Twisted Polytope Sheaves and Coherent-Constructible Correspondence for Toric Varieties

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by  Peng Zhou

TWISTED POLYTOPE SHEAVES AND COHERENT-CONSTRUCTIBLE CORRESPONDENCE FOR
TORIC VARIETIES
PENGZHOU
7 ABSTRACT. GivenasmoothprojectivetoricvarietyXΣ ofcomplexdimensionn,Fang-Liu-Treumann-Zaslow
1 [FLTZ] showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent
0 sheavesCoh(XΣ)intothedgderivedcategoryofconstructiblesheavesonatorusSh(Tn,ΛΣ).Recently,Kuwa-
2 gaki[Ku2]provedthatthequasi-embeddingisaquasi-equivalence,andgeneralizedtheresulttotoricstacks.
Herewegiveadifferentproofinthesmoothprojectivecase,usingnon-characteristicdeformationofsheavesto
n
findtwistedpolytopesheavesthatco-representthestalkfunctors.
a
J

3

]
G 1. INTRODUCTION
A
Toricvarietiesarecertaincompactificationsofthecomplextorus(C∗)n. Theyprovidemanyinteresting
.
h examples,andcanbestudiedinvariousways,usingalgebraicgeometry,symplecticgeometryorcombina-
t torics.
a
m
Forexample,letX beasmoothprojectivetoricvareitycorrespondingtoafanΣ,andLanampleline
Σ
[ bundlewithaliftingofthe(C∗)n-action. Thenthereisaconvexpolytope∆ inRn,whereRn isidentified
L
withthedualLiealgebraLie(Tn)∨ andTn =(U(1))n isthemaximalcompactrealsubgroupof(C∗)n. The
1
v convexpolytope∆Lcanbeunderstoodinthefollowingways,
9
8 (1) Algebraically, ∆L is the convex hull of the characters appearing in the weight decomposition of
6 H0(X,L)underthe(C∗)n-action.
0 (2) Symplectically, ∆ is the moment polytope of the Hamiltonian action Tn on (X,ω), where ω is a
L
0
symplectic2-formwith[ω]=c (L).
. 1
1 (3) Combinatorially, ∆ is the intersection of half-spaces Q = {x ∈ Rn | (cid:104)x,v (cid:105) ≤ a }, one for each
L ρ ρ ρ
0 compactifying divisor D of X given by a vector v ∈ Zn, and a is the vanishing order of the
ρ ρ ρ
7
invariant(meromorphic)sectionalongthedivisorD .
1 ρ
:
v InthecasewhereLisnotample, thecorrespondingpolytopebecomesa‘twistedpolytope’, asexplained
i inFigure1. ThenameoriginatesfromthepaperofKarshonandTolman[KT],wheretheygeneralizedthe
X
momentmaptothecasewhereωisdegenerate.
r
a
FIGURE1. Twistingsofaconvexpolytope.Asonepushestheedgesofthepolytopes,acer-
tainedgewillshrinktozero-lengththenreappearontheotherside. Notetheco-directions
oftheedges,indicatedbyhairs,remainfixed.Thecorrespondingtwistedpolytopesheaves
havestalksongreen,yellowandblueregionsasC,C[1],C[2],respectively.
Date:January4,2017.
1
2 PENGZHOU
Theabovecorrespondencebetweenequivariantlinebundlesandtwistedpolytopesenjoysacategorifi-
cationunderthenameof(equivariant)Coherent-Constructible-Correspondence(CCC).
Theorem1([FLTZ]). IfX isapropertoricvariety, thereisacorrespondingconicalLagrangianΛ ⊂ T∗Rn and
Σ
anequivalenceofderived(orrather,triangulateddgcategories)
κ:Perf (X )−∼→Sh (Rn,Λ )
T Σ cc Σ
where
• Perf (X ) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on
T Σ
X .
Σ
• Sh (Rn,Λ )isthetriangulateddgcategoryofconstructiblesheavesonRnwhicharecompactlysupported,
cc Σ
whosesingularsupportslieinΛ .
Σ
TheequivariantCCCimpliesthatthereisaquasi-embeddingforthenon-equivariantcase:
Proposition 1.1 ([Tr],Proposition 2.4, 2.7). Let π : Rn → Tn (cid:39) Rn/Zn be the projection. Then there exists a
functorκandcommutativediagrams
Perf (X ) ∼ κ Sh (Rn,Λ )
T Σ cc Σ
forget π!
Perf(X ) κ Sh(Tn,Λ ),
Σ Σ
Remark 1.2. When X is smooth, the homotopy category of Perf (X ) (resp. Perf(X )) coincide with
Σ T Σ Σ
theusualDbCoh (X )(resp. DbCoh(X )).
T Σ Σ
Remark 1.3. Under the quotient map π : Rn → Tn, all the upstairs objects in Rn are unadorned, and
downstairsobjectsinTnhaveoverlines.
