On the Bogomolov-Miyaoka-Yau inequality for Deligne-Mumford surfaces

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by  Jiun-Cheng Chen

ON THE BOGOMOLOV-MIYAOKA-YAU INEQUALITY FOR DELIGNE-MUMFORD
SURFACES
JIUN-CHENGCHENANDHSIAN-HUATSENG
1
1
ABSTRACT. We discuss a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-
0
Mumfordsurfacesofgeneraltype.
2

n
a
J

8 1. INTRODUCTION
1

] For a smooth complex projective surface S of general type, the Bogomolov-Miyaoka-Yau in-
G
equalityforS reads
A
. (1.1) 3c2(TS) c1(TS)2.
h ≥
t Together with Noether’s inequality, this puts constraints on the topology of surfaces of general
a
m types. Generalizations of (1.1) to singular surfaces and surface pairs have been found, see for
[ example[7], [4, 5]. In thispaper wediscussa generalizationof(1.1)to Deligne-Mumfordstacks.

1 We work over C. Let be a smooth proper Deligne-Mumford C-stack of dimension 2. Let
v X
π : X bethenaturalmaptothecoarsemodulispace. WeassumethatX isaprojectivevariety.
1
X →
8 Since is assumed to be smooth, it has a tangent bundle T . A good theory of Chern classes is
4 X X
availableforDeligne-Mumfordstacks,seeforexample[10], [3]. Weproposethefollowing
3
.
1 Conjecture1.1. Let beasabove. AssumethatthecanonicalbundleK := 2T isnumerically
∨
0 effective, then X X ∧ X
1
1 (1.2) 3c (T ) c (T )2.
v: 2 X ≥ 1 X
i
X Certainly (1.2) takes the same shape as (1.1). In what follows we give evidence for (1.2). In
r Section 2.1 we discuss (1.2) for stacks which non-trivial stack structures at generic points. In
a X
Section2.2weprove(1.2)foraclassofstacks withstackstructuresincodimension1. InSection
X
2.3 weprove(1.2)forGorensteinstacks withisolatedstack points.
X
Acknowledgment. J.-C.CisaGolden-JadeFellowofKendaFoundation,Taiwan. Heissupported
inpartbyNationalScienceCouncilandNationalCenterforTheoreticalSciences, Taiwan. H.-H.T
is supportedin part byNSF grantDMS-1047777.
2. EVIDENCE OF (1.2)
2.1. Codimension 0 stack structure. We examine(1.2) for stacks with non-trivialstack struc-
X
tures at generic points. In this case, is an e´tale gerbe over a stack with trivial generic stack
X
structure, see for example [2, Proposition 4.6]. More precisely, there is a finite group G, a stack
Date:January19,2011.
1
2 JIUN-CHENGCHENANDHSIAN-HUATSENG
with trivial generic stabilizers, and a morphismf : realizing as a G-gerbe over .
′ ′ ′
X X → X X X
Since T = f∗T ′, we see that (1.2) for is equivalent to (1.2) for ′. Therefore it suffices to
X X X X
consideronlythose withstack structuresin codimension 1.
X ≥
2.2. Codimension 1 stack structure. We will verify (1.2) for an example of stack with stack
X
structuresin codimension1.
Let X be a smoothcomplex projectivesurface and D a simplenormal crossing Q-divisorof the
form D = (1 1/r )D with r 2 integers. Let be the natural stack cover of the pair
i − i i i ≥ X
(X,D). ByPconstruction the coarse moduli space of is X. The natural map π : X is an
X X →
isomorphismoutsideπ 1(SuppD),whichiswhere hasnon-trivialstackstructures. Furthermore
−
X
wehavethefollowingformulaforthecanonical bundle:
(2.1) K = π (K +D).
∗ X
X
Wenowexamine(1.2)forthis . By (2.1),
X
c (T )2 = c (K )2 = (K +D)2.
1 1 X
X X
By Gauss-Bonnet theoremforDelignem-Mumfordstacks[8, Corollaire3.44]wehave
c (T ) = χ( ),
2
X X
theEulercharacteristicof asdefinedin[8,Definition3.43](notethatthenotationχorb isusedin
X
[8]). Put
:= π 1(D ), := ( ( )).
Di − i Di◦ Di \ ∪j6=i Di ∩Dj
Then wehave
χ( π 1(SuppD)) = χ( ) χ( ) χ(p).
