The Moment-Weight Inequality and the Hilbert–Mumford Criterion: GIT from the Differential Geometric Viewpoint

193 Pages·2021·2.312 MB·English

by  Valentina Georgoulas, Joel W. Robbin, Dietmar Arno Salamon

Lecture Notes in Mathematics 2297
Valentina Georgoulas
Joel W. Robbin
Dietmar Arno Salamon
The Moment-Weight
Inequality and
the Hilbert–Mumford
Criterion
GIT from the Differential Geometric
Viewpoint
Lecture Notes in Mathematics
Volume 2297
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
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DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA
IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK
AngelaKunoth,UniversityofCologne,Cologne,Germany
László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig,
Germany
ArianeMézard,IMJ-PRG,Paris,France
MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg
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Valentina Georgoulas (cid:129) Joel W. Robbin (cid:129)
Dietmar Arno Salamon
The Moment-Weight
Inequality
and the Hilbert–Mumford
Criterion
GIT from the Differential Geometric
Viewpoint
ValentinaGeorgoulas JoelW.Robbin
Zürich,Switzerland DepartmentofMathematics
UniversityofWisconsin–Madison
Madison,WI,USA
DietmarArnoSalamon
DepartmentofMathematics
ETHZürich
Zürich,Switzerland
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-89299-9 ISBN978-3-030-89300-2 (eBook)
https://doi.org/10.1007/978-3-030-89300-2
MathematicsSubjectClassification:53D20,14L24,32Q15
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Preface
Thisbookgivesanessentiallyself-containedexposition(exceptforanappealtothe
Lojasiewicz gradient inequality) of geometric invariant theory from a differential
geometricviewpoint.Centralingredientsare the moment-weightinequality(relat-
ingtheMumfordnumericalinvariantstothenormofthemomentmap),thenegative
gradientflowofthemomentmapsquared,andtheKempf–Nessfunction.
Thelastauthor,DAS,owesalottothelecturesbyandconversationswithSimon
Donaldson,GáborSzékelyhidi,XiuxiongChen,andSongSun.Thesecondauthor,
JWR, learned much from a course given by and conversations with Sean Paul at
the University of Wisconsin. Most of this book was written when JWR visited
theForschungsinstitutfürMathematikatETHZürichandhethanksthemfortheir
hospitality.ThefirstversionofthisbookwascompletedwhileDASvisitedtheIAS,
Princeton,andtheSCGP,StonyBrook;hethanksbothinstitutesfortheirhospitality.
ThankstoSamuelTrautweinforhelpfuldiscussions.ThankstoAmandaJennyfor
pointingouterrorsinearlierversionsofthemanuscript.
Zürich,Switzerland ValentinaGeorgoulas
Madison,WI,USA JoelW.Robbin
Zürich,Switzerland DietmarA.Salamon
August2021
v
Contents
1 Introduction................................................................. 1
2 TheMomentMap .......................................................... 7
3 TheMomentMapSquared................................................ 13
4 TheKempf–NessFunction ................................................ 19
5 µ-Weights.................................................................... 29
6 TheMoment-WeightInequality .......................................... 37
7 StabilityinSymplecticGeometry......................................... 51
8 StabilityinAlgebraicGeometry.......................................... 59
9 Rationality................................................................... 65
10 TheDominantµ-Weight................................................... 75
11 TorusActions................................................................ 89
12 TheHilbert–MumfordCriterion ......................................... 95
13 CriticalOrbits............................................................... 105
14 Examples .................................................................... 131
A NonpositiveSectionalCurvature.......................................... 151
B TheComplexifiedGroup .................................................. 159
C TheHomogeneousSpaceM =Gc/G .................................... 169
D ToralGenerators............................................................ 173
E ThePartialFlagManifoldGc/P≡G/C................................. 177
References......................................................................... 185
Index............................................................................... 189
vii
Chapter 1
Introduction
Many important problems in geometry can be reduced to a partial differential
equationoftheform
μ(x)=0,
where x ranges over a complexifed group orbit in an infinite-dimensional sym-
plectic manifold X and μ:X →g is an associated moment map (see Cal-
abi[6–8],Yau[79–81],Tian[75],Chen–Donaldson–Sun[18–20],Atiyah–Bott[2],
Uhlenbeck–Yau [76], Donaldson [27–30, 32]). Problems like this are extremely
difficult. The purpose of this book is to explain the analogous finite-dimensional
situation,whichisthesubjectofGeometricInvariantTheory.
GITwasoriginallydevelopedtostudyactionsofacomplexreductiveLiegroup
