    (L(m)2d )), ex-

    tend to the toroidal compactification of SF (m)2d constructed in Theorem 1.3.This proves the cases m ≥ 3 of the theorem.For the case m = 1 we use a cusp form of weight 9 with respect to S̃O

    +(L

    (1)2d )

    constructed in Proposition 3.1. 2

    We can obtain some information also for some of the remaining cases.

    Proposition 4.3 The spaces SF (1)8 and SF(1)12 have non-negative Kodaira di-

    mension.

    Proof. By Proposition 3.1 there are cusp forms of weight 11 for S̃O+

    (L(1)8 )

    and S̃O+

    (L(1)12 ). The weight of these forms is equal to the dimension. By the

    well-known observation of Freitag [F, Hilfssatz 2.1, Kap. III] these cusp forms

    determine canonical differential forms on the 11-dimensional varieties SF (1)8and SF (1)12 . 2

    These varieties may perhaps have intermediate Kodaira dimension as it seems

    possible that a reflective modular form of canonical weight exists for L(1)8 and

    L(1)12 .

    In [GHS2] we used pull-backs of the Borcherds modular form Φ12 on DII2,26 toshow that many moduli spaces of K3 surfaces are of general type. We can alsouse Borcherds products to prove results in the opposite direction.

    Theorem 4.4 The Kodaira dimension of F (m)II is −∞ for m = 0, 1 and 2.

    Proof. For m = 0 we can see immediately that the quotient is rational: a

    straightforward calculation gives that F (0)II = Γ\H1×H1 where H1 is the usualupper half plane and Γ is the degree 2 extension of SL(2,Z)× SL(2,Z) by theinvolution which interchanges the two factors. Compactifying this, we obtainthe projective plane P2.For m = 1, 2 we argue differently. There are modular forms similar to Φ12 forthe even unimodular lattices II2,10 and II2,18. They are Borcherds productsΦ252 and Φ127 of weights 252 and 127 respectively, defined by the automorphicfunctions

    ∆(τ)−1(τ)E4(τ)2 = q−1 + 504 + q(. . . )

    and

    ∆(τ)−1(τ)E4(τ) = q−1 + 254 + q(. . . ),

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  • 18 V. Gritsenko, K. Hulek and G. K. Sankaran

    where q = exp(2πiτ) and ∆(τ) and E4(τ) are the Ramanujan delta functionand the Eisenstein series of weight 4 (see [B1]). The divisors of Φ252 and Φ127coincide with the branch divisors of F (1)II and F

    (2)II defined by the (−2)-vectors.

    Moreover Φ252 and Φ127 each vanishes with order one along the respective divi-sor. Therefore if F10k(dZ)

    k (or F18k(dZ)k) defines a pluricanonical differential

    form on a smooth model of a toroidal compactification of F (1)II or F(2)II , then

    F10k (or F18k) is divisible by Φk252 (or Φ

    k127), since F10k or F18k must vanish

    to order at least k along the branch divisor. This is not possible, because thequotient would be a holomorphic modular form of negative weight. 2

    We have already remarked that the space F (2)II is the moduli space of K3 sur-faces with an elliptic fibration with a section. Using the Weierstrass equationsit is then clear that this moduli space is unirational (such K3 surfaces areparametrised by a linear system in the weighted projective space P(4, 6, 1, 1)).In fact it is even rational: see [Le].

    References

    [BHPV] W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact complex sur-faces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4. Springer-Verlag, Berlin, 2004.

    [B1] R.E. Borcherds, Automorphic forms on Os+2,2(R) and infinite prod-ucts. Invent. Math. 120 (1995), 161–213.

    [B2] R.E. Borcherds, Automorphic forms with singularities on Grassmani-ans. Invent. Math. 132 (1998), 491–562.

    [BB] W.L. Baily Jr., A. Borel, Compactification of arithmetic quotient ofbounded domains. Ann. Math 84 (1966), 442–528.

    [CM] A. Clingher, J.W. Morgan, Mathematics underlying the F-theory/heterotic string duality in eight dimensions. Comm. Math.Phys. 254 (2005), 513–563.

    [Dol] I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces. J.Math. Sci. 81 (1996), 2599–2630.

    [E] M. Eichler, Quadratische Formen und orthogonale Gruppen. DieGrundlehren der mathematischen Wissenschaften 63. Springer-Verlag, Berlin 1952.

    [EZ] M. Eichler, D. Zagier, The theory of Jacobi forms. Progress in Math-ematics 55. Birkhäuser, Boston 1985.

    [F] E. Freitag, Siegelsche Modulfunktionen. Grundlehren der mathe-matischen Wissenschaften 254. Springer-Verlag, Berlin–Göttingen–Heidelberg, 1983.

    [G1] V. Gritsenko, Fourier-Jacobi functions in n variables. (Russian) Zap.Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168(1988), Anal. Teor. Chisel i Teor. Funktsii. 9, 32–44, 187–188; trans-lation in J. Soviet Math. 53 (1991), no. 3, 243–252.

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    [G2] V. Gritsenko, Modular forms and moduli spaces of abelian and K3 sur-faces. Algebra i Analiz 6, 65–102; English translation in St. PetersburgMath. J. 6 (1995), 1179–1208.

    [GH] V. Gritsenko, K. Hulek, Minimal Siegel modular threefolds. Math.Proc. Cambridge Philos. Soc. 123 (1998), 461–485.

    [GHS1] V. Gritsenko, K. Hulek, G.K. Sankaran, The Hirzebruch–Mumfordvolume for the orthogonal group and applications. Doc. Math. 12(2007), 215–241.

    [GHS2] V. Gritsenko, K. Hulek, G.K. Sankaran, The Kodaira dimension ofthe moduli of K3 surfaces. Invent. Math. 169 (2007), 519–567.