Anditisconjecturedthatthisquasi-embeddingisaquasi-equivalence. Theconjecturehasbeenverified
incertaincasesbyTreumann[Tr],Scherotzke-Sibilia[SS]andKuwagaki[Ku1]. Recently,ithasbeenfully
proven by Kuwagaki [Ku2] in the generality of toric stacks, using gluing descriptions of ∞-categories on
bothsides.
In this paper, we prove the non-equivariant CCC for smooth projective toric varieties, by showing the
κ-imagesoflinebundlesgeneratetheconstructiblesheafcategory.
Theorem2. LetX beasmoothprojectivetoricvarietyofcomplexdimensionn,thenthereisanquasi-equivalence
Σ
ofcategory
κ:Coh(X )−∼→Sh(Tn,Λ )
Σ Σ
whereΛ isaconicalLagrangianinT∗Tn.
Σ
Thekeypartoftheproofisasfollowing. Foranypointθ ∈ Tn,thereisaconstructiblesheafP onTn
[θ]
astheκ-imageofacertainlinebundle(c.f. Definition4.1),suchthatforanysheafF ∈Sh(Tn,Λ ),itsstalk
Σ
atthepointθcanbecomputedby
F (cid:39)Hom(P [−n],F). (1)
θ [θ]
ThisimmediatelyimpliesthatifF satisfiesHom(κ(L),F)=0forallthelinebundlesLonX ,thenF =0.
Σ
Inotherwords,thestalkfunctorsinSh(Tn,Λ )areco-representedbyκ-imagesoflinebundlesonX . We
Σ Σ
thankDavidTreumannforthesuggestionofco-representingthestalkfunctors.
Thequasi-isomorphism(1)isduetoanon-characteristicdeformationargumentforconstructiblesheaf.
Wedefinea1-parameterfamilyofsheaves{P } ,suchthat
t t∈[0,1]
(1) P = j C , where B is a small enough convex open set around θ, such that F (cid:39) Γ(B,F) (cid:39)
0 B! B θ
Hom(P ,F).
0
TWISTEDPOLYTOPESANDCCC 3
FIGURE2. ExpandingfamilyoftwistedpolytopesheavesonR2.
(2) P =P [−n].
1 [θ]
(3) Fort∈(0,1),takethelinearinterpolationbetweenP andP ,andshowthatSS∞(P )∩Λ∞ =∅.
0 1 t Σ
By the non-characterstic deformation lemma1, Hom(P ,F) is invariant during the deformation, hence we
t
get(1).
Example1.4(Expandingfamilyoftwistedpolytopesheaves). TheexampleofHirzebruchsurfaceF , see
2
Figure 2. Here we describe the sheaf P upstairs in R2, where P := π P and π : R2 → T2 is the
[x] [π(x)] ∗ [x]
quotientmap. Thepointwewanttoprobeisatx = (−0.5,0), markedinblack. Thegreen, blue, redand
blackcurvesaretheboundariesofthetwistedpolytopesintheinterpolatingfamilyP . Thegreenandblue
t
ones are still open convex polytope, the red and black ones are twisted. We marked the direction of the
singularsupportforthesheafP correspondingredcurve,andnotethatSS∞(P )∩Λ∞ =∅.
red red
Remark1.5. Thecollectionoflinebundlesastheκ-preimagesof{P }isafinitecollection,sincesheaves
[θ]
in Sh(Tn,Λ ) admits a finite stratification depending only on Λ . In the case of Pn, they turn out to be
Σ Σ
O(1),··· ,O(n+1), and form an exceptional collection. However, for general toric variety, even smooth
Fanoones,thecollectionoflinebundlescannotalwaysbeanexceptionalcollection[E,HP].
However,foranycollectionoflinebundlesL ,··· ,L ,Craw-Smith[CS]consideredtheendomorphism
1 N
algebra A = End(⊕N L ) and defined a bound quiver of sections (Q,R) associated with A. It would be
i=1 i
interestingtostudyourcollectionoflinebundlesusingthisquiverapproach.
Theoutlineofthepaperisthefollowing.Insection2wereviewthebackgroundonconstructiblesheaves,
in particular state the non-characteristic deformation lemma. In section 3, we review toric geometry and
statetherelevantresultsfrom[FLTZ,Tr]ontheequivariantandnon-equivariantCCCfunctors. Insection
4,weproveourmaintheorem.
1.1. Acknowledgements. ItisapleasuretothankXinJin,LinhuiShen,LeiWu,EldenElmantoandDima
Tamarkinformanyhelpfuldiscussions, andDavidTreumannandDavidNadlerfortheirinterestsinthis
work.IamgratefulforEldenforcarefullyreadingthedraftandgivingmanyusefulcomments.Iamgrately
indebtedtomyadvisorEricZaslowforinspirationsandencouragements(andpatience!). Thediscussion
with David Treumann at IAS inspired the current approach, which uses the twisted polytope sheaves to
corepresentthestalkfunctor.
1Oneneedstobecarefulabouttheendpointt = 1,sinceSS∞(P1)∩Λ∞Σ (cid:54)= ∅. Thenon-characteristicdeformationlemmafor
sectionsoveropensets,Proposition2.7,avoidsthisproblem.