Gc onaprojectivealgebraicvariety1
X ⊂P(V).
Here Gc is the complexification of a compact Lie group G and the Hermitian
structureonV canbechosensothatGactsbyunitaryautomorphisms.Inthesmooth
caseXinheritsthestructureofaKählermanifoldfromthestandardKählerstructure
onP(V).TheG-actionisgeneratedbythestandardmomentmap
μ:X →g
(withvaluesintheLiealgebraofG).IntheoriginaltreatmentofMumford[63]the
symplecticformandthemomentmapwerenotused.Subsequentlyseveralauthors
1Mumford’s original theory even applies to varieties over general ground fields other than the
complexnumbers.Theseextensionsarenotdiscussedinourtreatment.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 1
V.Georgoulasetal.,TheMoment-WeightInequalityandtheHilbert–Mumford
Criterion,LectureNotesinMathematics2297,
https://doi.org/10.1007/978-3-030-89300-2_1
2 1 Introduction
discovered the connection between the theory of the moment map and GIT (see
Kirwan [53] and Ness [66]). In the article of Lerman [56] it is noted that both
Guilleman–Sternberg[42] andNess [66] creditMumfordfor the relationbetween
thecomplexquotientandtheMarsden–Weinsteinquotient
X//G:=μ−1(0)/G.
InourexpositionweassumethatXisaclosedKählermanifoldbutdonotassume
thatitisaprojectivevariety.Inthelattercasetheaforementionedstandardmoment
map satisfies certain rationality conditions (Chap.9) which we do not use in our
treatment.AsaresulttheMumfordnumericalinvariants
w (x,ξ):= lim(cid:5)μ(exp(itξ)x),ξ(cid:6)
μ
t→∞
associatedtoapointx ∈Xandanelementξ ∈g\{0}(Chap.5)neednotbeintegers
astheyareintraditionalGIT.IntheclassicaltheorytheLiealgebraelementbelongs
totheset
Λ:={ξ ∈g\{0}| exp(ξ)=1}
and thus generates a one-parameter subgroup of Gc. In our treatment ξ can be a
generalnonzeroelementofg.
Acentralingredientinourtreatmentisthemoment-weightinequality
−w (x,ξ)
sup μ ≤ inf |μ(gx)|. (1.1)
ξ∈g\{0} |ξ| g∈Gc
We give two proofs of this inequality in Chap.6, one due to Mumford [63] and
Ness [66, Lemma 3.1] and one due to Xiuxiong Chen [15]. (For an in depth
discussion of this inequality see Atiyah–Bott [2], in the setting of bundles over
Riemann surfaces, and Donaldson [34], Székelyhidi [72], Chen [15, 16], in the
setting of Kähler–Einstein geometry.) Following an argument of Chen–Sun [23]
we also provethatequality holdsin (1.1)wheneverthe righthandside is positive
(Theorem 10.4). We also prove that the supremum on the left is always attained
(Theorem 10.1) and that the supremum over all ξ ∈ g \ {0} agrees with the
supremum over all ξ ∈ Λ (Theorem 12.1). In the projective case the supremum
is attained at an element ξ ∈Λ by a theorem of Kempf [51], however, that need
not be the case in our more general setting. The Hilbert–Mumford numerical
criterion for μ-semistability is an immediate consequence of the aforementioned
results(Theorem12.2).Itassertsthat
Gc(x)∩μ−1(0)(cid:10)=∅ ⇐⇒ w (x,ξ)≥0∀ξ ∈Λ.
μ
1 Introduction 3
Further consequencesof the moment-weightinequalityinclude the Kirwan–Ness
Inequalitywhichassertsthatifxisacriticalpointofthemomentmapsquaredthen
|μ(x)|= inf |μ(gx)|
g∈Gc
(Corollary 6.2), the Moment Limit Theorem which asserts that each negative
gradient flow line of the moment map squared converges to a minimum of the
momentmapsquaredonthecomplexifiedgrouporbit(Theorem6.4),andtheNess
UniquenessTheoremwhichassertsthatanytwocriticalpointsofthemomentmap
squaredinthesameGc-orbitinfactbelongtothesameG-orbit(Theorem6.3)and,
moreover,thattheminimumofthemomentmapsquaredontheclosureofaGc-orbit
istakenonatauniqueG-orbit(Theorem6.5).
Acentralingredientintheproofsofthesetheoremsisthenegativegradientflow
ofthemomentmapsquared
f := 1|μ|2 :X →R.
2
Thegradientflowequationtakestheform
x˙ =−JL μ(x) (1.2)
x
whereL :g→T XdenotestheinfinitesimalactionoftheLiealgebraonX.Each
x x
Gc-orbitGc(x)⊂Xisinvariantunderthisflow,becauseeverysolutionof(1.2)has
theformx(t)=g(t)−1x,whereg :R→Gc satisfiesthedifferentialequation
g−1g˙ =iμ(g−1x). (1.3)
Equation(1.3)isthenegativegradientflowoftheKempf–Nessfunction
Φ :Gc/G→R.
x
ThehomogeneousspaceGc/Gissimplyconnectedandcompletewithnonpositive
sectional curvature, and the Kempf–Ness function is Morse–Bott and is convex
along geodesics (Theorem 4.3); its critical manifold may be empty. Moreover,
the Kempf–Ness Theorem characterizes the stability conditions in terms of the
propertiesofthe Kempf–Nessfunction(Theorem7.3);forexamplea pointx ∈X
isμ-semistable,i.e.theclosureofitsGc-orbitintersectsthezerosetofthemoment
map,ifandonlyiftheKempf–NessfunctionΦ isboundedbelow.
x
ThemomentmapsquaredisingeneralfarfrombeingMorse–Bottandmayhave
verycomplicatedcriticalpoints.However,theaforementionedtheorems(Kirwan–
Ness Inequality,Ness Uniqueness,MomentLimit Theorem)exhibita structureof
the gradient flow that resembles the stratification by stable manifolds associated
to a Morse–Bott function. More precisely, an element x ∈X is a critical point of
the moment map squared if it satisfies the equation L μ(x)=0. Critical points
x