    [GHS3] V. Gritsenko, K. Hulek, G.K. Sankaran, Moduli spaces of ir-reducible symplectic manifolds. MPI Preprint 2008-20 (2008),arXiv:0802.2078.

    [Ko] S. Kondo, Moduli spaces of K3 surfaces. Compositio Math. 89 (1993),251–299.

    [Le] P. Lejarraga, The moduli of Weierstrass fibrations over P1: rational-ity. Rocky Mountain J. Math. 23 (1993), 649–650.

    [Mum] D. Mumford, Hirzebruch’s proportionality principle in the non-compact case. Invent Math. 42 (1977), 239–277.

    [Nik1] V.V. Nikulin, Finite automorphism groups of Kähler K3 surfaces.Trudy Moskov. Mat. Obshch. 38(1979), 75–137. English translationin Trans. Mosc. Math. Soc. 2, 71-135 (1980).

    [Nik2] V.V. Nikulin, Integral symmetric bilinear forms and some of theirapplications. Izv. Akad. Nauk USSR 43 (1979), 105 - 167 (Russian);English translation in Math. USSR, Izvestiia 14 (1980), 103–166.

    [SZ] N.P. Skoruppa, D. Zagier Jacobi forms and a certain space of modularforms. Invent. Math. 94 (1988), 113–146.

    V. A. GritsenkoUniversité Lille 1Laboratoire Paul PainlevéF-59655 Villeneuve d’Ascq, [email protected]

    K. HulekInstitut für Algebraische GeometrieLeibniz Universität HannoverD-30060 [email protected]

    G. K. SankaranDepartment of Mathematical SciencesUniversity of BathBath BA2 [email protected]

    Documenta Mathematica 13 (2008) 1–19

  • 20

    Documenta Mathematica 13 (2008)

  • Documenta Math. 21

    The Euler Characteristic of a Category

    Tom Leinster1

    Received: May 11, 2007

    Revised: February 20, 2008

    Communicated by Stefan Schwede

    Abstract. The Euler characteristic of a finite category is defined andshown to be compatible with Euler characteristics of other types ofobject, including orbifolds. A formula is proved for the cardinality ofa colimit of sets, generalizing the classical inclusion-exclusion formula.Both rest on a generalization of Rota’s Möbius inversion from posetsto categories.

    2000 Mathematics Subject Classification: 18F99, 55U99, 05C50,57N65.Keywords and Phrases: Euler characteristic, finite category, inclusion-exclusion, Möbius inversion, cardinality of colimit.

    Introduction

    We first learn of Euler characteristic as ‘vertices minus edges plus faces’, andlater as an alternating sum of ranks of homology groups. But Euler character-istic is much more fundamental than these definitions make apparent, as hasbeen made increasingly explicit over the last fifty years; it is something akinto cardinality or measure. More precisely, it is the fundamental dimensionlessquantity associated with an object.

    The very simplest context for Euler characteristic is that of finite sets, andof course the fundamental way to assign a quantity to a finite set is to countits elements. Euler characteristic of topological spaces can usefully be thoughtof as a generalization of cardinality; for instance, it obeys the same laws withrespect to unions and products.

    1Partially supported by a Nuffield Foundation award NUF-NAL 04 and an EPSRC Ad-vanced Research Fellowship

    Documenta Mathematica 13 (2008) 21–49

  • 22 Tom Leinster

    In a more sophisticated context, integral geometry, Euler characteris-tic also emerges clearly as the fundamental dimensionless invariant. A sub-set of Rn is polyconvex if it is a finite union of compact convex subsets.Let Vn be the vector space of finitely additive measures, invariant underEuclidean transformations, defined on the polyconvex subsets of Rn. Had-wiger’s Theorem [KR] states that dimVn = n + 1. (See also [Sc2], and [MS]for an important application to materials science.) A natural basis con-sists of one d-dimensional measure for each d ∈ {0, . . . , n}: for instance,{Euler characteristic, perimeter, area} when n = 2. Thus, up to scalar multipli-cation, Euler characteristic is the unique dimensionless measure on polyconvexsets.

    Schanuel [Sc1] showed that for a certain category of polyhedra, Euler char-acteristic is determined by a simple universal property, making its fundamentalnature transparent.

    All of the above makes clear the importance of defining and understandingEuler characteristic in new contexts. Here we do this for finite categories.

    Categories are often viewed as large structures whose main purpose isorganizational. However, some different viewpoints will be useful here. Acombinatorial point of view is that a category is a directed graph (objects andarrows) equipped with some extra structure (composition and identities). Wewill concentrate on finite categories (those with only finitely many objects andarrows), which also suits the combinatorial viewpoint, and the composition andidentities will play a surprisingly minor role.

    A topological point of view is that a category can be understood through itsclassifying space. This is formed by starting with one 0-cell for each object, thengluing in one 1-cell for each arrow, one 2-cell for each commutative triangle,and so on.

    Both of these points of view will be helpful in what follows. The topologicalperspective is heavily used in the sequel [BL] to this paper.

    With topology in mind, one might imagine simply transporting the defini-tion of Euler characteristic from spaces to categories via the classifying spacefunctor, as with other topological invariants: given a category A, define χ(A)as the Euler characteristic of the classifying space BA. The trouble with thisis that the Euler characteristic of BA is not always defined. Below we give adefinition of the Euler characteristic of a category that agrees with the topo-logical Euler characteristic when the latter exists, but is also valid in a range ofsituations when it does not. It is a rational number, not necessarily an integer.

