Table Of ContentNonexistence of asymptotic GIT compactification
3 Xiaowei Wang and Chenyang Xu
1
0 January 8, 2013
2
n
a
J
We provide examples of families of (log) smooth canonically polarized varieties, in-
5 cluding smooth weighted pointed curves and smooth hypersurfaces in P3 with large
degree such that the Chow semistable limits under distinct pluricanonical embeddings
]
G do not stabilize.
A
Contents
.
h
t
a 1 Introduction 1
m
2 GIT height 3
[
2.1 GIT height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 2.2 Geometric height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
v 2.3 Height decreases along semistable replacement . . . . . . . . . . . . . . 10
3
7 3 MMP and KSBA-stable family 11
1
0 4 Examples 13
.
2 4.1 Log curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1 4.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2
1
1 Introduction
:
v
i
X We work over the field C of complex numbers. Ever since the geometric invariant
theory (GIT) was reinvented by Mumford, using it to construct the moduli space of
r
a polarizedvarieties(X,L)bytakingthegeometricinvarianttheoryquotientoftheChow
(or Hilbert) semistable points is one of the most important applications. When L is
a (positive) power of the canonical bundle ω , i.e., if X is a canonically polarized
X
manifold, it was shown by Mumford-Gieseker in dimension 1, Gieseker in dimension 2
andDonaldson-Yauinhigherdimensionthat(X,ω )isasymptoticallyChowstable(see
X
[MFK94, Gieseker77, Yau76, Donaldson01]), i.e., given a smooth canonically polarized
variety (X,ω ) we know that there exists an r such that (X,ω⊗r) is Chow stable for
X 0 X
any r≥r .
0
BythegeneralGITtheory[MFK94],weindeedgetaprojectivemodulispacecontain-
inganopenlocusparameterizingsmooth(X,ω⊗r). Tounderstandthiscompactification,
X
weneed tocharacterize singular objects that are includedin theGIT compactification.
SincebythenatureoftheGITconstructionthemembersincludedinthiscompactifica-
tionaprioridependonthemultipler ofthepluri-canonicalembedding. Afundamental
question, asked first by Koll´ar (cf. [VanOpstall04, Problem 3.1]), is to see whether or
not those members in the GIT compactification will eventually stabilize, i.e., whether
Chowsemistablecanonicallypolarizedvarietieswillbethesameforsufficientlydivisible
r like the case of moduli space of curves. However, in this note we show this fails in
general even for very natural examples.
1
Definition1. Apolarizedlogvariety(X,D;L)iscalledasymptoticallylogChowsemistable
(resp. weaklyasymptoticallylogChowstable)ifthecorrespondingChowpointsof(X,D)
under the embeddings given by |L⊗k| is semistable for k ≫ 1 (resp. infinitely many
k∈N). (see Section 2 for more background).
When D=0, we will just write (X,L) for (X,D;L).
Theorem 1. There exist projective flat families (X,D;L) which are families of
(1) smooth weighted pointed curve π : (X,Σn a s ;ω (Σn a s )) → B◦ for any
i=1 i i X i=1 i i
(a1,...,an)∈Qn∩[0,1]n, genus g≥3 satisfying iai > 2gg−−14,
(2) degree m smooth hypersurfaces (X,O(1))→B◦ iPn P3B◦ for any m>30, and
(3) smooth surfaces (X,ω )→B◦ whose fibers satisfy (K2 ,p )=(2,0),
X Xb g
over a punctured smooth curve B◦ =B\{b }, such that for any sufficiently divisible r,
0
the Chow semistable limitsof (X ,D ;L⊗r) forb→b are not weakly asymptotically log
b b b 0
Chow semistable.
The explicit families are given in Section 4. From this result, the fact that we can
use GIT to construct M seems to be completely a coincidence. In fact, Theorem 1
g
follows from thefollowing main technical results of this paper.
Theorem2. Let(X,D)→B beaKSBA-stablefamily(seeDefinition4)overasmooth
curve B. Let r be a positive integer such that r(K +D) is Cartier. Let b ∈ B such
X 0
thatforanyb∈B◦ =B\{b },thefiber(X ,D ;ω[r](rD)| )isasymptoticallylogChow
0 b b X Xb
semistable.