X \ − X − Di◦ −
Xi p Xi j
∈D ∩D
Similarly,put D = D ( (D D )), wehave
i◦ i \ ∪j6=i i ∩ j
χ(X SuppD) = χ(X) χ(D ) χ(p¯).
\ − i◦ −
Xi p¯ XDi Dj
∈ ∩
Since π 1(SuppD) X SuppD,wehaveχ( π 1(SuppD)) = χ(X SuppD). Equiva-
− −
X \ ≃ \ X \ \
lently,
χ( ) = χ(X) χ(D ) χ(p¯)+ χ( )+ χ(p).
X − i◦ − Di◦
Xi p¯ XDi Dj Xi p Xi j
∈ ∩ ∈D ∩D
Sincethemap D isofdegree1/r andthemap D D isofdegree1/r r , we
Di◦ → i◦ i Di∩Dj → i∩ j i j
have
1 1
χ( ) = χ(D ), χ( ) = χ(D D ).
i i i j i j
D r D ∩D r r ∩
i i j
Thisimpliesthat
(2.2) χ( ) = χ(X) (1 1/r )χ(D )+ (1/r r 1).
X − − i i◦ i j −
Xi p¯ XDi Dj
∈ ∩
By [5, Theorem 8.7], for p¯ D D the local orbifold Euler number of the pair (X,D) at p¯is
i j
∈ ∩
given by e (p¯;X,D) = 1/r r . Together with (2.2) this implies that χ( ) coincides with the
orb i j
X
orbifold Euler number e (X,D) of the pair (X,D), as defined in [5]. Thus if K is numerically
orb
X
effective,the(1.2)holdbecause itisequivalentto [5, Theorem0.1]appliedtothepair(X,D).
ONTHEBOGOMOLOV-MIYAOKA-YAUINEQUALITYFORDELIGNE-MUMFORDSURFACES 3
2.3. Condimension 2 stack structure. Let be a smooth proper Deligne-Mumford C-stack of
X
dimension 2 with isolated stack structures. Suppose that is Gorenstein. Let π : X be
X X →
the natural map to the coarse moduli space X, which we assume to be a projective surface with
canonical singularities. Let p ,p ,...,p be the stacky points. Since is Gorenstein, each p
1 2 k i
∈ X X
has a neighborhood p U of the form U [C2/G ] with G SU(2) a finite subgroup,
i i i i i
∈ ⊂ X ≃ ⊂
identifyingp with[0/G ] [C2/G ].
i i i
∈
Supposefurtherthat K isnumericallyeffective. Weprovethat(1.2)holdsforsuch .
X X
By assumptionwehaveK = π K . Thus
∗ X
X
c (T )2 = c (K )2 = c (K )2.
1 1 x X
X X
We now consider the term c (T ). The first step is to consider χ( ) by using Riemann-Roch
2
X OX
theorem for stacks [8]. We follow [9, Appendix A] for the presentation of the Riemann-Roch
theorem. We have
χ( ) = ch( )Td(T ).
OX Z OX X
I
Here I is the inertia stack of . By our assuXmeption onf , we have the following description of
X X X
I :
X
k
I = (Ip p ).
i i
X X ∪ \
i[=1
Here theterm Ip p istheinertiastack ofp BG withthemaincomponentremoved,namely
i i i i
\ ≃
Ip p BC (g).
i \ i ≃ Gi
[
(g)=(1):conjugacyclassofGi
6
By the definition of the Chern character ch, we have ch( ) = 1 on every component of I .
OX X
Hence
e e
k
(2.3) χ( ) = Td(T ) = Td(T ) + Td(T ) .
OX Z X Z X |X Z X |Ipi\pi
IX f X f Xi=1 Ipi\pi f
NotethatTd(T ) = Td(T ), and weonlyneed itsdegree2 component. Hence
X |X X
f 1
(2.4) Td(T ) = (c (T )+c (T )2).
2 1
Z X |X 12 Z X X
X f X
ThecontributioncomingfromIp p can bealsoevaluated.
i i
\
Lemma 2.4. Let E betheexceptionaldivisorof theminimalresolutionof C2/G . Then
i i
1 1
Td(T ) = (χ(E ) ).
ZIpi\pi f X |Ipi\pi 12 i − |Gi|
ThisLemmaisprovedintheAppendix.