    A version of the definition can be given very succinctly. Let A be a finitecategory; totally order its objects as a1, . . . , an. Let Z be the matrix whose(i, j)-entry is the number of arrows from ai to aj . Let M = Z

    −1, assumingthat Z is invertible. Then χ(A) is the sum of the entries of M . Of course, thereader remains to be convinced that this definition is the right one.

    The foundation on which this work rests is a generalization of Möbius–Rotainversion (§1). Rota developed Möbius inversion for posets [R]; we develop itfor categories. (A poset is viewed throughout as a category in which each hom-

    Documenta Mathematica 13 (2008) 21–49

  • The Euler Characteristic of a Category 23

    set has at most one element: the objects are the elements of the poset, andthere is an arrow a - b if and only if a ≤ b.) This leads, among otherthings, to a ‘representation formula’: given any functor known to be a sum ofrepresentables, the formula tells us the representation explicitly. This in turncan be used to solve enumeration problems, in the spirit of Rota’s paper.

    However, the main application of this generalized Möbius inversion is tothe theory of the Euler characteristic of a category (§2). We actually use a dif-ferent definition than the one just given, equivalent to it when Z is invertible,but valid for a wider class of categories. It depends on the idea of the ‘weight’of an object of a category. The definition is justified in two ways: by showingthat it enjoys the properties that the name would lead one to expect (behaviourwith respect to products, fibrations, etc.), and by demonstrating its compati-bility with Euler characteristics of other types of structure (groupoids, graphs,topological spaces, orbifolds). There is an accompanying theory of Lefschetznumber.

    The technology of Möbius inversion and weights also solves another prob-lem: what is the cardinality of a colimit? For example, the union of a familyof sets and the quotient of a set by a free action of a group are both examplesof colimits of set-valued functors, and there are simple formulas for their cardi-nalities. (In the first case it is the inclusion-exclusion formula.) We generalize,giving a formula valid for any shape of colimit (§3).

    Rota and his school proved a large number of results on Möbius inversionfor posets. As we will see repeatedly, many are not truly order-theoretic: theyare facts about categories in general. In particular, important theorems inRota’s original work [R] generalize from posets to categories (§4).

    (The body of work on Möbius inversion in finite lattices is not, however, soripe for generalization: a poset is a lattice just when the corresponding categoryhas products, but a finite category cannot have products unless it is, in fact, alattice.)

    Other authors have considered different notions of Möbius inversion for cat-egories; notably, there is that developed by Content, Lemay and Leroux [CLL]and independently by Haigh [H]. This generalizes both Rota’s notion for posetsand Cartier and Foata’s for monoids [CF]. (Here a monoid is viewed as a one-object category.) The relation between their approach and ours is discussedin §4. Further approaches, not discussed here, were taken by Dür [D] andLück [Lü].

    In the case of groupoids, our Euler characteristic of categories agrees withBaez and Dolan’s groupoid cardinality [BD]. The cardinality of the groupoidof finite sets and bijections is e = 2.718 . . ., and there are connections to ex-ponential generating functions and the species of Joyal [J, BLL]. Paré has adefinition of the cardinality of an endofunctor of the category of finite sets [Pa];I do not know whether this can be related to the definition here of the Lefschetznumber of an endofunctor.

    The view of Euler characteristic as generalized cardinality is promotedin [Sc1], [BD] and [Pr1]. The appearance of a non-integral Euler characteristic

    Documenta Mathematica 13 (2008) 21–49

  • 24 Tom Leinster

    is nothing new: see for instance Wall [Wl], Bass [Ba] and Cohen [Co], and thediscussion of orbifolds in §2.

    Ultimately it would be desirable to have the Euler characteristic of cate-gories described by a universal property, as Schanuel did for polyhedra [Sc1].For this, it may be necessary to relax the constraints of the present work, wherefor simplicity our categories are required to be finite and the coefficients arerequired to lie in the ring of rational numbers. Rather than asking, as below,‘does this category have Euler characteristic (in Q)?’, we should perhaps ask ‘inwhat rig (semiring) does the Euler characteristic of this category lie?’ However,this is not pursued here.

    Acknowledgements I thank John Baez, Andy Baker, Nick Gurski, IekeMoerdijk, Urs Schreiber, Ivan Smith and Stephen Watson for inspirationand useful discussions. I am grateful to the Centre de Recerca Matemàtica,Barcelona, for their hospitality. I also thank the very helpful referee.

    1 Möbius inversion

    We consider a finite category A, writing ob A for its set of objects and, when aand b are objects, A(a, b) for the set of maps from a to b.

    Definition 1.1 We denote by R(A) the Q-algebra of functions ob A ×ob A - Q, with pointwise addition and scalar multiplication, multiplica-tion defined by

    (θφ)(a, c) =∑

    b∈Aθ(a, b)φ(b, c)

    (θ, φ ∈ R(A), a, c ∈ A), and the Kronecker δ as unit.The zeta function ζA = ζ ∈ R(A) is defined by ζ(a, b) = |A(a, b)|. If

    ζ is invertible in R(A) then A is said to have Möbius inversion; its inverseµA = µ = ζ

    −1 is the Möbius function of A.

    If a total ordering is chosen on the n objects of A then R(A) can beregarded as the algebra of n × n matrices over Q. The defining equations ofthe Möbius function are

    b

    µ(a, b)ζ(b, c) = δ(a, c) =∑

    b

    ζ(a, b)µ(b, c)

    for all a, c ∈ A. By finite-dimensionality, µζ = δ if and only if ζµ = δ.The definitions above could be made for directed graphs rather than cat-

    egories, since they do not refer to composition. However, this generality seemsto be inappropriate. For example, the definition of Möbius inversion will lead toa definition of Euler characteristic, and if we use graphs rather than categoriesthen we obtain something other than ‘vertices minus edges’. Proposition 2.10clarifies this point.