4 PENGZHOU
2. CONSTRUCTIBLESHEAVES
Herewegiveaquickreviewonconstructiblesheaves,followingtheintroductionof[N]andtheAppen-
dixof[STW]veryclosely.ForathoroughaccountonconstructiblesheaftheoryanditsrelationwithFukaya
category,see[KS,S]and[NZ].
Wefirstgivethecategoricalbackground,especiallythedefinitionfordgenhancementofthetriangulated
derived categories. Then we give some useful formulae for practical computations. Then we take a brief
detourinsymplecticgeometrytodefineconicalLagrangianincotangentbundles,sothatwecandefinethe
singularsupport SS(F) ofaconstructible sheafF. Finally, wediscussvarious non-characteristic deforma-
tions results, where the sections Γ(U ,F), or more generally the hom-complexes Hom(P ,F), is invariant
t t
uptoquasi-isomorphismasU orP vary.
t t
2.1. Classical and differential graded derived categories of sheaves. Let X be a topological space. The
poset(viewedasacategory)Open(X)hasobjectsofopensubsets, andpartialorderings(morphisms)are
givenbyinclusionsofopensubsets. LetVectbetheabeliancategoryofC-vectorspaces.
• ApresheafF valuedinVectisafunctor
F :Open(X)o →Vect.
ApresheafF isasheaf,ifforanycollectionofopensubsets{U } ,wehaveanexactsequence
i i∈I
(cid:91) (cid:89) (cid:89)
0→F( U )→ F(U )→ F(U ∩U ).
i i i j
i i i,j
• LetC(X)betheabeliancategoryofcomplexesofsheavesonX,withmorphismsbeingdegree-zero
chainmaps.
• LetK(X)bethehomotopycategoryofC(X),whereobjectsarethesameasC(X),andsmorphisms
arechainmapsuptohomotopyequivalences
Hom (F•,G•):=Hom (F•,G•)/∼
K(X) C(X)
whereϕ ∼ϕ ifϕ −ϕ =d◦h−h◦dforsomedegree−1maph:F• →G•−1.
1 2 1 2
• ThederivedcategoryD(X)isobtainedfromK(X)byinvertingquasi-isomorphisms.Thebounded
derivedcategoryDb(X)isdefinedtobethefullsubcategoryofcomplexeswithboundedcohomolo-
gies.
Todefineconstructibility,letX bearealanalyticmanifold. Wefixanalgebro-geometriccategoryC,e.g.,
thecategoryofsubanalyticsets. AWhitneystratificationS ={S } ofX byC-submanifolds,isadecom-
α α∈A
positionX = (cid:116) S intodisjointstrata{S }indexedbyA, whereS arelocallyclosedC-submanifolds
α∈A α α α
andS ∩S (cid:54)=∅ifandonlyifS ⊂S ,andanypairofdistinctstrata(S ,S )satisfiestheWhitneycondition,
α β α β α β
thatis,ifasequence{x ∈ S }andasequence{y ∈ S }convergestoapointy ∈ S ,suchthatinsome
n α n β β
localcoordinatepatch,thesecantlinesx y convergestoalinelandthetangentplanesT S convergesto
i i xi α
aplaneτ,thenl⊂τ.
Let S = {S } be a Whitney stratification of X. An object F• ∈ D(X) is said to be S-constructible, if
α
therestrictionsHi(F•)| ofitscohomologysheavestothestrataofS arefinite-rankandlocallyconstant.
Sα
We denote by D (X) the full subcategory of D(X) spanned by S-constructible objects, and denote by
S
D (X) the full subcategory of D(X) spanned by constructible sheaves for some Whitney stratification. If
c
thestratificationS isfinite,thenbythefiniterankconditiononcohomologysheaves,D (X)⊂Db(X).
c
Next,wedefinethedifferentialgraded(dg)derivedcategory. Forareviewondgcategoryanddgquo-
tientconstruction,see[Ke]and[Dr].
TWISTEDPOLYTOPESANDCCC 5
• The (naive) dg category Sh (X) has objects as chain complexes of sheaves F•, same as C(X),
naive
andmorphismsarechaincomplexes,withthedegreenelementas
(cid:89)
Homn (F•,G•):= Hom (Fi,Gn+i)
Shnaive(X) X
i∈Z
anddifferentialsaregivenby
dn :Homn (F•,G•)→Homn+1 (F•,G•), ϕ(cid:55)→d ◦ϕ−(−1)nϕ◦d .
Shnaive(X) Shnaive(X) F G
• The dg derived category Sh(X) is the dg quotient of Sh (X) by the full subcategory spanned
naive
byacyclicobjects[Dr]. ThisisatriangulateddgcategorywhosecohomologycategoryH0(Sh(X))
iscanonicallyequivalenttothederivedcategoryD(X)asatriangulatedcategory.
• The dg derived category of bounded constructible categories Shb(X) is the full subcategory of
c
Sh(X), whose objects F• have bounded constructible cohomology sheaves. For a fixed Whitney
stratificationS ofX,theS-constructibledgderivedcategorySh (X)isthefulldgsubcategoryof
S
Sh(X)spannedbyobjectsprojectingtoD (X).