Then for any flat family (Xg,Dg;L) of polarized log varieties over B satisfying
(Xg,Dg;L)×BB◦ ∼=(X,D;ωX[r](rD))×BB◦ (1)
and with the special fiber (Xg,Dg ;L⊗k| ) being log Chow semistable for infinitely
b0 b0 Xb0
many k>0, we have
(Xg,Dg;L)∼=(X,D;ω[r](D)).
X
Asan immediateconsequence,weobtain acriterion totellwhen thereisnoasymp-
totically Chow semistable filling.
Theorem 3. Let us continue the notation as above. If we assume the polarized pair
(X ,D ;ω[r](rD)| )
b0 b0 X Xb0
isnot(weakly)asymptoticallylogChowsemistable. Thenforanyflatfamily(Xg,Dg;L)
satisfying(1)asabove,(Xg,Dg ;L| )isnot(weakly)asymptoticallyChowsemistable.
b0 b0 Xb0
Now we explain our strategy to prove Theorem 2. Let us recall one observation in
[LX11],whichsaysthatgivenafamilyofcanonicallypolarizedvarietiesoverapunctured
curve,theKSBA-stablecompactificationminimizesDFinvariantswhicharethedegrees
of theCM-line bundledefined in [PT10] (see Section 3). The new input we had in this
note is that a similar property holds for GIT stability as well. In fact, by applying the
theoryofheightintroducedin[Wang12](seeSubsection2.1),whichplaysasimilarrole
as the degree of the CM-line bundle but for Chow line bundle, we observed that the
Chow semistable filling minimizes the geometric height (see Subsection 2.3). However,
as the limit of the normalized geometric heights is identical to the DF invariant, we
immediately conclude the two special fibres for this two families must be isomorphic
using theuniquenessof KSBA-stablecompactification.
Finally,toproveTheorem3. Weassumethereisanewfamilywithaasymptotically
logChowsemistablespecialfiberandsatisfies(1). ThenbyTheorem2thecentralfiber
must beisomorphic to (X ,D ;ω[r](rD)| ), contradicting toourassumption that it
b0 b0 X Xb0
is not asymptotically log Chow semistable.
2
In conclusion, the moral of this paper is better being illustrated as the following
diagram
=
KSBA-stability K-stability
mit
' li
asymptotic Chow Stability .
Notations and Conventions See [KM98] for the definitions of discrepancies and
various singularities of log pairs, e.g., log canonical, klt etc.. For a proper morphism
f :Y →X and a Q-divisor D on Y, we denoteR(Y/X,D):= m=0f∗(mD).
Acknowledgement. The authors would like to express our gratitude to J´anos Koll´ar
L
and Baosen Wu for bring this question into our attention, also to J´anos Koll´ar for
providing us the hypersurfaces examples in Subsection 4.2. We thank Yi Hu and Yuji
Odakafortheirveryhelpfulcommentsontheearlyversion. WealsowanttothankProf.
Tian for point out the reference [PT10]. This collaboration started from XW’s visit to
the Mathematics Department of University of Utah in September 2012. He thanks the
faculty memberstherefor thehospitality and theresearch environmenttheyproduced.
CXispartiallysupportedbygrantDMS-1159175,WarnockPresidentialEndowedChair
and the Chinese government grant ’Recruitment Program of Global Experts’.
2 GIT height
In this section, to make our exposition self-contained we first review the theory of
GIT height for a family of GIT problem as introduced in [Wang12]. In particular,
we show GIT height decreases along semistable replacement. Then we will apply these
constructionstostudyChowstablepolarizedvarieties. Attheend,weprovethatamong
all the birational families over B that are isomorphic over an open set U ⊂ B, over
which thefibersareall Chow semistable, afamily with allfiberbeingChow semistable
minimizes theGIT height.
2.1 GIT height
To begin with, let (Z,O (1)) be a projective variety with an action of a reductive
Z
algebraic group G, and the polarization O (1) is a G-linearized ample line bundle on
Z
Z, that is, there is a G-action on thetotal space of O (1) that covers its action on Z.
Z
Definition 2. Let z ∈Z and λ:C∗ →G be a one parameter subgroup. Let
z := limλ(t)·z
0
t→0
Then the λ-weight of z, which will be denoted by wz(λ)∈Z, is the weight of C∗-action
on OZ(1)|z0 ∼=C. A point z is called (semi-)stable with respect to the G-linearization of
OZ(1) if wz(λ)(≥)>0 for all one parameter subgroups λ:C∗ →G.