    A different notion of Möbius inversion for categories has been considered;see §4.

    Documenta Mathematica 13 (2008) 21–49

  • The Euler Characteristic of a Category 25

    Examples 1.2 a. Any finite poset A has Möbius inversion; this special casewas investigated by Rota [R] and others. We may compute µ(a, c) byinduction on the number of elements between a and c:

    µ(a, c) = δ(a, c)−∑

    b:a≤b

  • 26 Tom Leinster

    Theorem 1.4 Let A be a finite skeletal category in which the only idempotentsare identities. Then A has Möbius inversion given by

    µ(a, b) =∑

    (−1)n/|Aut(a0)| · · · |Aut(an)|

    where Aut(a) is the automorphism group of a ∈ A and the sum runs over alln ≥ 0 and paths (1) for which a0, . . . , an are all distinct.

    Proof First observe that for a path (1) in A, if a0 6= a1 6= · · · 6= an then theais are all distinct. Indeed, if 0 ≤ i < j ≤ n and ai = aj then the sub-pathrunning from ai to aj is a circuit, so by Lemma 1.3, fi+1 is an isomorphism,and by skeletality, ai = ai+1.

    Now let a, c ∈ A and define µ by the formula above. We have∑

    b∈Aµ(a, b)ζ(b, c) = µ(a, c)ζ(c, c) +

    b:b6=cµ(a, b)ζ(b, c)

    = |Aut(c)|

    µ(a, c) +

    b:b6=c, g∈A(b,c)µ(a, b)/|Aut(c)|

    = |Aut(c)|{µ(a, c) +

    ∑(−1)n/|Aut(a0)| · · · |Aut(an)||Aut(c)|

    },

    where the last sum is over all n ≥ 0 and paths

    a = a0f1- · · · fn- an = b

    g- c

    such that a0 6= · · · 6= an 6= c. By definition of µ, the term in braces collapsesto 0 if a 6= c and to 1/|Aut(a)| if a = c. Hence ∑b µ(a, b)ζ(b, c) = δ(a, c), asrequired. �

    Corollary 1.5 Let A be a finite skeletal category in which the only endomor-phisms are identities. Then A has Möbius inversion given by

    µ(a, b) =∑

    n≥0(−1)n|{nondegenerate n-paths from a to b}| ∈ Z.

    When A is a poset, this is Philip Hall’s theorem (Proposition 3.8.5 of [St]and Proposition 6 of [R]).

    An epi-mono factorization system (E ,M) on a category A consists of aclass E of epimorphisms in A and a classM of monomorphisms in A, satisfyingaxioms [FK]. The axioms imply that every map f in A can be expressed as mefor some e ∈ E and m ∈ M, and that this factorization is essentially unique:the other pairs (e′,m′) ∈ E ×M satisfying m′e′ = f are those of the form(ie,mi−1) where i is an isomorphism. Typical examples are the categories ofsets, groups and rings, with E as all surjections and M as all injections.

    Documenta Mathematica 13 (2008) 21–49

  • The Euler Characteristic of a Category 27

    Theorem 1.6 Let A be a finite skeletal category with an epi-mono factorizationsystem (E ,M). Then A has Möbius inversion given by

    µ(a, b) =∑

    (−1)n/|Aut(a0)| · · · |Aut(an)|

    where the sum is over all n ≥ r ≥ 0 and paths (1) such that a0, . . . , ar aredistinct, ar, . . . , an are distinct, f1, . . . , fr ∈M, and fr+1, . . . , fn ∈ E.

    Proof The objects of A and the arrows in E determine a subcategory of A,also denoted E ; it satisfies the hypotheses of Theorem 1.4 and therefore hasMöbius inversion. The same is true of M.

    Any element α ∈ Qob A = ∏a∈A Q gives rise to an element of R(A), alsodenoted α and defined by α(a, b) = δ(a, b)α(b). This defines a multiplication-preserving map from Qob A to R(A), where the multiplication on Qob A is coor-dinatewise. We have elements |Aut| and 1/|Aut| of Qob A, where, for instance,|Aut|(a) = |Aut(a)|.

    By the essentially unique factorization property, ζA = ζE · 1|Aut| ·ζM. HenceA has Möbius function µA = µM · |Aut| · µE . Theorem 1.4 applied to µM andµE then gives the formula claimed. �

    Example 1.7 Let N ≥ 0 and write FN for the full subcategory of Set withobjects 1, . . . , N , where n denotes a (chosen) n-element set. Let E be the setof surjections in FN and M the set of injections; then (E ,M) is an epi-monofactorization system. Theorem 1.6 gives a formula for the inverse of the matrix(ij)i,j . For instance, take N = 3; then µ(1, 2) may be computed as follows:

    Paths Contribution to sum

    1-2- 2 −2/1!2! = −1

    1-3- 3

    6-- 2 3 · 6/1!3!2! = 3/21-

    2- 2-6- 3

    6-- 2 −2 · 6 · 6/1!2!3!2! = −3

    Here ‘1-2- 2’ means that there are 2 monomorphisms from 1 to 2, ‘3

    6-- 2’that there are 6 epimorphisms from 3 to 2, etc. Hence µ(1, 2) = −1+3/2−3 =−5/2.

    One of the uses of the Möbius function is to calculate Euler character-istic (§2). Another is to calculate representations. Specifically, suppose thatwe have a Set-valued functor known to be familially representable, that is, acoproduct of representables. The Yoneda Lemma tells us that the family ofrepresenting objects is unique (up to isomorphism). But if we have Möbiusinversion, there is actually a formula for it:

    Proposition 1.8 Let A be a finite category with Möbius inversion and letX : A - Set be a functor satisfying

    X ∼=∑

    a

    r(a)A(a,−)

    Documenta Mathematica 13 (2008) 21–49

  • 28 Tom Leinster

    for some natural numbers r(a) (a ∈ A). Then

    r(a) =∑

    b

    |X(b)|µ(b, a)

    for all a ∈ A.

