## Documenta Mathematica

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### Deutschen Mathematiker-Vereinigung

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### undet 1996

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ver¨offentlicht Forschungsarbeiten aus allen mathematischen Gebieten und wird in traditioneller Weise referiert. Es wird indiziert durch Mathematical Reviews, Science Citation Index Expanded, Zentralblatt f¨ur Mathematik.

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Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, publishes research manuscripts out of all mathematical fields and is refereed in the traditional manner. It is indexed in Mathematical Reviews, Science Citation Index Expanded, Zentralblatt f¨ur Mathematik.

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Alexander S. Merkurjev, Los Angeles merkurev@math.ucla.edu Anil Nerode, Ithaca anil@math.cornell.edu

Thomas Peternell, Bayreuth Thomas.Peternell@uni-bayreuth.de Eric Todd Quinto, Medford Todd.Quinto@tufts.edu

Takeshi Saito, Tokyo t-saito@ms.u-tokyo.ac.jp Stefan Schwede, Bonn schwede@math.uni-bonn.de Heinz Siedentop, M¨unchen (LMU) h.s@lmu.de

Wolfgang Soergel, Freiburg soergel@mathematik.uni-freiburg.de G¨unter M. Ziegler, Berlin (TU) ziegler@math.tu-berlin.de

ISSN 1431-0635 (Print), ISSN 1431-0643 (Internet)

SPARC

Leading Edge

Band 13, 2008 V. Gritsenko, K. Hulek and G. K. Sankaran

Hirzebruch-Mumford Proportionality and Locally Symmetric Varieties of Orthogonal

Type 1–19

Tom Leinster

The Euler Characteristic of a Category 21–49 Sergey Morozov

Essential Spectrum of Multiparticle

Brown–Ravenhall Operators in External Field 51–79 A. Vishik, K. Zainoulline

Motivic Splitting Lemma 81–96

J¨org Winkelmann

On Tameness and Growth Conditions 97–101 V. Bach, J. Hoppe, D. Lundholm

Dynamical Symmetries in

Supersymmetric Matrix 103–116

J¨urgen Ritter and Alfred Weiss

Equivariant Iwasawa Theory: An Example 117–129 Guido Kings

Degeneration of Polylogarithms and Special Values

ofL-Functions for Totally Real Fields 131–159 Alex Kumjian, David Pask, Aidan Sims

C∗-Algebras Associated to Coverings of k-Graphs 161–205 Alex Postnikov, Victor Reiner, Lauren Williams

Faces of Generalized Permutohedra 207–273 Denis Perrot

Secondary Invariants for Frechet Algebras

and Quasihomomorphisms 275–363

Paul S. Muhly and Baruch Solel Schur Class Operator Functions

and Automorphisms of Hardy Algebras 365–411 Viorel Costeanu

On the 2-Typical De Rham-Witt Complex 413–452 Ciro Ciliberto, Gerard van der Geer

K-Theory of Log-Schemes I 505–551 Eric A. Carlen and Elliott H. Lieb

Brascamp-Lieb Inequalities

for Non-Commutative Integration 553–584 Gereon Quick

Profinite Homotopy Theory 585–612

F. D´eglise

Around the Gysin Triangle II. 613–675

Matthias K¨unzer

Comparison of Spectral Sequences

Involving Bifunctors 677–737

Goro Shimura

Arithmetic of Hermitian Forms 739–774

Goro Shimura

The Critical Values of Certain Dirichlet Series 775–794 Christian Ausoni, Bjørn Ian Dundas and John Rognes

Divisibility of the Dirac Magnetic Monopole

as a Two-Vector Bundle over the Three-Sphere 795–801 Timothy Logvinenko

Natural G-Constellation Families 803–823 Eva Viehmann

The Global Structure of Moduli Spaces

### 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 there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19. In this paper we obtain the ﬁrst results in this direction. In particular the modular variety deﬁned by the orthogonal group of the even unimodular lattice of signature (2,8m+ 2) is of general type ifm≥5.

2000 Mathematics Subject Classiﬁcation: 14J15, 11F55

Keywords and Phrases: Locally symmetric variety; modular form; Hirzebruch-Mumford proportionality

1 Modular varieties of orthogonal type

LetLbe an integral indeﬁnite lattice of signature (2, n) and (,) the associated
bilinear form. By _{D}L we denote a connected component of the homogeneous

type IV complex domain of dimensionn

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) that
leaves _{D}L invariant. Any subgroup Γ of O+(L) of ﬁnite index determines a

modular variety

FL(Γ) = Γ\ DL.

For some special lattices L and subgroups Γ < O+(L) one obtains in this
way the moduli spaces of polarised abelian or Kummer surfaces (n = 3, see
[GH]), the moduli space of Enriques surfaces (n = 10, see [BHPV]), and the
moduli spaces of polarised or lattice-polarised K3 surfaces (0 < n _{≤} 19, see
[Nik1, Dol]). Other interesting modular varieties of orthogonal type include
the period domains of irreducible symplectic manifolds: see [GHS3].

It is natural to ask about the birational type of _{F}L(Γ). For many classical

moduli 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 we obtain the ﬁrst results in this direction. We determine the Kodaira dimension of many quasi-projective varieties associated with two series of even lattices. To explain what these varieties are, we ﬁrst introduce the stable orthogonal group O(L) of a nondegenerate even latticee L. This is deﬁned (see [Nik2] for more 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 we

write eΓ = Γ_{∩}O(L). Note that ife Lis unimodular then O(L) = O(L).e

The ﬁrst series of varieties we want to study, which we call the modular varieties ofunimodular type, is

FII(m)= O

+_{(II}

2,8m+2)\DII2,8m+2. (1)

FII(m) is of dimension 8m+ 2 and arises from the even unimodular lattice of

signature (2,8m+ 2)

II2,8m+2= 2U⊕mE8(−1),

where U denotes the hyperbolic plane and E8(−1) is the negative deﬁnite lattice associated to the root system E8. The varietyFII(2) is the moduli space

of elliptically ﬁbred K3 surfaces with a section (see e.g. [CM, Section 2]). The casem= 3 is of particular interest: it arises in the context of the fake Monster Lie algebra [B1].

The second series, which we call the modular varieties ofK3 type, is

F2(md)=Oe

+

(L(_{2}m_{d}))_{\D}_{L}(m)

2d . (2)

F2(md)is of dimension 8m+ 3 and arises from the lattice

L(_{2}m_{d})= 2U_{⊕}mE8(−1)⊕ h−2di,

where_{h−}2d_{i}denotes a lattice generated by a vector of square_{−}2d.

The ﬁrst three members of the series _{F}_{2}(m_{d}) have interpretations as moduli
spaces. _{F}_{2}(2)_{d} is the moduli space of polarisedK3 surfaces of degree 2d. For
m= 1 the 11-dimensional variety_{F}_{2}(1)_{d} is the moduli space of lattice-polarised
K3 surfaces, where the polarisation is deﬁned by the hyperbolic lattice _{h}2d_{i ⊕}
E8(−1) (see [Nik1, Dol]). Form= 0 anddprime the 3-foldF2(0)d is the moduli

Theorem 1.1 The modular varieties of unimodular andK3type are varieties of general type ifmanddare sufficiently large. More precisely:

(i) If m_{≥}5 then the modular varieties_{F}_{II}(m) and_{F}_{2}(m_{d})(for anyd_{≥}1) are
of general type.