Afamily GITproblem overasmooth propercurveB consistsof thefollowing data:
• Let (Z,O (1)) be a polarized projective manifold with an action of a connected
Z
reductive algebraic group G˜, where O (1) is very ample and G˜-linearized. We
Z
assume theinduced representation
ρ:G˜ −→GL(H0(Z,O (1))) (2)
Z
satisfies
Imρ⊃C∗·I, (3)
3
where I is theidentity in GL(H0(Z,O (1))). Let
Z
G:=ker(det◦ρ). (4)
Westudythe GIT problem for theG-action on (Z,O (1)).
Z
• Let
G˜ −−−−−→ FrG˜
π
B
y
bea principal G˜-bundleoverB.
Let π :Z :=FrG˜ × Z −→B be the fibration associated to FrG˜. By our assumption
Z G˜
that O (1) is veryample and G˜-linearized, we have
Z
O (1):=FrG˜× O (1) (5)
Z G˜ Z
is π-relative very ample. And for a fixed b∈B, thereis an isomorphism
ιb :(Zb =πZ−1(b),OZ(1)|Zb)∼=(Z,OZ(1)).
Moreover, thereis a bundlemorphism
Z −−−−−→ PE
πZ π
B B
y y
where E = π O (1) = FrG˜ × H0(Z,O (1)) →B. Over Z, there is a natural vector
Z∗ Z G˜ Z
bundleπZ∗E∨(1)=πZ∗E∨⊗OZ(1)=Hom(πZ∗E,OPE(1)|Z).
Definition 3. We define the height H for the family (Z,O (1)) to be the
(Z,OZ(1)) Z
determinant linebundledetπ∗E∨(1)∈Pic(Z),wherePic(Z)isthePicardvariety ofZ.
Z
For any section s of the fibration π:(Z,O (1))→B we define the height of s to be
Z
h (s) := degs∗H =deg(s∗detπ∗E∨(1))
(Z,OZ(1)) (Z,OZ(1)) Z
= (N+1)deg(s∗O (1))−deg(π O (1))
Z Z∗ Z
where N+1 is the rank of E.
Let s be a section of the fibration π :(Z,O (1))→B. It induces a G˜-equivariant
Z Z
Z-valued function of FrG˜ which is, by abuse of notation, still called s, i.e., s:FrG˜ →Z
is a morphism satisfying s(·g) = g−1 ·s(·),∀g ∈ G˜. For any k ≥ 1, suppose that
σ∈H0(Z,O (k))G is a G-invariant section, then it inducesa composite morphism
Z
σ◦s:FrG˜ −s→Z −σ→O (k).
Z
By our assumption (3), for any g ∈ G˜, there is a h ∈ G˜ such that ρ(h) = λI and
h−1·g∈G. These imply
detρ(g)=detρ(h)=λN+1 and ρ (g)·σ=ρ (h)·σ=λkσ
k k
where ρ : G˜ → GL(H0(Z,O (k))) is induced from the linearization of G˜ on O (1).
k Z Z
Hencefor any x∈FrG˜,
σ◦s(x·g)=σ(g−1·s(x))=ρk(g)−1(σ◦s)(x)=λ−kσ◦s(x)∈OZ(k)|s(x) ∼=C
Inparticular, ifk=(N+1)k′ thenσ◦s(x·g)=(detρ(g))−k′·σ◦s(x).Soσ◦sdefines
a section of theline bundle
(detE)⊗(−k′)⊗s∗O (k)=det(s∗(π∗E))⊗(−k′)⊗s∗O (k)
Z Z Z
4
since π ◦s=id, that is,
Z
s∗(σ)=σ◦s∈H0(B,s∗(det(π∗E∨(1)))⊗k′). (6)
Z
Thuswe obtain the following theorem, which refines [CH88, Theorem 1.1].
Theorem 4. Let B be a smooth projective curve and
Z //Z
π
(cid:15)(cid:15)
B
be the fiberation we constructed above. Let s ,s be two sections of the fibration Z over
0 1
B and b ,b ∈B are two distinct points
0 1
1. Supposes (b )∈Z isasemistablepointwithrespecttothelinebundleO (1)|
0 0 b0 Z Zb0
then h (s ) ≥ 0. Suppose further s (b ) ∈ Z is unstable with respect to
(Z,OZ(1)) 0 0 1 b1
O (1)| then h (s )>0.
Z Zb1 (Z,OZ(1)) 0
2. Suppose for any b∈B\{b } we have s (b)∈G·s (b)⊂Z and s (b ) is unstable
0 0 1 b 0 0
but s (b ) is semistable. Then
1 0
h (s )>h (s ).