Next, we reinterpret the term χ( ). By definition, χ( ) := ( 1)ldimHl( , ).
Sinceπ = (see e.g. [1, TheoOreXm 2.2.1]),wehaveHlO( X, ) =PHl≥0l(X−, ) and X OX
X X
∗OX O X OX O
(2.5) χ( ) = χ( ).
X
OX O
4 JIUN-CHENGCHENANDHSIAN-HUATSENG
Combining(2.3), (2.4), (2.5), and Lemma2.4, weobtainthefollowingexpressionofc (T ):
2
X
k
(2.6) c (T ) = 12χ( ) c (T )2 (χ(E ) 1/ G ).
2 X 1 i i
Z X O −Z X − − | |
X X Xi=1
Usingthis,weseethatin thepresent situation,(1.2)isequivalentto
k
4 1
(2.7) 12χ( ) c (K )2 + (χ(E ) ).
X 1 X i
O ≥ 3 − G
Xi=1 | i|
On the other hand, it is clear that (2.7) is a special case of [7, Corollary 1.3]. This completes the
proof.
APPENDIX A. PROOF OF LEMMA 2.4
In this Appendix we prove Lemma 2.4. By our assumption on , for g G , the g-action on
i
X ∈
the tangent space T has two eigenvalues ξ and ξ 1, where ξ is a certain root of unity. By the
piX g g− g
definitionofTd(T )wehave
X
1 1
f
(A.1) Td(T ) = .
ZIpi\pi f X |Ipi\pi (g)6=(1):conjXugacyclassofGi |CGi(g)|2−ξg −ξg−1
Wenowevaluate(A.1)usingtheADE classification ofC2/G .
i
A.1. Type A. If C2/G is of typeA , then G Z and theaction on C2 is givenas follows. If
i n 1 i n
weidentifyZ withthegroupofn-th−rootsof1, t≃henan elementξ Z acts onC2 viathematrix
n n
∈
ξ 0
.
(cid:18) 0 ξ−1 (cid:19)
It followsthat (A.1)is givenby
n 1
1 − 1
(A.2) .
n 2 exp(2π√ 1l/n) exp(2π√ 1l/n) 1
Xl=1 − − − − −
By [6, Lemma3.3.2.1],(A.2)is equalto
n2 1 1
− = (n 1/n).
12n 12 −
Since the exceptional divisor of the minimal resolution of C2/Z is a chain of (n 1) copies of
n
−
CP1, itsEulercharacteristicisn. ThisprovestheLemmaintypeA case.
A.2. Type D. If C2/G is of typeD (here n 2), then G is isomorphicto thebinary dihedral
i n+2 i
≥
groupDic . ThegroupDic is oforder 4nandmay bepresentedas follows:
n n
Dic = a,x a2n = 1,x2 = an,x 1ax = a 1 .
n − −
|
(cid:10) (cid:11)
TheactionofDic on C2 isgivenasfollows:
n
exp(π√ 1/n) 0 0 1
(A.3) a − , x .
7→ (cid:18) 0 exp( π√ 1/n) (cid:19) 7→ (cid:18) 1 0 (cid:19)
− − −
ONTHEBOGOMOLOV-MIYAOKA-YAUINEQUALITYFORDELIGNE-MUMFORDSURFACES 5
Anelementcalculationshowsthattheconjugacyclasses ofDic andtheorders oftheircentralizer
n
subgroupsaregivenas follows:
1 , an (orderofcentralizer group = 4n)
{ } { }
(A.4) al,a l ,1 l n 1, (orderofcentralizer group = 2n)
−
{ } ≤ ≤ −
xa,xa3,xa5,...,xa2n 1 , x,xa2,xa4,...,xa2n 2 (orderofcentralizergroup = 4).
− −
{ } { }
Using (A.3) and (A.4) it is easy to identify the contribution from each conjugacy class. It follows
that (A.1)is givenby
n 1
1 − 1 1 1 1
(A.5) + + + .
2n 2 exp(π√ 1k/n) exp(π√ 1k/n) 1 16n 8 8
Xk=1 − − − − −
We need to evaluate the sum n 1 1 . Again by [6, Lemma 3.3.2.1],
k=−1 2 exp(π√ 1k/n) exp(π√ 1k/n)−1
wehave P − − − −
(2n)2 1 2n−1 1
− =
12 2 exp(2π√ 1k/(2n)) exp(2π√ 1k/(2n)) 1
Xk=1 − − − − −
n 1
− 1 1
= +
2 exp(π√ 1k/n) exp(π√ 1k/n) 1 4
Xk=1 − − − − −
n 1
− 1
+ .