(ii) Form= 4the varietiesF2(4)d are of general type ifd≥3and d6= 4. (iii) Form= 3the varietiesF2(3)d are of general type ifd≥1346. (iv) Form= 1the varietiesF2(1)d are of general type ifd≥1537488.

Remark. The methods of this paper are also applicable ifm= 2. Using them, one can show that the moduli spaceF2(2)d of polarised K3 surfaces of degree 2d

is of general type ifd≥231000. This case was studied in [GHS2], where, using a diﬀerent method involving special pull-backs of the Borcherds automorphic form Φ12on the domainDII2,26, we proved thatF2(2)d is of general type ifd >61

or d= 46, 50, 54, 57, 58, 60.

The methods of [GHS2] do not appear to be applicable in the other cases
studied here. Instead, the proof of Theorem 1.1 depends on the existence of a
good toroidal compactiﬁcation of_{F}L(Γ), which was proved in [GHS2], and on

the exact formula for the Hirzebruch-Mumford volume of the orthogonal group found in [GHS1].

We shall construct pluricanonical forms on a suitable compactiﬁcation of the
modular variety _{F}L(Γ) by means of modular forms. Let Γ < O+(L) be a

subgroup 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 weightk and characterχ: Γ _{→}C∗ _{with}

respect to the groupΓis a holomorphic function

F :D•

L→C which has the two properties

F(tz) = t−k_{F(z)}

∀t_{∈}C∗_{,}

F(g(z)) = χ(g)F(z) _{∀}g_{∈}Γ.

The space of modular forms is denoted byMk(Γ, χ). The space of cusp forms,

i.e. modular forms vanishing on the boundary of the Baily–Borel
compactiﬁ-cation of Γ_{\D}L, is denoted by Sk(Γ, χ). We can reformulate the deﬁnition of

modular forms in geometric terms. LetF _{∈}Mkn(Γ,detk) be a modular form,

wherenis the dimension of_{D}L. Then

F(dZ)k _{∈}H0(_{F}L(Γ)◦,Ω⊗k),

where dZ is a holomorphic volume form on _{D}L, Ω is the sheaf of germs of

canonical n-forms on _{F}L(Γ) and FL(Γ)◦ is the open smooth part of FL(Γ)

The main question in the proof of Theorem 1.1 is how to extend the form
F(dZ)k _{to}

FL(Γ) and to a suitable toroidal compactiﬁcationFL(Γ)tor. There

are three possible kinds of obstruction to this, which we call (as in [GHS2])
elliptic, reﬂective and cusp obstructions. Elliptic obstructions arise if_{F}L(Γ)tor

has non-canonical singularities arising from ﬁxed loci of the action of the group
Γ. Reﬂective obstructions arise because the projection π is branched along
divisors whose general point is smooth in_{F}L(Γ). Cusp obstructions arise when

we extend the form fromFL(Γ) toFL(Γ)tor.

The problem of elliptic obstructions was solved forn_{≥}9 in [GHS2].

Theorem 1.3 ([GHS2, Theorem 2.1]) Let L be a lattice of signature (2, n)

withn≥9, and letΓ<O+(L)be a subgroup of finite index. Then there exists
a toroidal compactificationFL(Γ)tor ofFL(Γ) = Γ\DLsuch thatFL(Γ)tor has
canonical singularities and there are no branch divisors in the boundary. The
branch divisors in_{F}L(Γ)arise from the fixed divisors of reflections.

Reﬂective obstructions, that is branch divisors, are a special problem related to the orthogonal group. They do not appear in the case of moduli spaces of polarised abelian varieties of dimension greater than 2, where the modular group is the symplectic group. There are no quasi-reﬂections in the symplectic group even for g= 3.

The branch divisor is deﬁned by special reﬂective vectors in the latticeL. This
description is given in _{§}2. To estimate the reﬂective obstructions we use the
Hirzebruch-Mumford proportionality principle and the exact formula for the
Hirzebruch-Mumford volume of the orthogonal group found in [GHS1]. We do
the 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 form
trick (see [G2] and [GHS2]).

2 The branch divisors

To estimate the obstruction to extending pluricanonical forms to a smooth
projective model of _{F}L(Oe

+

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

π:_{D}L→ FL(Oe

+

(L)) =Oe+(L)_{\ D}L. (3)

According to [GHS2, Corollary 2.13] these divisors are deﬁned by reﬂections σr∈O+(L), where

σr(l) =l−2(l, r)

(r, r)r,

coming from vectorsr ∈L with r2 _{<}_{0 that are}_{stably reflective}_{: by this we}
mean thatris primitive andσr or−σris inOe

+

(L). By a (k)-vector fork_{∈}Z

Let D be the exponent of the ﬁnite abelian group AL and let the divisor

div(r) of r _{∈} L be the positive generator of the ideal (l, L). We note that
r∗ _{=}_{r/}_{div(r) is a primitive vector in} _{L}∨_{. In [GHS2, Propositions 3.1–3.2] we}

proved the following.

Lemma 2.1 LetLbe an even integral lattice of signature(2, n). Ifσr∈Oe

+ (L)

thenr2 _{=}_{−}_{2}_{. If}_{−}_{σ}

r∈Oe

+

(L), then r2_{=}_{−}_{2D} _{and}_{div(r) =}_{D}_{≡}_{1} _{mod}_{2}_{or}
r2_{=}_{−}_{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 sublatticeM of an even latticeLthe groupO(Me )can be considered as a subgroup ofO(L)e .

Proof. LetM⊥ _{be the orthogonal complement of}_{M} _{in} _{L. We have as usual}

M_{⊕}M⊥_{⊂}L_{⊂}L∨_{⊂}M∨_{⊕}(M⊥)∨.

We can extendg _{∈}O(Me ) to M _{⊕}M⊥ _{by putting} _{g}_{|}

M⊥ ≡id. It is clear that

g _{∈} O(Me _{⊕}M⊥_{). For any} _{l}∨ _{∈} _{L}∨ _{we have} _{g(l}∨_{)} _{∈} _{l}∨_{+ (M} _{⊕}_{M}⊥_{). In}

particular,g(l)∈Lfor anyl∈Landg∈O(L).e ✷ We can describe the components of the branch locus in terms of homogeneous domains. Forra stably reﬂective vector inLwe put

Hr={[w]∈P(L⊗C)|(w, r) = 0},

and let N be the union of all hyperplane sections Hr∩ DL over all stably

reﬂective vectorsr.

Proposition 2.3 Letr_{∈}Lbe a stably reflective vector: suppose thatrand

L do not satisfy D = 4, r2 _{=}

−4, div(r) = 2. Let Kr be the orthogonal
complement of r in L. Then the associated component π(Hr∩ DL) of the
branch locus_{N} is of the formOe+(Kr)\DKr.

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

ΓKr ={ϕ∈Oe

+

(L)|ϕ(Kr) =Kr}. (4)

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

(Lemma 2.2) preserves the spinor norm (see [GHS1, §3.1]), because Kr has

signature (2, n_{−}1) and soOe+(Kr) becomes a subgroup ofOe

+ (L). Therefore the image of ΓKr containsOe

+

(Kr) for any r. Now we prove that

this image coincides with Oe+(Kr) for allr, except perhaps ifD = 4,r2=−4

Let us consider the inclusions

hri ⊕Kr⊂L⊂L∨⊂ hri∨⊕Kr∨.