    In the first formula,∑

    denotes coproduct of Set-valued functors.

    Proof Follows from the definition of Möbius function. �

    In the spirit of Rota’s programme, this can be applied to solve countingproblems, as illustrated by the following standard example.

    Example 1.9 A derangement is a permutation without fixed points. We cal-culate dn, the number of derangements of n letters.

    Fix N ≥ 0. Take the category DinjN of Example 1.2(c) and the functorS : DinjN

    - Set defined as follows: S(n) is Sn, the underlying set of the nthsymmetric group, and if f ∈ DinjN (m,n) and τ ∈ Sm, the induced permutationSf (τ) ∈ Sn acts as τ on the image of f and fixes all other points. Any permu-tation consists of a derangement together with some fixed points, so there isan isomorphism of sets

    Sn ∼=∑

    m

    dmDinjN (m,n)

    where∑

    denotes disjoint union. Then by Proposition 1.8 and Example 1.2(c),

    dn =∑

    m

    |Sm|µ(m,n) =∑

    m

    m!(−1)n−m(n

    m

    )= n!

    (1

    0!− 1

    1!+ · · ·+ (−1)

    n

    n!

    ).

    To set up the theory of Euler characteristic we will not need the fullstrength of Möbius invertibility; the following suffices.

    Definition 1.10 Let A be a finite category. A weighting on A is a functionk• : ob A - Q such that for all a ∈ A,

    b

    ζ(a, b)kb = 1.

    A coweighting k• on A is a weighting on Aop.

    Note that A has Möbius inversion if and only if it has a unique weighting,if and only if it has a unique coweighting; they are given by

    ka =∑

    b

    µ(a, b), kb =∑

    a

    µ(a, b).

    Documenta Mathematica 13 (2008) 21–49

  • The Euler Characteristic of a Category 29

    Examples 1.11 a. Let L be the category

    b1

    a

    -

    b2.-

    Then the unique weighting k• on L is (ka, kb1 , kb2) = (−1, 1, 1).

    b. Let M be a finite monoid, regarded as a category with unique object ⋆.Again there is a unique weighting k•, with k⋆ = 1/|M |.

    c. If A has a terminal object 1 then δ(−, 1) is a weighting on A.

    d. A finite category may admit no weighting at all. (This can happen evenwhen the category is Cauchy-complete, in the sense defined in the Ap-pendix.) An example is the category A with objects and arrows

    a1 a2

    a3

    a4

    --��HHHHHHHjHH

    HHHH

    HY ������������

    ���*

    ?

    @@

    @@

    @@

    @R

    ��

    ��

    ��

    ��

    ��

    ��

    jj ��

    K

    f12,g12

    f21,g21f13

    f31

    f23

    f32

    f34f14 f24

    g24

    1f11 1 f22

    f44=1

    f33=1

    where if aip- aj

    q- ak and neither p nor q is an identity thenq ◦ p = fik.

    e. A category may certainly have more than one weighting: for instance,if A is the category consisting of two objects and a single isomorphismbetween them, a weighting on A is any pair of rational numbers whose sumis 1. But even a skeletal category may admit more than one weighting.Indeed, the full subcategories B = {a1, a2} and C = {a1, a2, a3} of thecategory A of the previous example both have a 1-dimensional space ofweightings.

    In contrast to Möbius invertibility, the property of admitting at least oneweighting is invariant under equivalence:

    Lemma 1.12 Let A and B be equivalent finite categories. Then A admits aweighting if and only if B does.

    Documenta Mathematica 13 (2008) 21–49

  • 30 Tom Leinster

    Proof Let F : A - B be an equivalence. Given a ∈ A, write Ca for thenumber of objects in the isomorphism class of a. Take a weighting l• on B andput ka = (

    ∑b:b∼=F (a) l

    b)/Ca. I claim that k• is a weighting on A.

    To prove this, choose representatives a1, . . . , am of the isomorphism classesof objects of A; then F (a1), . . . , F (am) are representatives of the isomorphismclasses of objects of B. Let a′ ∈ A. For any a ∈ A, the numbers ζ(a′, a) and kadepend only on the isomorphism class of a. Hence

    a∈Aζ(a′, a)ka =

    m∑

    i=1

    a:a∼=aiζ(a′, a)ka

    =

    m∑

    i=1

    Caiζ(a′, ai)k

    ai

    =m∑

    i=1

    b:b∼=F (ai)ζ(a′, ai)l

    b

    =∑

    b∈Bζ(F (a′), b)lb

    = 1,

    as required. �

    Weightings and Möbius functions are compatible with sums and productsof categories. We write

    ∑i∈I Ai for the sum of a family (Ai)i∈I of categories,

    also called the coproduct or disjoint union and written∐i∈I Ai. The following

    lemma is easily verified.

    Lemma 1.13 Let n ≥ 0 and let A1, . . . ,An be finite categories.a. If each Ai has a weighting k•i then

    ∑i Ai has a weighting l

    • given byla = kai whenever a ∈ Ai. If each Ai has Möbius inversion then so does∑i Ai, where for a ∈ Ai and b ∈ Aj,

    µ∑Ak (a, b) ={µAi(a, b) if i = j0 otherwise.

    b. If each Ai has a weighting k•i then∏i Ai has a weighting l

    • given byl(a1,...,an) = ka11 · · · kann . If each Ai has Möbius inversion then so does∏i Ai, with

    µ∏Ai((a1, . . . , an), (b1, . . . , bn)) = µA1(a1, b1) · · ·µAn(an, bn).

