Table Of ContentLecture 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
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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