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

|detKr|=|detL| · |r

2_{|}

div(r)2 and [L:hri ⊕Kr] =

|r2_{|}

div(r) = 1 or 2.

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

+

(Kr). Let us

assume that the index is equal to 2. In this case the lattice _{h}r_{i}∨ is generated
byr∨_{=}_{−}_{r/(r, r) =}_{r}∗_{/2, where}_{r}∗_{=}_{r/}_{div(r) is a primitive vector in}_{L}∨_{. In}

particular r∨ _{represents a non-trivial class in} _{h}_{r}_{i}∨

⊕K∨

r modulo L∨. Let us

takek∨_{∈}_{K}∨

r such thatk∨6∈L∨. Thenk∨+r∨∈L∨ and

ϕ(k∨)_{−}k∨_{≡}r∨_{−}ϕ(r∨) modL.
We note that ifϕ∈ΓKr thenϕ(r) =±r. Hence

ϕ(k∨)_{−}k∨_{≡}

(

0 modL ifϕ(r) =r r∗ modL ifϕ(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}_{K}

r (Kr =Kr∨∩L), except

possibly ifD= 4, r2_{=}

−4, div(r) = 2. ✷

The groupOe+(L) acts on_{N}. We need to estimate the number of components
of Oe+(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 latticeII2,8m+2any primitive vectorrhas div(r) = 1.
Consequentlyris stably reﬂective if and only ifr2_{=}

−2.

For L(_{2}m_{d}) the reﬂections and the corresponding branch divisors arise in two
diﬀerent ways, according to Lemma 2.1. We shall classify the orbits of such
vectors.

Proposition 2.4 Supposedis a positive integer.

(i) Any two (_{−}2)-vectors in the lattice II2,8m+2 are equivalent modulo
O+(II2,8m+2), and the orthogonal complement of a(−2)-vectorr is

iso-metric to

K_{II}(m)=U_{⊕}mE8(−1)⊕ h2i.

(ii) There is oneOe+(L(_{2}m_{d}))-orbit of(_{−}2)-vectorsrinL(_{2}m_{d})withdiv(r) = 1. If

d_{≡}1mod4then there is a second orbit of(_{−}2)-vectors, withdiv(r) = 2.
The orthogonal complement of a(_{−}2)-vectorrinL(_{2}m_{d})is isometric to

ifdiv(r) = 1, and to

N_{2}(m_{d})=U⊕mE8(−1)⊕

1 2

1−d

2 1

,

ifdiv(r) = 2.

(iii) The orthogonal complement of a(−2d)-vectorrin L(2md) is isometric to

II2,8m+2= 2U⊕mE8(−1)

ifdiv(r) = 2d, and to

K_{2}(m)=U_{⊕}mE8(−1)⊕ h2i⊕ h−2i or T2,8m+2=U⊕U(2)⊕mE8(−1)

ifdiv(r) =d.

(iv) Suppose d > 1. The number of O(Le (_{2}m_{d}))-orbits of (−2d)-vectors with

div(r) = 2dis2ρ(d)_{. The number of}_{O(L}_{e} (m)

2d )-orbits of(−2d)-vectors with

div(r) =dis

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.}

Hereρ(d)is the number of prime divisors ofd.

Proof. If the lattice L contains two hyperbolic planes then according to the
well-known result of Eichler (see [E,§10]) theOe+(L)-orbit of a primitive vector
l ∈ L is completely deﬁned by two invariants: by its length (l, l) and by its
imagel∗_{+}_{L}_{in the discriminant group}_{A}

L, wherel∗=l/div(l).

i) Ifuis a primitive vector of an even unimodular latticeII2,8m+2then div(u) =
1 and there is only one O(II2,8m+2)-orbit of (−2)-vectors. Therefore we can
takerto be a (_{−}2)-vector inU, and the form of the orthogonal complement is
obvious.

ii) In the lattice L(_{2}m_{d}) we ﬁx a generator h of its _{h−}2d_{i}-part. Then for any
r_{∈}L(_{2}m_{d})we can write r=u+xh, whereu_{∈}II2,8m+2 and x∈Z. It is clear
that div(r) dividesr2_{. If} _{f}_{|}_{div(r), where}_{f} _{= 2,} _{d} _{or 2d, then the vector} _{u}
is also divisible by f. Therefore the (−2)-vectors form two possible orbits of
vectors with divisor equal to 1 or 2. Ifr2 _{=}_{−}_{2 and div(r) = 2 then} _{u}_{= 2u}

0
withu0∈2U⊕mE8(−1) and we see that in this cased≡1 mod 4. This gives
us two diﬀerent orbits for such d. In both cases we can ﬁnd a (−2)-vector r
in the sublattice U_{⊕ h−}2d_{i}. Elementary calculation gives us the orthogonal
complement ofr.

iv) To ﬁnd the number of orbits of (_{−}2d)-vectors we have to consider two cases.
a) Let div(r) = 2d. Thenr= 2du+xh andr∗ _{≡}_{(x/2d)h} _{mod}_{L, where}_{u}_{∈}

II2,8m+2andxis modulo 2d. Moreover (r, r) = 4d2(u, u)−x22d=−2d. Thus x2

≡1 mod 4d. This congruence has 2ρ(d)_{solutions modulo 2d. For any such}
x mod 2dwe can ﬁnd a vectoruin 2U_{⊕}mE8(−1) with (u, u) = (x2−1)/2d.
Thenr= 2du+xhis primitive (becauseuis not divisible by any divisor ofx)
and (r, r) =_{−}2d.

b) Let div(r) =d. Thenr=du+xh, whereuis primitive,r∗≡(x/d)h modL
and xis modulo d. We have (r∗, r∗)≡ −2x2_{/d} _{mod 2}_{Z} _{and}_{x}2 _{≡}_{1 mod} _{d.}
For any solution modulodwe can ﬁnd as aboveu∈2U⊕mE8(−1) such that
r=du+xhis primitive and (r, r) =−2d. It is easy to see that the number of
solutions{x modd|x2_{≡}_{1 mod}_{d}_{}} _{is as stated.} _{✷}

Remark. To calculate the number of the branch divisors arising from vectors
r with r2 _{=}

−2done has to divide the corresponding number of orbits found
in Proposition 2.4(iv) by 2 if d > 2. This is because _{±}r determine diﬀerent
orbits but the same branch divisor. For d= 2 the proof shows that there is
one 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) with
two hyperbolic planes. We choose a primitive isotropic vector c1 in L. This
vector determines a 0-dimensional cusp and a tube realisation of the domain

DL. The tube domain (see the deﬁnition ofH(L1) below) is a complexiﬁcation
of the positive cone of the hyperbolic latticeL1=c⊥1/c1. If div(c1) = 1 we call
this cuspstandard (as above, by [E] there is only one standard cusp). In this
caseL1=U⊕L0. In [GHS2,§4] we proved that any 1-dimensional boundary
component of Oe+(L)_{\ D}L contains the standard 0-dimensional cusp if every

isotropic (with respect to the discriminant form: see [Nik2, _{§}1.3]) subgroup of
AL is cyclic.