    Thinking of R(A) as a matrix algebra (as described after Definition 1.1),the part of (a) concerning Möbius inversion merely says that the inverse of ablock sum of matrices is the block sum of the inverses.

    Documenta Mathematica 13 (2008) 21–49

  • The Euler Characteristic of a Category 31

    To every Set-valued functor X there is assigned a ‘category of elements’E (X). (See the Appendix for a review of definitions.) This is also true offunctors X taking values in Cat, the category of small categories and functors,even when X is only a weak or ‘pseudo’ functor. We say that a Set- or Cat-valued functor X is finite if E (X) is finite. When the domain category A isfinite, this just means that each set or category X(a) is finite.

    Lemma 1.14 Let A be a finite category and X : A - Cat a finite weakfunctor. Suppose that we have weightings on A and on each X(a), all writtenk•. Then there is a weighting on E (X) defined by k(a,x) = kakx (a ∈ A,x ∈ X(a)).

    Proof Let a ∈ A and x ∈ X(a). Then

    (b,y)∈E(X)ζ((a, x), (b, y))kbky =

    b

    f∈A(a,b)

    y∈X(b)ζ((X(f))x, y)ky

    kb

    =∑

    b

    ζ(a, b)kb = 1.

    This result will be used to show how Euler characteristic behaves withrespect to fibrations.

    2 Euler characteristic

    In this section, the Euler characteristic of a category is defined and its basicproperties are established. The definition is justified by a series of propositionsshowing its compatibility with the Euler characteristics of other types of object:graphs, topological spaces, and orbifolds. There follows a brief discussion ofthe Lefschetz number of an endofunctor.

    Lemma 2.1 Let A be a finite category, k• a weighting on A, and k• a coweight-ing on A. Then

    ∑a k

    a =∑

    a ka.

    Proof

    b

    kb =∑

    b

    (∑

    a

    kaζ(a, b)

    )kb =

    a

    ka

    (∑

    b

    ζ(a, b)kb

    )=∑

    a

    ka.

    If A admits a weighting but no coweighting then∑

    a ka may depend on

    the weighting k• chosen: see Example 4.8 of [BL].

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  • 32 Tom Leinster

    Definition 2.2 A finite category A has Euler characteristic if it admits botha weighting and a coweighting. Its Euler characteristic is then

    χ(A) =∑

    a

    ka =∑

    a

    ka ∈ Q

    for any weighting k• and coweighting k•.

    With the Gauss–Bonnet Theorem in mind, one might think of weight asan analogue of curvature: summed over the whole structure, it yields the Eulercharacteristic.

    Any category A with Möbius inversion has Euler characteristic, χ(A) =∑a,b µ(a, b), as in the Introduction.

    Examples 2.3 a. If A is a finite discrete category then χ(A) = |ob A|.

    b. If M is a finite monoid then χ(M) = 1/|M |. (We continue to viewmonoids as one-object categories.) When M is a group, this can beunderstood as follows: M acts freely on the contractible space EM , whichhas Euler characteristic 1; one would therefore expect the quotient spaceBM to have Euler characteristic 1/|M |. (Compare [Wl] and [Co].)

    c. By Corollary 1.5, a finite poset A has Euler characteristic∑n≥0(−1)ncn ∈ Z, where cn is the number of chains in A of length

    n. (See [Pu], [Fo], [R] and [Fa] for connections with poset homology,and §4 for further comparisons with the Rota theory.) More generally,the results of §1 lead to formulas for the Euler characteristic of any fi-nite category that either has no non-trivial idempotents or admits anepi-mono factorization system.

    For example, let A be a category with no non-trivial idempotents. LetB be a skeleton of A, that is, a full subcategory containing exactly oneobject from each isomorphism class of A. Theorem 1.4 tells us that B hasMöbius inversion and gives us a formula for its Möbius function, hencefor its Euler characteristic. Proposition 2.4(b) below then implies that Ahas Euler characteristic, equal to that of B.

    d. By 1.11(c) and its dual, if A has Euler characteristic and either an initialor a terminal object then χ(A) = 1; moreover, if A has both an initial anda terminal object then it does have Euler characteristic. This applies, forinstance, to the category C of 1.11(e). Hence having Möbius inversionis a strictly stronger property than having Euler characteristic, even forskeletal categories.

    e. Euler characteristic is not invariant under Morita equivalence. Recallthat categories A and B are Morita equivalent if their presheaf categories[Aop,Set] and [Bop,Set] are equivalent; see [Bo], for instance. Equiva-lent categories are Morita equivalent, but not conversely. For instance,

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  • The Euler Characteristic of a Category 33

    take A to be the two-element monoid consisting of the identity and anidempotent, and B to be the category generated by objects and arrows

    b1i-�s

    b2

    subject to si = 1. Then A and B are Morita equivalent but not equivalent.Moreover, their Euler characteristics are different: χ(A) = 1/2 by (b), butχ(B) = 1 by (d).

    Clearly χ(Aop) = χ(A), one side being defined when the other is. Thenext few propositions set out further basic properties of Euler characteristic.

    Proposition 2.4 Let A and B be finite categories.

    a. If there is an adjunction A -� B and both A and B have Euler charac-teristic then χ(A) = χ(B).

    b. If A ≃ B then A has Euler characteristic if and only if B does, and inthat case χ(A) = χ(B).

    In (a), it may be that one category has Euler characteristic but the otherdoes not: consider, for instance, the unique functor from the category Aof 1.11(d) to the terminal category.