2 exp(2π√ 1(n+k)/(2n)) exp(2π√ 1(n+k)/(2n)) 1
Xk=1 − − − − −
Notethat
2 exp(2π√ 1(n+k)/(2n)) exp(2π√ 1(n+k)/(2n)) 1
−
− − − −
=2+exp(π√ 1k/n)+exp(π√ 1k/n) 1
−
− −
=2+2cos(πk/n) = 4cos2(πk/(2n)) = 4sin2((π(k+n)/(2n));
2 exp(π√ 1k/n) exp(π√ 1k/n) 1
−
− − − −
=2 2cos(πk/n) = 4sin2(πk/(2n)).
−
Sincesin(π(k +n)/(2n)) = sin(π(k n)/(2n)),weseethat
− −
n 1
− 1
2 exp(π√ 1k/n) exp(π√ 1k/n) 1
Xk=1 − − − − −
n 1
− 1
= ,
2 exp(2π√ 1(n+k)/(2n)) exp(2π√ 1(n+k)/(2n)) 1
Xk=1 − − − − −
from whichitfollowsthat
n−1 1 1 (2n)2 1
2 + = − .
2 exp(π√ 1k/n) exp(π√ 1k/n) 1 4 12
Xk=1 − − − − −
6 JIUN-CHENGCHENANDHSIAN-HUATSENG
Thisshowsthat
n−1 1 n2 1
= −
2 exp(π√ 1k/n) exp(π√ 1k/n) 1 6
Xk=1 − − − − −
and (A.1)isgivenby
n2 1 1 1 1 1 1
− + + + = (n+3 ).
12n 16n 8 8 12 − 4n
Since the exceptional divisor of the minimal resolution of C2/Dic is a tree of CP1 whose dual
n
graphistheDynkindiagramD ,itsEulercharacteristicisn+3andtheLemmaisprovedinthis
n+2
case.
A.3. Type E. If C2/G is of type E, then there are three possibilities: E ,E ,E . The group G
i 6 7 8 i
is isomorphic to the binary tetrahedral group (for E ), the binary octahedral group (for E ), or the
6 7
binary icosahedral group (for E ). In each case the group and its action on C2 can be explicitly
8
described, and theLemmacan beproved by computing(A.1) using this information. We work out
thedetailsforE and leavetheothertwocases tothereader.
6
In the E case, the group G is isomorphic to the binary tetrahedral group 2T. This group is of
6 i
order24 and itselementscan beidentifiedwiththefollowingquaternionnumbers:
1
( 1 i j k), i, j, k, , 1.
2 ± ± ± ± ± ± ± ±
Thegroup2T has7 conjugacyclasses:
ConjugacyClass (1) ( 1) (i) (1(1+i+j +k))
− 2
Size 1 1 6 4
ConjugacyClass (1(1+i+j k)) (1( 1+i+j +k)) (1( 1+i+j k))
2 − 2 − 2 − −
Size 4 4 4
Theactionof2T onC2 can bedescribedusingthefollowingidentification
x+yi z +wi
x+yi+zj +wk .
7→ (cid:18) z +wi x yi (cid:19)
− −
Nowit isstraightforwardtosee that(A.1)isgivenby
1 1 1 1 1 1 1 1 1 1 1 1 167 1 1
+ + + + + = = (7 ).
242 ( 2) 42 0 62 1 62 1 62 ( 1) 62 ( 1) 288 12 − 24
− − − − − − − − −
Since 7 is the Euler characteristic of the exceptional divisor of the minimal resolution of C2/2T,
theresultfollows.
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DEPARTMENTOFMATHEMATICS,THIRDGENERALBUILDING,NATIONALTSINGHUAUNIVERSITY,NO. 101
SEC. 2 KUANG FU ROAD, HSINCHU, TAIWAN 30043,TAIWAN
E-mailaddress:[email protected], [email protected]
DEPARTMENT OF MATHEMATICS, OHIO STATE UNIVERSITY, 100 MATH TOWER, 231 WEST 18TH AVE.,
COLUMBUS, OH 43210,USA
E-mailaddress:[email protected]