Let us ﬁx a 1-dimensional cusp by choosing two copies of U in L. (One has
to add to c1 a primitive isotropic vector c2 ∈ L1 with div(c2) = 1). Then
L=U _{⊕}L1=U⊕(U⊕L0) and the construction of the tube domain may be
written down simply in coordinates. We have

H(L1) =Hn={Z = (zn, . . . , z1)∈H1×Cn−2×H1; (ImZ,ImZ)L1 >0},

where Z_{∈}L1⊗Cand (zn−1, . . . , z2)∈L0⊗C. (We representZ as a column
vector.) An isomorphism between_{H}n andDL 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_{H}nis given by the usual fractional linear

deﬁned byg_{∈}O+(L_{⊗}R) is equal to det(g)j(g, Z)−n_{, where}_{j(g, Z) is the last}

((n+ 2)-nd) coordinate ofg p(Z)_{∈ D}L. Using this we deﬁne the automorphic

factor

J: O+(L_{⊗}R)_{× H}n+2 → C∗
(g, Z) _{7→} (detg)−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} _{H}

n, for Γ a subgroup

of ﬁnite index of O+(L), ifF g(Z)=J(g, Z)k_{F}_{(Z}_{) for any}_{g} _{∈}_{Γ; in other}

words ifF _{∈}Mnk(Γ,detk) (see Deﬁnition 1.2). To prove Theorem 1.1 we need

cusp forms of weight smaller than the dimension of the corresponding modular variety.

Proposition 3.1 For unimodular type, cusp forms of weight12 + 4m exist: that is

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

ForK3type we have the bounds

dimS11+4m(Oe

+

(L(_{2}m_{d}))) > 0 if d >1;
dimS10+4m(Oe

+

(L(_{2}m_{d}))) > 0 if d_{≥}1;
dimS7+4m(Oe

+

(L(_{2}m_{d}))) > 0 if d_{≥}4;
dimS6+4m(Oe

+

(L(_{2}m_{d}))) > 0 if d= 3 or d≥5;
dimS5+4m(Oe

+

(L(2md))) > 0 if d= 5 or d≥7;

dimS2+4m(Oe

+

(L(_{2}m_{d}))) > 0 if d >180.

Proof. For anyF(Z)∈Mk(Oe

+

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

F(Z) =f0(z1) +

X

m≥1

fm(z1;z2, . . . zn−1) exp(2πimzn).

A lifting construction of modular formsF(Z)∈Mk(Oe

+

The dimension ofJk,1(L0) depends only on the discriminant form and the rank ofL0(see [G2, Lemma 2.4]). In particular, for the special cases ofL=II2,8m+2 andL=L(2md)we have

J_{k}cusp_{+4}_{m,}_{1}(mE8(−1))∼=Sk(SL2(Z))
and

J_{k}cusp_{+4}_{m,}_{1}(mE8(−1)⊕ h−2di)∼=J_{k,d}cusp,

where J_{k,d}cusp is the space of the usual Jacobi cusp forms in two variables of
weightkand indexd(see [EZ]) andSk(SL2(Z)) is the space of weightkcusp
forms 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 Oe

+

(L(_{2}m_{d})) with trivial character. The fact
that we get a cusp form was proved in [G2] for maximal lattices, i.e., if dis
square-free. In [GHS2, _{§}4] we extended this to all lattices L for which the
isotropic subgroups of the discriminant AL are all cyclic, which is true in all

cases considered here.

To prove the unimodular type case of Proposition 3.1 we can take the Jacobi
form corresponding to the cusp form ∆12(τ). Using the Jacobi lifting
construc-tion we obtain a cusp form of weight 12 + 4mwith respect to O+(II2,8m+2).
For the K3 type case we need the dimension formula for the space of Jacobi
cusp formsJ_{k,d}cusp (see [EZ]). For a positive integerl one sets

{l_{}}12=

⌊ l

12⌋ ifl6≡2 mod 12

⌊l

12⌋ −1 ifl≡2 mod 12. Then ifk >2 is even

dimJk,dcusp= d X

j=0

{k+ 2j}12−

j2 4d

,

and ifkis odd

dimJ_{k,d}cusp=

d−1

X

j=1

{k_{−}1 + 2j_{}}12−

_{j}2
4d

.

This gives the bounds claimed. For k = 2, using the results of [SZ] one can
also calculate dimJ_{2}cusp_{,d} : there is an extra term,_{⌈}σ0(d)/2⌉, whereσ0(d) denotes
the number of divisors ofd. This gives dimJ_{2}cusp_{,d} >0 ifd >180 and for some

smaller values ofd. ✷

4 Kodaira dimension results

particular all singularities are canonical and there is no ramiﬁcation divisor
which is contained in the boundary. Then the canonical divisor (as aQ-divisor)
is given by K_{F}L(Γ)tor =nM −V −D whereM is the line bundle of modular

forms of weight 1,nis the dimension of_{F}L(Γ),V is the branch locus (which is

given by reﬂections) andDis the boundary. Hence in order to constructk-fold pluricanonical forms we must ﬁnd modular forms of weight kn which vanish of orderkalong the branch divisor and the boundary. This also suﬃces since

FL(Γ)tor has canonical singularities.

Our strategy is the following. For Γ_{⊆}Oe+(L) we choose a cusp formFa ∈Sa(Γ)

of low weight a, i.e. astrictly less than the dimension. Then we consider
ele-mentsF _{∈}Fk

aMk(n−a)(Γ,detk): for simplicity we assume thatkis even. Such
anF vanishes to order at least kon the boundary of any toroidal
compactiﬁ-cation. Hence if dZ is the volume element on_{D}L deﬁned in§3 it follows that

F(dZ)k _{extends as a} _{k-fold pluricanonical form to the general point of every}

boundary component of_{F}L(Γ)tor. Since we have chosen the toroidal

compact-iﬁcation so that all singularities are canonical and that there is no ramcompact-iﬁcation
divisor which is contained in the boundary the only obstructions to extending
F(dZ)k _{to a smooth projective model are the reﬂective obstructions, coming}

from the ramiﬁcation divisor of the quotient map π: DL → FL(Γ) studied in

§2.

LetDK be an irreducible component of this ramiﬁcation divisor. Recall from

Proposition 2.3 thatDK =P(K⊗C)∩ DL where K =Kr is the orthogonal

complement of a stably reﬂective vector r. For the lattices chosen in The-orem 1.1 all irreducible components of the ramiﬁcation divisor are given in Proposition 2.4.

Proposition 4.1 We assume that k is even and that the dimensionn _{≥} 9.
For Γ _{⊆} Oe+(L), the obstruction to extending forms F(dZ)k _{where} _{F}

∈

Fk

aMk(n−a)(Γ)toFL(Γ)tor lies in the space

B=M

K

B(K) =M

K k/2−1

M

ν=0

Mk(n−a)+2ν(Γ∩Oe

+ (K)),

where the direct sum is taken over all irreducible components_{D}K of the
rami-fication divisor of the quotient mapπ: _{D}L→ F(Γ).

Proof. Letσ∈Γ be plus or minus a reﬂection whose ﬁxed point locus isDK.

We can extend the diﬀerential form provided thatF vanishes of orderk along every irreducible componentDK of the ramiﬁcation divisor.

IfFa vanishes alongDK thenK gives no restriction on the second factor of the

modular formF.