S
• AdgfunctorF :C →Dofdgcategoriesisaquasi-embedding(resp. quasi-equivalence)ifandonly
iftheinducedcohomologicalfunctorH(F):H(C)→H(D)isanembedding(resp. equivalence).
Inthispaper,wewillonlyworkwithconstructiblesheaves,andwillomitthesubscriptcforconstructibil-
ity. To simplify notation, we use sheaf F to mean complex of sheaves F•, Hom to mean hom-complex
X
Hom• .
Sh(X)
2.2. UsefulFormulaeforComputations. Inspiteoftheabstractcategoricaldefinitions,constructiblesheaves
enjoymanyfunctorialpropertieswhichfaciliatesactualcomputations. Herewegivesomeusefulformulae
andexamples.
Weusef∗,f ,f!,f,Hom,⊗tomeanthecorrespondingdgderivedfunctors:
∗ !
−⊗F : Sh(X)↔Sh(X) :Hom(F,−)
f∗ : Sh(X)↔Sh(Y) :f
∗
f : Sh(Y)↔Sh(X) :f!
!
wheref :Y →X isamapofrealanalyticmanifolds.
TheVerdierdualityD:Sh(X)o →Sh(X)isananti-involution. Itinterchangesshriekwithstar
DD=id, f =Df D, f! =Df∗D.
! ∗
The shrieks and stars are directly related in two cases: when f is proper f = f ; when f is a smooth
! ∗
morphism of relative dimension d , f!(−) (cid:39) f∗(−) ⊗ ω (cid:39) f∗(−) ⊗ or [d ], where or is the
f Y/X Y/X f Y/X
orientationsheafofthefiber.
GivenanopensubsetU ofX anditsclosedcomplementZ,
j i
openinclusion: U (cid:44)→− X ←−(cid:45)Z, closedinclusion,
wehavej∗ =j!andi =i. Furthermore,thereareexacttriangles
∗ !
ii! →id→j j∗ −[1→] , jj! →id→i i∗ −[1→] .
! ∗ ! ∗
These are sheaf-theoretic incarnations of excisions: applied to the constant sheaf on X and taking global
sections,weget
H∗(Z,i!C)→H∗(X,C)→H∗(U,C)−[1→] , H∗(U,C)→H∗(X,C)→H∗(Z,C)−[1→] .
c c c
IfY isalocallyclosedC-submanifoldofX,weusej :Y (cid:44)→X todenotetheinclusion. LetC ∈Sh(Y)
Y Y
denote the constant sheaf on Y, and ω = DC be the Verdier dualizing complex of Y, then ω is the
Y Y Y
canonicallyisomorphictotheshiftedorientationsheafor [dimY]onY. ThestandardsheafonY isj C ,
Y Y∗ Y
andthecostandardsheafonY isj ω .
Y! Y
6 PENGZHOU
Theconstructiblesheavescanbe‘constructed’bytakingshiftsandmappingconesofcertainfinitecol-
lection of sheaves. Let T = {τ } be a triangulation of X by simplices j : τ (cid:44)→ X, where each τ is
α α α α
theembeddingimageofsomeopensimplex. WedenotebyC (T)thethefulldgsubcategoryofSh (X)
∗ T
spannedbystandardobjectsj C .Themorphismsbetweenstandardobjectsarequasi-isomorphictocom-
α∗ τα
plexesconcentratedatdegreezero.
(cid:40)
C ifβ ≥α
Hom (j C ,j C )(cid:39)
ShT(X) β∗ τβ α∗ τα 0 else
whereα≤β ifτ ⊂τ .
α β
ApplyingVerdierdualitytothestandardsheaves, wegetthecostandardsheavesD(j C ) = j ω .
α∗ τα α! τα
Wehave
(cid:40)
C ifα≤β
Hom (j ω ,j ω )(cid:39)
ShT(X) α! τα β1 τβ 0 else.
WedenotebyC(T)thethefulldgsubcategoryofSh (X)spannedbycostandardobjectsj ω
! T α! τα
Lemma2.1([N],Lemma2.3.1). Sh (X)isthetriangulatedenvelopeofC (T)(resp. C(T)).
T ∗ !
Example2.2. LetY =(0,1)⊂R,thenj C istheconstantsheafwithstalkCsupportedontheclosedin-
Y∗ Y
terval[0,1],andj ω isthecostandardsheafwithstalkisomorphictoC[1]supportedontheopeninterval
Y! Y
(0,1).
Example2.3. Letf :{0}(cid:44)→Rn,then
f!(CRn)=Df∗DCRn =Df∗(CRn[n])=D(C{0}[n])=C{0}[−n]
wherewehaveidentifiedtheorientationsheafonRn withtheconstantsheafbychosinganorientationon
Rn.
2.3. Conical Lagrangian and Singular Support. In this subsection, we define singular supports of con-
structible sheaves. Roughly speaking, singular supports encode the ‘positions and directions’ where sec-
tions‘failtopropagate’. Wefirstneedtointroducenotationsfromsymplecticandcontactgeometry.