    Proof

    a. Suppose that AF-�G

    B with F ⊣ G. Then ζ(F (a), b) = ζ(a,G(b)) for alla ∈ A, b ∈ B; write ζ(a, b) for their common value. Take a coweighting k•on A and a weighting k• on B. Then

    ∑a ka =

    ∑b k

    b by the same proofas that of Lemma 2.1.

    b. The first statement follows from Lemma 1.12 and its dual, and the secondfrom (a). �

    Example 2.5 If B is a category with an initial or a terminal object thenχ(AB) = χ(A) for all A, provided that both Euler characteristics exist. In-deed, if 0 is initial in B then evaluation at 0 is right adjoint to the diagonalfunctor A - AB.

    Proposition 2.6 Let n ≥ 0 and let A1, . . . ,An be finite categories that allhave Euler characteristic. Then

    ∑i Ai and

    ∏i Ai have Euler characteristic,

    with

    χ

    (∑

    i

    Ai

    )=∑

    i

    χ(Ai), χ

    (∏

    i

    Ai

    )=∏

    i

    χ(Ai).

    Proof Follows from Lemma 1.13. �

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  • 34 Tom Leinster

    Example 2.7 Let A be a finite groupoid. Choose one object ai from eachconnected-component of A, and write Gi for the automorphism group of ai.Then A ≃ ∑iGi, so by 2.3(b), 2.4(b) and 2.6, we have χ(A) =

    ∑i 1/|Gi|.

    This is what Baez and Dolan call the cardinality of the groupoid A [BD].

    One might also ask whether χ(AB) = χ(A)χ(B). By 2.3(d), 2.5 and 2.6, theanswer is yes if every connected-component of B has an initial or a terminalobject (and all the Euler characteristics exist). But in general the answer isno: for instance, take A to be the 2-object discrete category and B to be thecategory of 3.4(b). See also Propp [Pr2], Speed [Sp], and §5, 6 of Rota [R].

    An important property of topological Euler characteristic is its behaviourwith respect to fibre bundles (or more generally, fibrations). Take a spaceA with connected-components A1, . . . , An, take a fibre bundle E over A, andwrite Xi for the fibre in the ith component. Then under suitable hypotheses,χ(E) =

    ∑i χ(Ai)χ(Xi).

    There is an analogy between topological fibrations and categorical fibra-tions, which are functors satisfying a certain condition. (In this discussion Iwill use ‘fibration’ to mean what is usually called an opfibration; the differ-ence is inessential.) The crucial property of fibrations of categories is that forany category A, the fibrations with codomain A correspond naturally to theweak functors A - Cat. Given a fibration P : E - A, define a functorX : A - Cat by taking X(a), for each a ∈ A, to be the fibre over a: thesubcategory of E whose objects e are those satisfying P (e) = a and whosearrows f are those satisfying P (f) = 1a. Conversely, given a weak functorX : A - Cat, the corresponding fibration is the category of elements E (X)together with the projection functor to A. For details, see [Bo], for instance.

    The formula for the Euler characteristic of a fibre bundle has a categoricalanalogue. Since in general the fibres of a fibration over A vary within eachconnected-component of A, the formula for categories is more complicated. Westate the result in terms of Cat-valued functors rather than fibrations; theproof follows from Lemma 1.14.

    Proposition 2.8 Let A be a finite category and X : A - Cat a finite weakfunctor. Let k• be a weighting on A and suppose that E (X) and each X(a)have Euler characteristic. Then

    χ(E (X)) =∑

    a

    kaχ(X(a)).

    Examples 2.9 a. When X is a finite Set-valued functor, χ(E (X)) =∑a k

    a|X(a)|. For example, let M be a finite monoid. A finite functorX : M - Set is a finite set S with a left M -action. Following [BD],we write E (X) as S//M , the lax quotient of S by M . (Its objects arethe elements of S, and the arrows s - s′ are the elements m ∈ Msatisfying ms = s′.) Then χ(S//M) = |S|/|M |.

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  • The Euler Characteristic of a Category 35

    b. Define a sequence (Sn)n≥−1 of categories inductively as follows. S−1 isempty. Let L be the category of 1.11(a); define X : L - Cat byX(a) = Sn−1 and X(b1) = X(b2) = 1 (the terminal category); putSn = E (X). Then explicitly, Sn is the poset

    c0 - c1 - · · · - cn

    d0 -

    -

    d1 -

    -

    - -· · · -

    -

    dn.

    -

    (If we take the usual expression of the topological n-sphere Sn as a CW-complex with two cells in each dimension ≤ n then Sn is the set of cellsordered by inclusion; Sn is the classifying space of Sn.)

    Each Sn is a poset, so has Euler characteristic. By Proposition 2.8,

    χ(Sn) = −χ(Sn−1) + 2χ(1) = 2− χ(Sn−1)

    for all n ≥ 0; also χ(S−1) = 0. Hence χ(Sn) = 1 + (−1)n.

    The next three propositions show how the Euler characteristics of varioustypes of structure are compatible with that of categories.

    First, Euler characteristic of categories extends Euler characteristic ofgraphs. More precisely, let G = (G1

    -- G0) be a directed graph, whereG1 is the set of edges and G0 the set of vertices. We will show that if F (G)is the free category on G then χ(F (G)) = |G0| − |G1|. This only makes senseif F (G) is finite, which is the case if and only if G is finite and circuit-free;then F (G) is also circuit-free. (A directed graph is circuit-free if it contains nocircuits of non-zero length, and a category is circuit-free if every circuit consistsentirely of identities.)

    Proposition 2.10 Let G be a finite circuit-free directed graph. Then χ(F (G))is defined and equal to |G0| − |G1|.

    Proof Given a, b ∈ G0, write ζG(a, b) for the number of edges from a to bin G. Then ζF (G) =

    ∑n≥0 ζ

    nG in R(F (G)), the sum being finite since G is

    circuit-free. Hence µF (G) = δ − ζG, and the result follows. �

    This suggests that in the present context, it is more fruitful to view agraph as a special category (via F ) than a category as a graph with structure.Compare the comments after Definition 1.1.