(Z,OZ(1)) 0 (Z,OZ(1)) 1
Proof. It follows from GIT (cf. [MFK94]) that if s0(b) ∈ (Zb,OZ(1)|Zb) ∼= (Z,OZ(1))
is semistable with respect theG-action then thereis a G-invariant section
σ∈H0(Z,O (k))G
Z
for some k≫1 suchthat σ(s (b))6=0. Withoutloss of generality,we may assume that
0
k=(N +1)k′.
By theconstruction above (c.f. (6)), we have
s∗(σ)∈H0(B,s∗(det(π∗E∨(1)))⊗k′)
0 0
such that σ◦s (b)6=0, this implies
0
0 ≤ deg(s∗(det(π∗E∨(1)))⊗k′)
0
= k′((N +1)deg(s∗O (1))−deg(E))
0 Z
= k′[(N +1)deg(s∗O (1))−deg(detπ O (1))],
0 Z ∗ Z
which is exactly what we want to prove. For the further part, we only need to notice
thats (b )beingunstableimpliesthatσ(s (b ))=0,i.e.,s∗(σ)isanonvanshingsection
0 1 0 1 0
which has at least one zero.
For the second part, it follows from our assumption that s∗(σ) has the same zero
1
section as s∗(σ) along B\{b }, but at b loses at least one zero by replacing s (b ) by
0 0 0 0 0
s (b ) in Z . Hencewe obtain
1 0 b0
deg(s∗(det(π∗E∨(1)))⊗k′)≤degs∗(det(π∗E∨(1)))⊗k′ −1.
1 Z 0 Z
Remark 1. It follows from the semistable replacement property of GIT (see [Mum77,
5.3]), after a possible base change of the family Z → B, we can always decrease the
height by replacing the unstable value (e.g. s (b ) in Theorem 4) by a semistable one
0 0
(e.g. s (b ) ).
1 0
5
2.2 Geometric height
Nowweapplytheset-upintheprevioussubsection tothestudyaproperflatfamily of
polarized varieties with respect to Chow stability. Recall for a subvariety X ⊂ PN of
degree d, thecorresponding Chow point
Chow(X)∈Pd,n:=P(Symd(CN+1)⊗(n+1)).
is the degree (d,··· ,d)-hypersurface in (PˇN)n+1 consisting of points (H1,··· ,Hn+1)∈
(PˇN)n+1, where PˇN is the dual of PN, such that Hi ⊂ PN are hyperplanes satisfying
H1∩···∩Hn+1∩X 6= ∅. The group SL(N +1) naturally acts on Pd,n and OPd,n(1)
is SL(N +1)-linearized. We say X is Chow (semi-)stable if Chow(X) ∈ Pd,n is (semi-
)stable with respect to the SL(N +1)-linearization of the hyperplane bundle OPd,n(1).
Similarly, let D=Σl a D be a divisor of X with each D ⊂X being a prime divisor
i=1 i i i
and (a1,··· ,al)∈Ql>0. Welet
l
Chow(X,D):=(Chow(X),Chow(D1),··· ,Chow(Dl))∈Pd,n× Pdi,n−1
i=1
Y
denote the Chow point corresponding to the log pair (X,D), where d =deg(D ). Let
i i
usconsider thediagonal action of SL(N+1) on Pd,n× l Pdi,n−1. Wesay (X,D) is
i=1
log Chow (semi-)stable if Chow(X,D)∈Pd,n×Πli=1Pdi,Qn−1, where d:=(d,d1,··· ,dl),
is GIT (semi-)stable (cf. Definition 2) with respect to the SL(N +1)-linearization of
integral multipleof theQ-line bundle
π0∗OPd,n(1)⊗π1∗OPd1,n−1(cna1)⊗···⊗πl∗OPdl,n−1(cnal)
covering the diagonal action of SL(N +1), where π denote the projection to the ith
i
factor and c = n+1. Wedenote
n 2n
Pd,n :=Pd,n×Πli=1Pdi,n−1, where d:=(d,d1,··· ,dl).
Example 1 (cf. [Hasset03,LW11]). A weighted pointednodalcurve(X,a·x) consists
ofareduced, connected nodalcurveX,l ordered(notnecessarilydistinct)smoothpoints
x=(x ,··· ,x)∈Xl
1 l
of X, and weights a = (a1,··· ,al), ai ∈ Q>0, of x, such that the total weight at any
point is no more than one (i.e. for any p∈X, a ≤1). If in addition, ω (a·x)
xi=p i X
is ample then we call (X,a·x) a weighted pointed stable curve.