Now let _{{}w = 0_{}} be a local equation for _{D}K. Then σ∗(w) = −w (this is

independent of whether σ or _{−}σ is the reﬂection). For every modular form
F _{∈} Mk(Γ) of even weight we have F σ(z) = F(z). This implies that if

We denote byM2b(Γ)(−νDK) the space of modular forms of weight 2b which

vanish of order at least ν along _{D}K. Since the weight is even we have

M2b(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 containsOe

+

(K) as sub-group of Oe+(L)(with equality in almost all cases), so we may replace Γ∩ΓK

by Γ_{∩}Oe+(K). In this way we obtain an exact sequence

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

(K)),

where the last map is given by F7→F/w2ν_{. This gives the result.} _{✷}

Now we proceed with the proof of Theorem 1.1.

Let L be a lattice of signature (2, n) and Γ <Oe+(L): recall that k is even. According to Proposition 4.1 we can ﬁnd pluricanonical diﬀerential forms on

FL(Γ)tor if

CB(Γ) = dimMk(n−a)(Γ)−

X

K

dimB(K)>0, (6)

where summation is taken over all irreducible components of the ramiﬁcation
divisor (see the remark at the end of _{§}2). It now remains to estimate the
dimension ofB(K) for each of the ﬁnitely many components of the ramiﬁcation
locus in the cases we are interested in, namely Γ = O+(II2,8m+2) and Γ =

e

O+(L(2md)).

According to the Hirzebruch-Mumford proportionality principle

dimMk(Γ) =

2

n!volHM(Γ)k

n_{+}_{O(k}n−1_{).}

The exact formula for the Hirzebruch-Mumford volume volHM for any

indef-inite orthogonal group was obtained in [GHS1]. It depends mainly on the determinant and on the local densities of the lattice L. Here we simply quote the 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)-vectorr. According to Proposition 2.4
the orthogonal complement Kr is KII(m). This lattice diﬀers from the lattice

L(2m), whose Hirzebruch-Mumford volume was calculated in [GHS1,§3.5], only by one copy of the hyperbolic plane. Therefore

volHMOe

+

(L(_{2}m)) = (B8m+4/(8m+ 4)) volHMOe

+

(K_{II}(m)),
and hence, for evenk,

dimMk(Oe

+

(KII(m))) =

21−4m

(8m+ 1)!·

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

where the Bi are Bernoulli numbers. Assume thatm≥3. Let us take a cusp

form

F _{∈}S4m+12(O+(II2,8m+2))

from Proposition 3.1. In this case the dimension of the obstruction spaceB of Proposition 4.1 for the pluricanonical forms of orderk= 2k1is given by

k1_{X}−1

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

Using this estimate we easily obtain that (7) holds ifm_{≥}5. Therefore we have
proved Theorem 1.1 for the latticeII2,8m+2.

Next we consider the latticeL(_{2}m_{d})of K3 type. For this lattice the branch divisor
ofF2(md)is calculated in Proposition 2.4. It contains one or two (ifd≡1 mod 4)

components deﬁned by (−2)-vectors and some number of components deﬁned by (−2d)-vectors. To estimate the obstruction constantCB(Γ) in (6) we use the

dimension formulae for the space of modular forms with respect to the group

e

O+(M), where M is one of the following lattices from Proposition 2.4: L(_{2}m_{d})
(the main group);K_{2}(m_{d})andN_{2}(m_{d})(the (_{−}2)-obstruction);M2,8m+2,K2(m)and
T2,8m+2 (the (−2d)-obstruction). The corresponding dimension formulae were
found in [GHS1] (see §§3.5, 3.6.1–3.6.2, 3.3 and 3.4). The branch divisor of
(−2d)-type appears only ifd >1. We note that
Therefore in order to estimateCB(Γ) we can assume that all (−2d)-divisors

where

B(−2)= dimB(K2(md)) + dimB(N

(m) 2d ),

B(−2d)= 2ρ(d)−1(dimB(M2,8m+2) + 2hddimB(T2,8m+2))

andB(K) is the obstruction space from Proposition 4.1. Byhd we denote the

sumδ0,d(8)−δ2,d(4), where d(n) isdmodnand δ is the Kronecker delta (see Proposition 2.4 and the remark following it).

For any lattice considered above

All terms in (10) contain a common factor. First

dimM2k1w(Oe From [GHS1, (16)] it follows that

volHM(Oe

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

where

Hereχ0,f denotes the principal Dirichlet character modulof.

We note that |PK(n)|<1 and|PN(n)|<1 for anyd. We conclude that

B(−2)< Cm,dk1,wEw(8m+ 3)b(−2)

whereb(−2)= 28m+3−1.

The (−2d)-contribution is calculated according to [GHS1, 3.3–3.4]. We note that Oe+(T2,8m+2) is a subgroup ofOe

As a result we see that that the obstruction constantCB(Oe

For m _{≤} 3 we choose Fa of weight 4m+ 2, i.e. we take w = 4m+ 1. Such

a cusp form exists ifd >180 according to Proposition 3.1. For such dwe see that β(−2d)<1. Then the obstruction constantCB(Oe

+

(L(_{2}m_{d}))) is positive if

B_{B}4_{8}m_{m}+2_{+2}

(1 +_{4m}1_{+ 1})8m+3(28m+3+ 2)<

√

d.

This inequality gives us the bound ondin 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 branch
divisor forms the most important reﬂective obstruction to the extension of the

e

O+(L(_{2}m_{d}))-invariant diﬀerential forms to a smooth compact model ofF2(md). Let

us consider the double covering _{SF}(_{2}m_{d}) of _{F}_{2}(m_{d}) for d >1 determined by the
special orthogonal group:

SF(2md)=SOf

+

(L(_{2}m_{d}))_{\D}_{L}(m)
2d → F

(m) 2d .

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

The variety SF2(2)d can be interpreted as the moduli space of K3 surfaces of

degree 2d with spin structure: see [GHS2, _{§}5]. The three-fold _{SF}_{2}(0)_{d} is the
moduli space of (1, t)-polarised abelian surfaces.

Theorem 4.2 The variety_{SF}(_{2}m_{d})is of general type for anyd >1ifm_{≥}3. If

m= 2thenSF2(2)d is of general type ifd≥3. Ifm= 1thenSF

(1)

2d is of general type if d= 5 ord≥7.

Proof. The case m = 2 is [GHS2, Theorem 5.1], and the result form _{≥}5 is
immediate from Theorem 1.1. For m = 1, 3 and 4 we can prove more than
what follows from Theorem 1.1.

The branch divisor ofSF(2md) is deﬁned by the reﬂections in vectorsr ∈L

(m) 2d

such that _{−}σr ∈ SOf

+

(L(_{2}m_{d})), because the rank of L(_{2}m_{d}) is odd. Therefore
r2_{=}

−2d, by Proposition 2.1.
If F _{∈} M2k+1(SOf

+

(L(_{2}m_{d}))) is a modular form (note that the character det is
trivial),d >1 andz∈ D•

L(m)_{2d} is such that (z, r) = 0, then

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

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 dim_{SF}(_{2}m_{d}) = 8m+ 3. By
Proposition 3.1 there exists a cusp form F11+4m ∈ S11+4m(SOf

+

(L(_{2}m_{d}))). For
m _{≥} 3 we have that 11 + 4m < 8m+ 3. Therefore the diﬀerential forms
Fk

11+4mF(4m−8)k(dZ)k, for arbitrary F(4m−8)k ∈ M(4m−8)k(SOf

+

(L(_{2}m_{d}))),
ex-tend to the toroidal compactiﬁcation of SF(2md) constructed in Theorem 1.3.