Let X be a smooth manifold, T∗X its cotangent bundle with the canonical one-form λ = pdq and the
canonicalsympletictwoformω = dλ = dp∧dq. LetT˙∗X = T∗X\X,whereX isidentifiedwiththezero
sectioninT∗X. LetT∞X = T˙∗X/R ,whereR actsbyfiberwisedilation. Thereisanaturalfiberwise
>0 >0
compactifiationofT∗X toT∗X,whereT∞X correspondstothedivisoratinfinityT∗X\T∗X ([NZ],§5.1.1).
A contact manifold (M,ξ) is a smooth manifold of odd dimension 2m + 1, with a smooth rank 2m
subbundleξ ⊂ TM, calledahyperplanedistribution, suchthatlocallyξ = ker(α)forsomeone-formαand
α∧(dα)m (cid:54)= 0. Such a one-form α is called a contact form. The Reeb vector field with respect to a contact
formαistheuniquevectorfieldRsuchthatι α=1andι dα=0. Acontactomorphismbetweencontact
R R
manifoldsisadiffeomorphismthatpreservesthehyperplanedistributions. ALegendriansubmanifoldL
ofM isanm-dimensionalsubmanifoldsuchthatTL⊂ker(α)∩ker(dα).
ThedivisorT∞X atinfinityofthecompactificationT∗X hasanaturalcontactstructuredefinedinthe
followingway: FixanysmoothsectionH oftheR -bundleT˙∗X →T∞X,thenT∞X isdiffeomorphicto
>0
H bythesectionmap; thecanonicalone-formλofT∗X restrictstoacontactformαonH, henceinduces
acontactstructureξ onT∞X. IfwefixaRiemmanianmetriconX,thenthesectionH canbetakenasthe
unitcospherebundle
S∗X ={(x,η)∈T∗X |(cid:107)η(cid:107)=1}.
TheReebflowonS∗X istheunitgeodesicflow. WewillidentifyS∗X andT∞X.
Example2.4. Thesimplestexamplecontactmanifoldisthe1-jetbundleonRn: J1Rn :=T∗ Rn×R ,and
(x,y) z
onechoiceofthecontactformcanbetakenasα=z−(cid:80)n y dx andthecorrespondingReebflowis∂ .
i=1 i i z
TWISTEDPOLYTOPESANDCCC 7
A conical Lagrangian Λ ⊂ T∗X is a Lagrangian (possibly singular) invariant under the R -action. A
>0
homogenousconicalLagrangianisaonecontainedinT˙∗X. GivenaconicalLagrangianΛ, wedefinethe
associatedLegendriansas
Leg(Λ)=Λ∞ =(Λ\X)/R ⊂T∞X.
>0
Conversely,givenaLegendrianL⊂T∞X,weuseLag(L)todenotethehomogeneousconicalLagrangian
inT˙∗X asthepreimageofthequotientT˙∗X →T∞X.
LetS ={S } beaWhitneystratificationofX,thenthereisacanonicalconicalLagrangianassociated
α α∈A
toS,
(cid:91)
Λ := T∗ X
S Sα
α∈A
whereT∗X ={(x,η)∈T∗X |x∈N,η| =0}denotestheconormalbundleofasubmanifoldN ⊂X.
N TN
LetF beaS-constructiblesheafinSh (X)foraWhitneystratificationS. ThesingularsupportSS(F)
S
is a (singular) conical Lagrangian contained in Λ defined in the following way: a point (x,η) ∈ T∗X is
S
not in the singular support SS(F), if there is a small open ball B(x,(cid:15)) around x, and a Morse function
f : B(x,(cid:15)) → R, with f(x) = 0 and df(x) = η, such that for any 0 < δ (cid:28) 1, the canonical restriction
morphism
Γ(f−1(−∞,δ),F)→Γ(f−1(−∞,−δ),F)
isaquasi-isomorphism. Weuse
SS∞(F):=(SS(F))∞ =(SS(F)\X)/R
>0
todenotetheLegendrianinT∞X associatedtotheconicalLagrangianSS(F)inT∗X.
Example2.5. Letj :U =B(0,1)(cid:44)→R2 betheinclusionofanopenunitballinR2. Thenj C issupported
∗ U
ontheclosedsetU,withsingularsupportatinfinityas
SS∞(j C )={(x,η)∈S∗R2 |x∈∂U,η =−d|x|}= .
∗ U
AndjC issupportedontheopensetU,withsingularsupportatinfinityas
! U
SS∞(jC )={(x,η)∈S∗R2 |x∈∂U,η =d|x|}= .
! U
Here the Legendrians are represented by co-oriented hypersurfaces in R2 with hairs indicating the co-
orientation.
ThefollowingLemmafrom[KS]isusefulincharacterisingthesingularsupportunder⊗andHom.