P
A polarized weighted pointed nodal curve is a weighted pointed nodal curve together
with a very ample polarization L. Then (X,a·x;L) is Chow (semi-)stable iff
Chow(X,a·x)∈Pd,1
obtained from the embedding X ⊂ PH0(L) is GIT (semi-)stable with respect to the
SL(N+1)-linearization of integral multiple of the Q-line bundle
π0∗OPd,1(1)⊗π1∗OP1,0(a1)⊗···⊗πl∗OP1,0(al)
since n=1 and c =1.
1
To fit the above geometric GIT construction into a family version over a proper
smooth curve B, let us briefly summarize the construction of Chow section due to
Mumford(fordetailspleasesee[MFK94,section5.4],[Zhang96,section1.3]or[BGS94,
section4.3]). LetBbeanintegralschemeE bealocallyfreesheafofrankN+1overB,
and X be an effective cycle of PE :=Proj(Sym∗E), the projective space over B, whose
components are flat and of relative dimension n overB. Thus we havediagram
6
X //PE
π π
(cid:15)(cid:15) (cid:15)(cid:15)
B B
Let OPE(1) and OPE∨(1) denote the hyperplane line bundle of PE and PE∨ respec-
tively. Then the canonical section of E ⊗E∨, which is dual to the canonical pairing
E∨⊗E → OB, gives a section ∆ of OPE(1)⊠OPE∨(1) over PE×PE∨. Let πi denote the
i-th projection
πi:(PE∨)n+1 −→PE∨
and ∆i’s bethe corresponding sections of OPE(1)⊗π∗iOPE∨(1) on PE⊗(PE∨)n+1. Let
n
Γ:= ∆−1(0).
i
i=0
\
If we regard thepoints of PE∨ as hyperplanesof PE then
Γ={(x,H0,··· ,Hn)∈PE⊗(PE∨)n+1 | x∈Hi,∀i}.
Andif we regard Γ as a correspondence from PE to (PE∨)n+1
Γ⊂PE⊗(PE)n+1
xxqqqqqpq1qqqqqq PPPPPPPpP2PPPP''
PE (PE∨)n+1
then
Y :=p2∗(p∗1(X)∩Γ)⊂(PE∨)n+1
will be a divisor of degree (d,··· ,d) of (PE∨)n+1 whose components are flat (c.f.
[BGS94],Lemma4.3.1) overB,wheredisthedegreeofXb ⊂PEb:=π−1(b)forgeneral
b∈B.
Let OPd,n(1) be thehyperplanebundleof
Pd,n :=P[(SymdE)⊗(n+1)]−→B .
Thenthecanonical pairingof (SymdE)⊗(n+1)⊗(SymdE∨)⊗(n+1) givesrise toasection
∆′ of the line bundle
OPd,n(1)⊗π0∗OPE∨(d)⊗···⊗πn∗OPE∨(d)
onPd,n×(PE∨)n+1.BytreatingpointsofP[(SymdE)⊗(n+1)]ashypersurfacesof(PE∨)n+1
of degree (d,··· ,d),we can regard
Γ′={∆′ =0}={(H,y ,··· ,y )|(y ,··· ,y )∈H}.
0 n 0 n
as a correspondence
Γ′ ⊂Pd,n⊗(PE∨)n+1
wwoooooopo1oooooo QQQQQQQpQ2QQQQQ((
Pd,n (PE∨)n+1
and section
s =p (p∗Y ∩Γ′) (7)
(X,L) 1∗ 2
of Pd,n overB corresponds to Y(X) viaΓ′ is theChow section for X.
Let us apply the construction above to the relative Chow varieties of a family of
n-dimensional log varieties satisfying thefollowing
7
Assumption 1. Suppose that X is a separated scheme of dimension n+1 and D =
Σl a D with D ⊂X be prime divisors. Let
i=1 i i i
π:(X,D)−→B
be a flat proper morphism over a smooth projective curve B. Let L → X be a line
bundle such that E := π L is locally free of rank N +1. Suppose that the following
∗
conditions are satisfied:
1. L is relatively very ample and H0(Xb,Lb)∼=Eb for every b∈B;
2. Thereisanonempty openset U ⊂B suchthat Chow(X ,D ) isasemistable point
b b
with respect to the polarization of L| for all b∈U.