This proves the casesm≥3 of the theorem.

For the casem= 1 we use a cusp form of weight 9 with respect to SOf+(L(1)_{2}_{d})

constructed in Proposition 3.1. ✷

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

Proposition 4.3 The spaces_{SF}8(1)andSF
(1)

12 have non-negative Kodaira

di-mension.

Proof. By Proposition 3.1 there are cusp forms of weight 11 for SOf+(L(1)_{8} )
and SOf+(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}_{8}(1)

and_{SF}_{12}(1). ✷

These varieties may perhaps have intermediate Kodaira dimension as it seems possible that a reﬂective modular form of canonical weight exists forL(1)8 and L(1)12.

In [GHS2] we used pull-backs of the Borcherds modular form Φ12onDII2,26 to

show that many moduli spaces of K3 surfaces are of general type. We can also use Borcherds products to prove results in the opposite direction.

Theorem 4.4 The Kodaira dimension ofFII(m) is−∞form= 0,1and2.
Proof. For m = 0 we can see immediately that the quotient is rational: a
straightforward calculation gives that_{F}_{II}(0) = Γ_{\}H1×H1whereH1 is the usual
upper half plane and Γ is the degree 2 extension of SL(2,Z)×SL(2,Z) by the
involution which interchanges the two factors. Compactifying this, we obtain
the projective planeP2.

Form= 1, 2 we argue diﬀerently. There are modular forms similar to Φ12for the even unimodular lattices II2,10 and II2,18. They are Borcherds products Φ252and Φ127of weights 252 and 127 respectively, deﬁned by the automorphic functions

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

where q = exp(2πiτ) and ∆(τ) andE4(τ) are the Ramanujan delta function
and the Eisenstein series of weight 4 (see [B1]). The divisors of Φ252 and Φ127
coincide with the branch divisors of_{F}_{II}(1)and_{F}_{II}(2) deﬁned by the (_{−}2)-vectors.
Moreover Φ252and Φ127each vanishes with order one along the respective
divi-sor. Therefore ifF10k(dZ)k (orF18k(dZ)k) deﬁnes a pluricanonical diﬀerential

form on a smooth model of a toroidal compactiﬁcation of FII(1) or F

(2)

II , then

F10k (or F18k) is divisible by Φk252 (or Φk127), since F10k or F18k must vanish

to order at leastkalong the branch divisor. This is not possible, because the quotient would be a holomorphic modular form of negative weight. ✷

We have already remarked that the space_{F}_{II}(2) is the moduli space of K3
sur-faces with an elliptic ﬁbration with a section. Using the Weierstrass equations
it is then clear that this moduli space is unirational (such K3 surfaces are
parametrised by a linear system in the weighted projective spaceP(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 Grenzgebiete4. 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 of bounded domains. Ann. Math84(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. Die Grundlehren der mathematischen Wissenschaften 63. Springer-Verlag, Berlin 1952.

[EZ] M. Eichler, D. Zagier,The theory of Jacobi forms. Progress in Math-ematics55. Birkh¨auser, Boston 1985.

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

[G2] V. Gritsenko,Modular forms and moduli spaces of abelian and K3 sur-faces.Algebra i Analiz6, 65–102; English translation in St. Petersburg Math. 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–Mumford volume for the orthogonal group and applications. Doc. Math. 12

(2007), 215–241.

[GHS2] V. Gritsenko, K. Hulek, G.K. Sankaran, The Kodaira dimension of the 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 K¨ahler K3 surfaces.

Trudy Moskov. Mat. Obshch. 38(1979), 75–137. English translation in Trans. Mosc. Math. Soc.2, 71-135 (1980).

[Nik2] V.V. Nikulin, Integral symmetric bilinear forms and some of their applications. Izv. Akad. Nauk USSR43(1979), 105 - 167 (Russian); English translation in Math. USSR, Izvestiia14(1980), 103–166. [SZ] N.P. Skoruppa, D. ZagierJacobi forms and a certain space of modular

forms. Invent. Math.94(1988), 113–146.

V. A. Gritsenko Universit´e Lille 1

Laboratoire Paul Painlev´e

F-59655 Villeneuve d’Ascq, Cedex France

valery.gritsenko@math.univ-lille1.fr

K. Hulek

Institut f¨ur Algebraische Geometrie Leibniz Universit¨at Hannover D-30060 Hannover

Germany

hulek@math.uni-hannover.de

G. K. Sankaran

Department of Mathematical Sciences University of Bath

Bath BA2 7AY England

### 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 and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Rota’s M¨obius inversion from posets to categories.

2000 Mathematics Subject Classiﬁcation: 18F99, 55U99, 05C50, 57N65.

Keywords and Phrases: Euler characteristic, ﬁnite category, inclusion-exclusion, M¨obius inversion, cardinality of colimit.

Introduction

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

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

1_{Partially supported by a Nuffield Foundation award NUF-NAL 04 and an EPSRC }

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 under

Euclidean 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}whenn= 2. Thus, up to scalar multipli-cation, Euler characteristic is the unique dimensionless measure on polyconvex sets.

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

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

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

A topological point of view is that a category can be understood through its classifying space. This is formed by starting with one 0-cell for each object, then gluing 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 topological perspective 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 space functor, as with other topological invariants: given a category A, deﬁne χ(A) as the Euler characteristic of the classifying space BA. The trouble with this is that the Euler characteristic of BAis not always deﬁned. Below we give a deﬁnition 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 of situations when it does not. It is a rational number, not necessarily an integer. A version of the deﬁnition can be given very succinctly. LetAbe a ﬁnite category; 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, assuming

that Z is invertible. Thenχ(A) is the sum of the entries ofM. Of course, the reader remains to be convinced that this deﬁnition is the right one.

hom-set has at most one element: the objects are the elements of the pohom-set, and
there is an arrow a ✲ b if and only if a _{≤} b.) This leads, among other
things, to a ‘representation formula’: given any functor known to be a sum of
representables, the formula tells us the representation explicitly. This in turn
can be used to solve enumeration problems, in the spirit of Rota’s paper.

However, the main application of this generalized M¨obius inversion is to
the theory of the Euler characteristic of a category (_{§}2). We actually use a
dif-ferent deﬁnition than the one just given, equivalent to it whenZ 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 showing
that it enjoys the properties that the name would lead one to expect (behaviour
with 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 Lefschetz
number.

The technology of M¨obius inversion and weights also solves another
prob-lem: what is the cardinality of a colimit? For example, the union of a family
of sets and the quotient of a set by a free action of a group are both examples
of 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 M¨obius inversion
for posets. As we will see repeatedly, many are not truly order-theoretic: they
are facts about categories in general. In particular, important theorems in
Rota’s original work [R] generalize from posets to categories (_{§}4).

(The body of work on M¨obius inversion in ﬁnite lattices is not, however, so ripe for generalization: a poset is a lattice just when the corresponding category has products, but a ﬁnite category cannot have products unless it is, in fact, a lattice.)