Proposition2.6(Proposition5.4.14of[KS]). LetF andGbelongtoShb(X),then
(1)ifSS∞(F)∩(SS(G)a)∞ =∅,thenSS(F ⊗G)⊂SS(F)+SS(G).
(2)ifSS∞(F)∩SS∞(G)=∅,thenSS(Hom(F,G))⊂SS(G)−SS(F).
where(−)aisthefiberwiseanti-podalmapinT∗X and±isthefiberwisesum/substractioninT∗X.
ForaversionwithoutassumingSS∞(F)∩SS∞(G)=∅,seeCorollary6.4.5and6.2.4inloc.cit.
2.4. Non-characteristicDeformationLemma. JustasinMorsetheory,wherealevelsetsf−1(t)ofaMorse
function f on M has constant diffeomorphism type when t varies in the connected components of the
complementofthecriticalvaluesoff,thenon-characteristicdeformationresultsforconstructiblesheaves
are about the invariance of the hom-complexes Hom(F ,G ) for families of sheaves {F } and {G }, when
t t t t
SS∞(F)andSS∞(G)aredisjoint.
Wefirststatetheversionregardingsectionsofasheafoveranincreasingsequenceofopensets.
8 PENGZHOU
Proposition 2.7 (Proposition 2.7.2 in [KS] ). Let X be a real analytic manifold, F a bounded complex of con-
structiblesheavesinSh(X),andlet{Ut}t∈RbeafamilyofopensubsetsofX. Weassumethefollowingconditions:
(1)U =(cid:83) U forallt∈R.
t s<t s
(2)Forallpairs(s,t)withs≤t,thesetU \U ∩Supp(F)iscompact.
t s
(3)SettingZ =∩ U \U 2,wehaveforallpairs(s,t)withs≤tandallx∈Z \U ,that
s t>s t s s t
Hom(j C ,F)| (cid:39)0.
X\Ut∗ X\Ut x
Thenwehaveforallt∈R,thequasi-isomorphism
∼ (cid:91)
Γ(U,F)−→Γ(U ;F), whereU = U .
t s
s∈R
Remark2.8. ThesectionfunctorcanbeviewedasΓ(U ,F)=Hom(j C ,F). Hencethisisaspecialcase
t Ut! Ut
for Hom(G ,F ). The advantage for this version is that the results holds for the section over union of the
t t
opensets{U },insteadofjustbetweenpairsofopensetsU ,U forsomefinitet,s.
s t s
Thefollowingtwoversionsofthenon-characteristicdeformationresultsarenotgoingtobeusedinthe
paper. Itispresentedheresincetheirconceptualpicturesaresomewhatclearer.
Proposition2.9(Corollary2.10,[S]). LetI beanopenintervalofR,letq : M ×I → I betheprojection,andlet
ι betheembeddingM ×{s}(cid:44)→M ×I. LetF ∈Sh(M ×I),suchthatSS∞(F)∩(T∗ M ×T∗I)∞ =∅andqis
s M
properonSupp(F). SetF =ι∗F. Thenwehaveisomorphisms
s s
Γ(M,F )(cid:39)Γ(M,F ) forall s,t∈I.
s t
Since the hom-complex can be obtained by taking the global section of hom-sheaf, we have a non-
characteristicdeformationresultforHom(F ,G ). Firstwestatealemma:
t t
Lemma2.10(PetrowskytheoremforSheaves,Corollary4.6[S]). LetF,Gbeboundedconstructiblessheavesin
Sh(X). IfSS∞(F)∩SS∞(G)=∅,thenthenaturalmorphism
Hom(F,C )⊗G→Hom(F,G)
X
isanisomorphism.
Proposition2.11. LetI beanopenintervalofR,letq : M ×I → I betheprojection,andletι betheembedding
s
M =M ×{s}(cid:44)→M ×I. LetF,G∈Sh(M ×I),suchthat
s
(1)SS∞(F),SS∞(G)and(T∗ M ×T∗I)∞arepairwisedisjoint,and
M
(2)SS∞(G )∩SS∞(F )=∅,whereF =ι∗F,G =ι∗G,and
s s s s s s
(3)qisproperonSupp(F)andSupp(G).
Thenwehaveisomorphisms
Hom(F ,G )(cid:39)Hom(F ,G ) forall s,t∈I.
s s t t
Proof. Considerthehom-sheafHom(F,G). Weclaimthat
SS∞(Hom(F,G))∩(T∗ M ×T∗I)∞ =∅.
M
Indeed,ifthereexists((x,t),(p,τ))∈T∗(M×I)inSS(Hom(F,G))∩T∗ M×T∗Iforwhich(p,τ)(cid:54)=0,then
M
p=0,τ (cid:54)=0sinceT∗ M ={(x,0):x∈M}. Andthereexists((x,t),(p ,τ ))∈SS(F)and((x,t),(p ,τ ))∈
M 1 1 2 2
SS(G), such that p −p = 0 and τ −τ = τ (Proposition 2.6). By condition (1), p (cid:54)= 0 and p (cid:54)= 0. By
1 2 1 2 1 2
condition(2),ifp ,p arenon-zerothenp (cid:54)= p . Henceitisimpossibletohavep −p = 0,andtheclaim
1 2 1 2 1 2
isproven.