Xb
From the above assumption, we may view the pair (X,D) as a family of effective
cycles of PE with E =π∗L,
(X,D) //PE
π π
(cid:15)(cid:15) (cid:15)(cid:15)
B B
whose components are flat and of dimension n over B. Denote by d = degX ,d =
0 b i
degD | , 1≤i≤l, and we define
i b
l
Pd,n :=Pd0,n× Pdi,n−1 with d=(d ,d¯):=(d ,d ,··· ,d).
0 0 1 l
i=1
Y
For a=(a1,··· ,al)∈(Q∩[0,1])l, we introduce a Q-linebundle
OPd,n(1,a):=π0∗OPd,n(1)⊗π1∗OPd1,n−1(cna1)⊗···⊗πl∗OPdl,n−1(cnal) (8)
on Pd,n, where π denote the projection of Pd,n to the ith factor. ApplyingMumford’s
i
construction of Chow section outlined above we obtain section s .
(X,D;L)
Pd,n
{==
s(X,D;L){{{
{{ Π
{{ (cid:15)(cid:15)
B B
By subsection 2.1, there is a height (cf. Definition 3) attached to the above family of
GIT problem with G = SL(N +1) and G˜ = GL(N +1). To unwind the underlying
meaning of theheight in terms of the geometry of (X,D;L),we havethe following:
Proposition 1. For the section s constructed above, we have
(X,D;L)
1. Choosing m∈Z such that cnm·a∈Zl+1, the Chow line is defined to be
Pic(B) ∋ ΛChow(L):=H(Pd,n,OPd,n(m,ma))
= dets∗(X,D;L) Π∗(Π∗OPd,n(m,ma))∨⊗OPd,n(m,ma) ⊗N+1
= (cid:0) (Lhn+1i)⊗m(cid:0) ⊗(L|hni)⊗cnma ⊗(N+1)⊗(detE∨)⊗(m(n(cid:1)+(cid:1)1)d+cnmna·d¯) ⊗Kd,a,m
D
(cid:18)(cid:16) (cid:17) (cid:19)
whereLhn+1i andL|hni denote theDelignepairingsforthe family(X,L)→B and
D
(D,L| ) → B respectively (cf. [Deligne87, Chapter 6]), E = π (L), a = a ,
D ∗ i i
and
P
l
Kd,a,m =dim Symm(Symd(CN+1))⊕ Symcnmai(Symd(CN+1)) .
!
i=1
M
8
2. We define the geometric height to be
h(X,D;L)
1
:= m·Kd,a,mh(Pd,n,OPd,n(m,ma))(s(X,D;L))=deg(ΛChow(L))
1
= {(N +1)L−detπ∗(π L)}n+1
(N+1)n ∗
l
a
+c i {(N +1)L| −(πDi)∗detπ L}n
n (N+1)n−1 Di ∗
i=1
X
3. Let us define the Donaldson-Futaki invariant for the family (X,D;L)→B as
DF((X,D)/B,L):=(n+1)Ln·(K +D)−nLn+1µ
X/B
with µ:= Ln−1·(KX+D). Then we have for k≫1,
Ln
h(X,D;Lk)
degX
= (n+1)Ln·(K +D)−nLn+1µ k2n+O(k2n−1)
2n! X/B
= degXD(cid:0)F((X,D)/B,L)k2n+O(k2n−1). (cid:1)
2n!
Proof. Itisastraightforwardcalculationwhichcompletelyparallelsto[Wang12,Propo-
sition 11]. Weomit the details.
For thelater application, we will need thefollowing technical result.
Proposition2. Let(X,D;L)beaflatfamilysatisfyingAssumption1andletν :Xν →
X be a finite morphism, which does not change the generic fiber over B. Assume Xν is
flat over B. Then
1. There is a k0 ∈N so that for k≥k0 the new family(Xν,ν∗−1D;(ν∗L)⊗k) satisfies
Assumption 1;
2. We have
h(Xν,ν−1D;(ν∗L)⊗k)≤h(X,D;L⊗k)
∗
with the equality holds if and only if Xν ∼=X and
DF((Xν,ν−1D)/B;ν∗L)≤DF((X,D)/B;L).