Other authors have considered diﬀerent notions of M¨obius 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 posets and 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 discussed in §4. Further approaches, not discussed here, were taken by D¨ur [D] and L¨uck [L¨u].

In the case of groupoids, our Euler characteristic of categories agrees with Baez and Dolan’s groupoid cardinality [BD]. The cardinality of the groupoid of ﬁnite sets and bijections is e = 2.718. . ., and there are connections to ex-ponential generating functions and the species of Joyal [J, BLL]. Par´e has a deﬁnition 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 Lefschetz number of an endofunctor.

is nothing new: see for instance Wall [Wl], Bass [Ba] and Cohen [Co], and the
discussion 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, where for simplicity our categories are required to be ﬁnite and the coeﬃcients are required to lie in the ring of rational numbers. Rather than asking, as below, ‘does this category have Euler characteristic (inQ)?’, we should perhaps ask ‘in what 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, Ieke Moerdijk, Urs Schreiber, Ivan Smith and Stephen Watson for inspiration and useful discussions. I am grateful to the Centre de Recerca Matem`atica, Barcelona, for their hospitality. I also thank the very helpful referee.

1 M¨obius inversion

We consider a ﬁnite categoryA, writing obAfor its set of objects and, whena andb are objects,A(a, b) for the set of maps fromatob.

Definition 1.1 We denote by R(A) the Q-algebra of functions obA _{×}

obA ✲ Q, with pointwise addition and scalar multiplication, multiplica-tion deﬁned by

(θφ)(a, c) =X

b∈A

θ(a, b)φ(b, c)

(θ, φ_{∈}R(A),a, c_{∈}A), and the Kroneckerδas 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 M¨obius inversion; its inverse
µA=µ=ζ−1 _{is the}_{M¨}_{obius function} _{of}_{A}_{.}

If a total ordering is chosen on the n objects of A then R(A) can be
regarded as the algebra of n_{×}n matrices over Q. The deﬁning equations of
the M¨obius function are

X

b

µ(a, b)ζ(b, c) =δ(a, c) =X

b

ζ(a, b)µ(b, c)

for alla, c∈A. By ﬁnite-dimensionality,µζ=δ if and only ifζµ=δ.

The deﬁnitions above could be made for directed graphs rather than cat-egories, since they do not refer to composition. However, this generality seems to be inappropriate. For example, the deﬁnition of M¨obius inversion will lead to a deﬁnition of Euler characteristic, and if we use graphs rather than categories then we obtain something other than ‘vertices minus edges’. Proposition 2.10 clariﬁes this point.

Examples 1.2 a. Any ﬁnite posetAhas M¨obius inversion; this special case was investigated by Rota [R] and others. We may compute µ(a, c) by induction on the number of elements betweenaandc:

µ(a, c) =δ(a, c)− X

b:a≤b<c

µ(a, b).

In particular,µ(a, c) = 0 unlessa≤c, and µ(a, a) = 1 for alla. See also Theorem 1.4 and Corollary 1.5.

b. LetM be a ﬁnite monoid, regarded as a category with unique object⋆.
(The arrows of the category are the elements of the monoid, and
composi-tion in the category is multiplicacomposi-tion in the monoid.) Thenζ(⋆, ⋆) =_{|}M_{|},
soµ(⋆, ⋆) = 1/|M|.

c. LetN≥0. WriteDinj

N for the category with objects 0, . . . , N whose maps

a ✲ b are the order-preserving injections {1, . . . , a} ✲ {1, . . . , b}.
Thenζ(a, b) = _{a}b, and it is easily checked that µ(a, b) = (−1)b−a b

a

. If
we use surjections instead of injections thenζ(a, b) = a_{b}−_{−}_{1}1andµ(a, b) =
(_{−}1)a−b a−1

b−1

.

A category with M¨obius inversion must be skeletal (isomorphic objects must be equal), for otherwise the matrix of ζ would have two identical rows. The property of having M¨obius inversion is not, therefore, invariant under equivalence of categories.

In general we cannot hope to just spot the M¨obius function of a category. In 1.3–1.7 we make tools for computing M¨obius functions. These cover large classes of categories, although not every ﬁnite skeletal category has M¨obius inversion (1.11(d), (e)).

Letn_{≥}0, letA be a category or a directed graph, and leta, b_{∈}A. An
n-path fromatob is a diagram

a=a0

f1_{✲}

a1

f2_{✲}

· · · fn✲ an=b (1)

inA. It is acircuit ifa=b, and (whenAis a category)nondegenerate if nofi

is an identity.

Lemma 1.3 The following conditions on a finite categoryAare equivalent: a. every idempotent inAis an identity

b. every endomorphism inAis an automorphism c. every circuit inAconsists entirely of isomorphisms.

Proof (a) _{⇒} (b) follows from the fact that if f is an element of a ﬁnite
monoid then some positive power off is idempotent. The other implications

Theorem 1.4 LetAbe a finite skeletal category in which the only idempotents are identities. Then Ahas M¨obius inversion given by

µ(a, b) =X(_{−}1)n_{/}

|Aut(a0)| · · · |Aut(an)|

where Aut(a) is the automorphism group of a_{∈}A and the sum runs over all

n_{≥}0 and paths (1) for which a0, . . . , an are all distinct.

Proof First observe that for a path (1) inA, ifa06=a16=· · · 6=an then the

ais are all distinct. Indeed, if 0 ≤i < j ≤n and ai = aj then the sub-path

running from ai to aj is a circuit, so by Lemma 1.3, fi+1 is an isomorphism, and by skeletality,ai=ai+1.

Now leta, c_{∈}Aand deﬁneµby the formula above. We have

X

b∈A

µ(a, b)ζ(b, c) = µ(a, c)ζ(c, c) + X

b:b6=c

µ(a, b)ζ(b, c)

= _{|}Aut(c)_{|}

µ(a, c) +

X

b:b6=c, g∈A(b,c)

µ(a, b)/_{|}Aut(c)_{|}

= |Aut(c)|nµ(a, c) +X(−1)n/|Aut(a0)| · · · |Aut(an)||Aut(c)| o

,

where the last sum is over alln_{≥}0 and paths

a=a0

f1_{✲}

· · · fn✲ an=b
g_{✲}

c

such thata0 6=· · · 6=an 6=c. By deﬁnition ofµ, the term in braces collapses

to 0 if a6=c and to 1/|Aut(a)| ifa=c. Hence P_{b}µ(a, b)ζ(b, c) =δ(a, c), as

required.

Corollary 1.5 LetAbe a finite skeletal category in which the only endomor-phisms are identities. ThenAhas M¨obius inversion given by

µ(a, b) =X

n≥0

(_{−}1)n_{|{}nondegenerate n-paths fromatob_{}| ∈}Z.

WhenAis 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 a
classE of epimorphisms inAand a classMof monomorphisms inA, satisfying
axioms [FK]. The axioms imply that every mapf inAcan be expressed asme
for 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}

Theorem 1.6 LetAbe a finite skeletal category with an epi-mono factorization
system(_{E},_{M}). Then Ahas M¨obius inversion given by

µ(a, b) =X(_{−}1)n/_{|}Aut(a0)| · · · |Aut(an)|

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

Proof The objects ofAand the arrows inE determine a subcategory of A, also denoted E; it satisﬁes the hypotheses of Theorem 1.4 and therefore has M¨obius inversion. The same is true ofM.