    The second result compares the Euler characteristics of categories andtopological spaces. We show that under suitable hypotheses, χ(BA) = χ(A),where BA is the classifying space of a category A (that is, the geometric re-alization of its nerve NA). To ensure that BA has Euler characteristic, weassume that NA contains only finitely many nondegenerate simplices; then

    χ(BA) =∑

    n≥0(−1)n|{nondegenerate n-simplices in NA}|.

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  • 36 Tom Leinster

    An n-simplex in NA is just an n-path in A, and is nondegenerate in the senseof simplicial sets if and only if it is nondegenerate as a path, so A must containonly finitely many nondegenerate paths. This is the case if and only if A iscircuit-free, if and only if A is skeletal and contains no endomorphisms exceptidentities. So by Corollary 1.5, we have:

    Proposition 2.11 Let A be a finite skeletal category containing no endomor-phisms except identities. Then χ(BA) is defined and equal to χ(A). �

    For the final compatibility result, consider the following schematic dia-grams:

    {triangulated manifolds}

    {posets}?

    χ- Z

    χ

    -

    {triangulated orbifolds}

    {categories}?

    χ- Q.

    χ

    -

    On the left, we start with a compact manifold M equipped with a finite trian-gulation. As shown in §3.8 of [St], the topological Euler characteristic of M isequal to the Euler characteristic of the poset of simplices in the triangulation,ordered by inclusion. We generalize this result from manifolds to orbifolds,which entails replacing posets by categories and Z by Q.

    Let M be a compact orbifold equipped with a finite triangulation.(See [MP] for definitions.) The simplices in the triangulation form a posetP , and if p ∈ P is a d-dimensional simplex then ↓ p = {q ∈ P | q ≤ p} isisomorphic to the poset Pd+1 of nonempty subsets of {1, . . . , d+1}, with p ∈↓pcorresponding to {1, . . . , d + 1} ∈ Pd+1. Every p ∈ P has a stabilizer groupG(p), and

    χ(M) =∑

    p∈P(−1)dim p/|G(p)|.

    On the other hand, the groups G(p) fit together to form a complex of finitegroups on P op, that is, a weak functor G : P op - Cat taking values in finitegroups (regarded as one-object categories) and injective homomorphisms; see §3of [M]. This gives a finite category E (G). For example, when M is a manifold,each group G(p) is trivial and E (G) ∼= P .

    The following result is joint with Ieke Moerdijk.

    Proposition 2.12 Let M be a compact orbifold equipped with a finite trian-gulation. Let G be the resulting complex of groups. Then χ(E (G)) is definedand equal to χ(M).

    Proof Every arrow in E (G) is monic, so by Theorem 1.4, E (G) has Eulercharacteristic. Moreover, P is a finite poset, so has a unique coweighting k•,and χ(E (G)) =

    ∑p kp/|G(p)| by the dual of Proposition 2.8.

    The coweight of p in P is equal to the coweight of p in ↓p ∼= Pd+1, whered = dim p. The unique coweighting k• on Pd+1 is given by kJ = (−1)|J|−1, sokp = (−1)(d+1)−1 = (−1)dim p. The result follows. �

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  • The Euler Characteristic of a Category 37

    We now turn to the theory of Lefschetz number. Let F : A - A be anendofunctor of a category A. The category FixF has as objects the (strict)fixed points of F , that is, the objects a ∈ A such that F (a) = a; a map a -b in FixF is a map f : a - b in A such that F (f) = f .

    Definition 2.13 Let F be an endofunctor of a finite category. Its Lefschetznumber L(F ) is χ(FixF ), when this exists.

    The Lefschetz number is, then, the sum of the (co)weights of the fixedpoints. This is analogous to the standard Lefschetz fixed point formula,(co)weight playing the role of index. The following results further justify thedefinition.

    Proposition 2.14 Let A be a finite category.

    a. L(1A) = χ(A), one side being defined if and only if the other is.

    b. If B is another finite category and AF-�G

    B are functors then L(GF ) =

    L(FG), one side being defined if and only if the other is.

    c. Let F : A - A and write BF : BA - BA for the induced map onthe classifying space of A. If A is skeletal and contains no endomorphismsexcept identities then L(F ) = L(BF ), with both sides defined.

    In the special case that A is a poset, part (c) is Theorem 1.1 of [BB].

    Proof For (a) and (b), just note that Fix 1A ∼= A and FixGF ∼= FixFG.For (c), recall from the proof of Proposition 2.11 that NA has only finitelymany nondegenerate simplices; then

    L(BF ) =∑

    n≥0(−1)n|{nondegenerate n-simplices in NA fixed by NF}|

    =∑

    n≥0(−1)n|{nondegenerate n-paths in FixF}|

    = L(F ),

    using Corollary 1.5 in the last step. �

    An algebra for an endofunctor F of A is an object a ∈ A equipped witha map h : F (a) - a. With the evident structure-preserving morphisms,algebras for F form a category AlgF . There is a dual notion of coalgebra(where now h : a - F (a)), giving a category CoalgF .

    Proposition 2.15 Let F be an endofunctor of a finite skeletal category Acontaining no endomorphisms except identities. Then χ(AlgF ) = L(F ) =χ(CoalgF ), with all three terms defined.

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  • 38 Tom Leinster

    Proof First observe that A is circuit-free. Now, the inclusion FixF -AlgF has a right adjoint R: given an algebra (a, h), circuit-freeness impliesthat FN (a) is a fixed point for all sufficiently large N , and R(a, h) = FN (a).The Euler characteristics of AlgF and FixF exist