∗
Proof. The first part follows from the assumption that ν is a finite morphism. For the
second part, if we write πν =π◦ν then for any k∈N we have
1 n+1
(N +1)k(ν∗L)−πν∗det πνν∗L⊗k
(N +1)n k ∗
k
n (cid:16) (cid:17)o
= (N +1)kn+1Ln+1−kn(n+1)degX·deg det π ν ν∗L⊗k
k ∗ ∗
(cid:16) (cid:16) (cid:17)(cid:17)
where degX = [L| ]n for a general fiber X = X and N +1 = dimH0(X ,L| ),
Xb b k b Xb
thankstotheprojection formula ν (ν∗L)n+1 =Ln+1. Forthesecond term we consider
∗
theexact sequence
0 −−−−−→ OX −−−−−→ ν∗OXν −−−−−→ Q −−−−−→ 0.
Since by our assumption the support of Q does not dominate B, π (Q⊗ L⊗k) is a
∗ OX
torsion sheaf on B. So for k sufficiently large, we have c (π (Q⊗ L⊗k)) ≥ 0 and
1 ∗ OX
hence
c1(π∗ν∗(ν∗L⊗k)) = c1(π∗(ν∗(OXν)⊗OX L⊗k))
= c (π (Q⊗ L⊗k))+c (det(π (L⊗k)))
1 ∗ OX 1 ∗
≥ c (π (L⊗k)).
1 ∗
9
with the equality holds if and only if Q = 0. Putting these together, we immediately
conclude that
h(Xν,ν−1D;(ν∗L)⊗k)≤h(X,D;L⊗k),
∗
with the equality holding if and only if Q=0. Finally, the decreasing of DF invariants
undernormalizaition follows immediately from part 3 of Proposition 1.
Remark 2. This argument combined with [Wang12, Theorem 7] simplify the proof of
[RT07, Proposition 5.1]. We can apply this to the normalization Xν of X when the
generic fiber is normal, or more generally, the S -hull Xν of X when the generic fiber
2
satisfies the Serre condition S .
2
2.3 Height decreases along semistable replacement
Let
(X,D;L)_ _ _ _f _ _ _//(X′,D′;L′)
HHHHHHHHH## zzuuuuuuuuuu
B
betwo birational families satisfying Assumption 1 and
(X◦,D◦;L|X◦)∼=(X′◦,D′◦;L′|X′◦) overB◦ =B\{b0}
where (X◦,D◦)=(X,D)× B◦ and (X′◦,D′◦)=(X′,D′)× B◦. Supposethereis an
B B
openU ⊂B containing b suchthat(X ,D ;L| )is Chow semistable forall b∈U but
0 b b Xb
(Xb′0,Db′0;L′|Xb′0) is Chow unstable. By the construction we performed in the previous
subsection we obtain a section s for the fibration
(X,D;L)
Pd,n −→B.
Now we pick two points b ,b ∈U and a section
0 1
σ ∈ H0(P[(SymdEb0)⊗(n+1)]×Πli=1P[(SymdiEb0)⊗n],OPd,n(m,ma)|b0)SL(N+1)
∼= H0(P[(SymdEb1)⊗(n+1)]×Πli=1P[(SymdiEb1)⊗n],OPd,n(m,ma)|b1)SL(N+1)
such that σ◦s (b ) 6= 0 for i = 0,1, which can be achieved by the semi-stability
(X,D;L) i
of (X ,D ;L| ) for i=0,1. By Theorem 4, we obtain
bi bi Xbi
0≤h(X,D;L)<h(X′,D′;L′). (9)
Hencewe havethe following
Proposition 3. Let (X,D;L)→B be a family satisfying the Assumption 1. We fix a
point b ∈ U. Then for any family (X′,D′;L′) flat over B satisfying the first part of
0
Assumption 1 and
(X◦,D◦;L|X◦)∼=(X′◦,D′◦;L′|X′◦) over B◦ =B\{b0},
we have
0≤h(X,D;L)≤h(X′,D′;L′)
with‘=’holdsifandonlyifChow(Xb′0,Db′0;L′|Xb′0)isS-equivalenttoChow(Xb0,Db0;L|Xb0)
in P[(SymdCN+1)⊗(n+1)].
Proof. By the discussion in the beginning of this subsection, we only to prove the ‘=’
case. SupposethatChow(Xb′0,Db′0;L′|Xb′0)isnotS-equivalenttoChow(Xb0,Db0;L|Xb0)
then it must be unstable and hence we have strict inequality (9). Thus our proof is
completed.
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