Any element α_{∈}QobA _{=}Q

a∈AQgives rise to an element of R(A), also
denotedαand deﬁned by α(a, b) =δ(a, b)α(b). This deﬁnes a
multiplication-preserving map fromQobA_{to}_{R(}_{A}_{), where the multiplication on}_{Q}obA_{is }
coor-dinatewise. We have elements _{|}Aut_{|}and 1/_{|}Aut_{|}ofQobA_{, where, for instance,}

|Aut_{|}(a) =_{|}Aut(a)_{|}.

By the essentially unique factorization property,ζA=ζE·_{|}Aut1 |·ζM. Hence

Ahas M¨obius functionµA=µ_{M}_{· |}Aut_{| ·}µ_{E}. Theorem 1.4 applied to µ_{M}and

µ_{E} then gives the formula claimed.

Example 1.7 LetN _{≥} 0 and writeFN for the full subcategory ofSet with

objects 1, . . . , N, where ndenotes a (chosen) n-element set. Let _{E} be the set
of surjections in FN and Mthe set of injections; then (E,M) is an epi-mono

factorization system. Theorem 1.6 gives a formula for the inverse of the matrix
(ij_{)}

i,j. For instance, take N = 3; thenµ(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! = 3/2
1✲ 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. Henceµ(1,2) =_{−}1 + 3/2_{−}3 =

−5/2.

One of the uses of the M¨obius function is to calculate Euler
character-istic (_{§}2). Another is to calculate representations. Speciﬁcally, suppose that
we have a Set-valued functor known to be familially representable, that is, a
coproduct of representables. The Yoneda Lemma tells us that the family of
representing objects is unique (up to isomorphism). But if we have M¨obius
inversion, there is actually a formula for it:

Proposition 1.8 Let A be a finite category with M¨obius inversion and let

X :A ✲ Setbe a functor satisfying

X ∼=X

a

for some natural numbersr(a)(a_{∈}A). Then

r(a) =X

b

|X(b)|µ(b, a)

for all a_{∈}A.

In the ﬁrst formula,Pdenotes coproduct ofSet-valued functors.

Proof Follows from the deﬁnition of M¨obius function.

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

Example 1.9 Aderangement is a permutation without ﬁxed points. We cal-culatedn, the number of derangements ofnletters.

Fix N _{≥} 0. Take the category Dinj_{N} of Example 1.2(c) and the functor
S:Dinj_{N} ✲ Setdeﬁned as follows: S(n) isSn, the underlying set of thenth

symmetric group, and iff _{∈}Dinj

N(m, n) andτ ∈Sm, the induced permutation

Sf(τ)∈Sn acts asτ on the image of f and ﬁxes all other points. Any

permu-tation consists of a derangement together with some ﬁxed points, so there is an isomorphism of sets

Sn∼= X

m

dmDinjN(m, n)

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

dn= X

m

|Sm|µ(m, n) = X

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 full strength of M¨obius invertibility; the following suﬃces.

Definition 1.10 LetA be a ﬁnite category. Aweighting on Ais a function
k• _{: ob}_{A} ✲ _{Q}_{such that for all}_{a}

∈A,

X

b

ζ(a, b)kb_{= 1.}

Acoweighting k• onAis a weighting onAop.

Note thatAhas M¨obius inversion if and only if it has a unique weighting, if and only if it has a unique coweighting; they are given by

ka=X

b

µ(a, b), kb= X

a

Examples 1.11 a. LetLbe the category

b. LetM be a ﬁnite monoid, regarded as a category with unique object⋆.
Again there is a unique weightingk•_{, with}_{k}⋆_{= 1/}

|M_{|}.

c. If Ahas a terminal object 1 thenδ(_{−},1) is a weighting onA.

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

a1 a2

ak and neither p nor q is an identity then

q◦p=fik.

e. A category may certainly have more than one weighting: for instance, ifA is the category consisting of two objects and a single isomorphism between them, a weighting onAis any pair of rational numbers whose sum is 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 the categoryA of the previous example both have a 1-dimensional space of weightings.

In contrast to M¨obius invertibility, the property of admitting at least one weighting is invariant under equivalence:

Proof Let F : A ✲ B be an equivalence. Given a_{∈}A, write Ca for the

number of objects in the isomorphism class ofa. Take a weightingl•_{on}_{B}_{and}

put ka_{= (}P

b:b∼=F(a)lb)/Ca. I claim thatk• is a weighting onA.

To prove this, choose representativesa1, . . . , amof the isomorphism classes

of objects of A; then F(a1), . . . , F(am) are representatives of the isomorphism

classes of objects ofB. Let a′ _{∈}_{A}_{. For any}_{a}_{∈}_{A}_{, the numbers}_{ζ(a}′_{, a) and}_{k}a

depend only on the isomorphism class ofa. Hence

X

a∈A

ζ(a′, a)ka =

m X

i=1

X

a:a∼=ai

ζ(a′, a)ka

=

m X

i=1

Caiζ(a′, ai)kai

=

m X

i=1

X

b:b∼=F(ai)

ζ(a′, ai)lb

= X

b∈B

ζ(F(a′), b)lb

= 1,

as required.

Weightings and M¨obius functions are compatible with sums and products
of categories. We write P_{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≥0and letA1, . . . ,An be finite categories. a. If each Ai has a weighting k•i then

P

iAi has a weighting l• given by

la _{=}_{k}a

i whenever a∈Ai. If each Ai has M¨obius inversion then so does P

iAi, where fora∈Ai andb∈Aj,

µP_{A}_{k}(a, b) =

µAi(a, b) if i=j

0 otherwise.

b. If each Ai has a weighting k•i then QiAi has a weighting l• given by

l(a1,...,an) _{=} _{k}a1

1 · · ·kann . If each Ai has M¨obius inversion then so does Q

iAi, with

µQ_{A}_{i}((a1, . . . , an),(b1, . . . , bn)) =µA1(a1, b1)· · ·µAn(an, bn).

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 of functorsX taking values inCat, the category of small categories and functors, even when X is only a weak or ‘pseudo’ functor. We say that aSet- orCat -valued functor X is finite ifE(X) is ﬁnite. When the domain category Ais ﬁnite, this just means that each set or categoryX(a) is ﬁnite.

Lemma 1.14 Let A be a finite category and X : A ✲ Cat a finite weak functor. Suppose that we have weightings on A and on eachX(a), all written

k•_{. Then there is a weighting on} _{E}_{(X)} _{defined by} _{k}(a,x) _{=} _{k}a_{k}x _{(}_{a} _{∈} _{A}_{,}

x_{∈}X(a)).

Proof Leta∈Aandx∈X(a). Then

X

(b,y)∈E(X)

ζ((a, x),(b, y))kbky = X

b X

f∈A(a,b)

X

y∈X(b)

ζ((X(f))x, y)ky

kb

= X

b

ζ(a, b)kb_{= 1.}

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

2 Euler characteristic

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

Lemma 2.1 LetAbe a finite category,k• _{a weighting on}_{A, and}_{k}

•a

coweight-ing on A. ThenP_{a}ka_{=}P
aka.
Proof

X

b

kb=X

b X

a

kaζ(a, b) !

kb=X

a

ka X

b

ζ(a, b)kb

!

=X

a

ka.

If A admits a weighting but no coweighting thenP_{a}ka _{may depend on}