HencebyProposition2.9,wehave
Γ(M,Hom(F,G)| )(cid:39)Γ(M,Hom(F,G)| ) forall t,s∈I.
s t
2Seeerrataathttps://webusers.imj-prg.fr/˜pierre.schapira/books/Errata.pdffortheneedofclosure.
TWISTEDPOLYTOPESANDCCC 9
Finally,weclaimthat
Hom (ι∗F,ι∗G)(cid:39)Γ(M,Hom(F,G)| ) forall s∈I.
M s s s
Let C := ι ι∗C be the constant sheaf support on the slice M ×{t}, for any t ∈ I. Then SS∞(C ) is
t t∗ t M×I t
disjointfromSS∞(F),SS∞(G)andSS∞(Hom(F,G))byassumptionandthefirstclaim. Hencewehave
C ⊗F (cid:39)Hom(Hom(C ,C ),C )⊗F (cid:39)Hom(C [−1],C )⊗F −∼→Hom(C [−1],F)
s s M×I M×I s M×I s
wherePetrowskyTheoremisappliedinthelastisomorphism. ThesameistruebyreplacingF withGand
Hom(F,G). Then
Hom (ι∗F,ι∗G)(cid:39)Hom (F,ι ι∗G)(cid:39)Hom (F,C ⊗G)
M s s M×I s∗ s M×I s
(cid:39) Hom (F,Hom(C [−1],G))(cid:39)Hom (C [−1],Hom(F,G))
M×I s M×I s
(cid:39) Γ(M ×I,Hom(C [−1],C )⊗Hom(F,G))=Γ(M ×I,C ⊗Hom(F,G))
s M×I s
(cid:39) Γ(M,Hom(F,G)| )
s
Thisfinishestheproofoftheproposition. (cid:3)
3. TORICGEOMETRY
Ann-dimensionalsmoothprojectivecomplexmanifoldX istoricifthereisaholomorphic(C∗)n-action
withanopendenseorbitXo onwhich(C∗)n actsfreely. ThecomplementoftheopenorbitD =X\Xo isa
simplenormalcrossingdivisorwithirreducibletorus-invariantcomponents.
Wefirstreviewthestandardsetupandnotationforthecombinatorialdatausedfordefiningatoricva-
riety. Thenweexplaintherelationshipbetweenequivariantlinebundles,toricdivisors,andtwistedpoly-
topes(asacollectionoflabeledvertices). Finally,wereview‘twistedpolytopesheaves’,thecorresponding
constructiblesheavesforequivariantlinebundlesundertheequivariantCCC.
3.1. FanData. Thedataofatoricmanifoldcanbeexpressedcombinatoricallyusingafan. LetN (cid:39) Zn be
aranknlattice,withNR =N ⊗ZR. LetM =Hom(N,Z)betheduallatticeandMR =M ⊗ZRbethedual
vector space. Let (cid:104)−,−(cid:105) : MR ×NR → R be the dual pairing. Let TM = MR/M be a real n-dimensional
torus,andπ :MR →TM bethequotientmap. Werecallthefollowingdefinitions.
(cid:80)
(1) Aconvexpolyhedralconeσ ⊂ NR isasetoftheformσ = cone(S) = { u∈Sλuu | λu ≥ 0},where
theconegeneratorS ⊂ NR isafinitesubset. Aconeσ isrationalifthereisageneratorS forσ such
thatS ⊂N. Aconeisstronglyconvexifitdoesnotcontainanynon-triviallinearsubspaceofNR.
(2) Letσ ∈Σbeacone,wedefinethedual(closed)coneσ∨as
σ∨ :={x∈MR |(cid:104)x,y(cid:105)≥0,∀y ∈σ}.
Wealsodefineσ⊥ ={x∈M |(cid:104)x,y(cid:105)=0,∀y ∈σ}⊂MR,andσo (resp. (σ∨)o)astherelativeinterior
ofσ(resp. σ∨).
(3) AfaceofaconeσisthesubsetH ∩σforsomem ∈ σ∨ andH = m⊥. Weuseσ(r)todenotethe
m m
collectionofr-dimensionalfacesofσ.
(4) AfanΣinNR isafinitecollectionofstronglyconvexrationalpolyhedralconesσ ⊂ NR,suchthat
(a)ifσ ∈ΣthenanyfaceofσisinΣ,and(b)ifσ ,σ areconesinΣthenσ ∩σ isafaceinbothσ
1 2 1 2 1
andσ . WeuseΣ(r)todenotethecollectionofr-dimensionalconesinΣ.
2
(5) AfanΣinNRiscomplete,ifitssupport|Σ|:=∪σ∈ΣσistheentireNR. AcompletefanΣissmooth,if
eachmaximalconeσ ∈Σ(n)isgeneratedbyalatticebasisofN.
(6) AsmoothcompletefanΣisprojective,ifthereexistsaconvexpiecewiselinearfunctionϕ:NR →R,
suchthatthemaximallinearitydomainsofϕarethemaximalconesofΣ. (cf. Proposition3.6)
SeeExample3.17forafanofP2.
Assumption: WewillalwaysassumeΣtobeasmoothprojectivefan.
10 PENGZHOU
TheaffinetoricvarietyX isthendefinedby
σ
X =Spec(C[σ∨∩M])
σ
whereC[σ∨∩M])isthegroupringoftheabeliansemi-groupσ∨∩M. Ifτ ⊂σisafaceofσ,thenσ∨ ⊂τ∨,
hence C[σ∨ ∩M]) (cid:44)→ C[τ∨ ∩M]), and X (cid:44)→ X is an open inclusion. We may equip Σ with a partial
τ σ
ordering,forτ,σ ∈ Σ,τ ≤ σ ⇐⇒ τ ⊂ σ. ThenX canbegluedtogetherfromaffineopenpiecesX ,asa
Σ σ
colimitofschemes
X =colim X .
Σ σ∈Σ σ
3.2. ToricDivisors,SupportFunctionsandTwistedPolytopes. Foreachrayρ∈Σ(1),letv ∈ρ∩N bea
ρ
minimalraygenerator,λvρ :C∗ →N⊗ZC∗astheone-parametersubgroup,andDρ ={limt→0λvρ(t)·x|x∈Xo}
(cid:80)
the torus-invariant divisor, or, a toric divisor. We write for a summation over the rays ρ ∈ Σ(1) when
ρ
thereisnodangerofconfusion.
LetD =(cid:80) a D beatoricR-divisoronX ,a ∈R.Ifa ∈Zforallρ,thenDisanintegraltoricdivisor,
ρ ρ ρ Σ ρ ρ
ortoricZ-divisor. Therearetwoequivalentwaystodescribeatoricdivisor,eitherusingasupportfunction
ϕD onNR,oratwistedpolytopeχD onMR.
Definition 3.1 (Support function). A support function for Σ is a continuous piecewise linear function ϕ :
NR →R,suchthatforeachmaximalconeσ ∈Σ(n),therestrictionϕ|σ islinear.
• AsupportfunctionϕisintegralifitsendsN toZ.
• Asupportfunctionϕisconvex,ifforanyx,y ∈NR
ϕ(tx+(1−t)y)≤tϕ(x)+(1−t)ϕ(y).
• Furthermore,wesayϕisstrictlyconvex,ifthestrictinequalityholdswheneverx,yisnotcontained
inthesamecone.
Definition 3.2 (Twisted polytope). A twisted polytope for Σ is an assignment of element in MR to top-
dimensionalconesinΣ,
χ:Σ(n)→MR, σ (cid:55)→χσ,
suchthatifσ,τ ∈Σ(n)then(cid:104)χ ,·(cid:105)=(cid:104)χ ,·(cid:105)onσ∩τ.
σ τ
• Atwistedpolytopeχisintegralifthefunctionχ ∈M forallσ ∈Σ(n).
σ
• Ifχisatwistedpolytope,thenforanyconeσ ∈Σ,wedefineχσ ∈MR/σ⊥by
χσ = AffineHull({χτ | τ isamaximalconecontainingσ})⊂MR.
• For any x ∈ MR, let χ+x denote the translated twisted polytope that sends σ (cid:55)→ χσ +x for any
σ ∈Σ(n).
Remark3.3. Thedataforatwistedpolytopeisacollectionofthevertices,labelledbyΣ(n).
Proposition3.4. LetΣbeann-dimensionalsmoothprojectivefan. Thenwehaveacanonicalequivalencesamong
thefollowingthreetypesofobjects. (1)AtoricR-divisorD =(cid:80) a D ,a ∈R.
ρ ρ ρ ρ
(2)Atwistedpolytope,χ:Σ(n)→MR.
(3)Asupportfunction,ϕ:NR →R.
Inparticular,integraltoricdivisorscorrespondstointegraltwistedpolytopesandintegralsupportfunctions.
Proof. (2) ⇔ (3). Givenχ, wemaydefineϕbyϕ(x) = (cid:104)χ ,x(cid:105)ifx ∈ σ forsomemaximalconeσ ∈ Σ(n).
σ
Thisiswell-definedsinceifx ∈ σ∩τ,then(cid:104)χ ,x(cid:105) = (cid:104)χ ,x(cid:105). Conversely,givenϕ,thenforeachmaximal
σ τ
coneσ, thelinearfunctionϕ|σ determinesanelementinMR, denotedbyχσ. Thecontinuityofϕensures
(cid:104)χ ,·(cid:105)=(cid:104)χ ,·(cid:105)onσ∩τ.
σ τ
(1) ⇔ (3). GivenatoricR-divisorD = (cid:80) a D ,foreachρ ∈ Σ(1),wedefineϕ| : ρ → Rbyv (cid:55)→ a .
ρ ρ ρ ρ ρ ρ
SincetheconesofΣaresimplicial,thereisauniquepiecewiselinearextensionofϕtoNR thatislinearin
eachconeofΣ. Conversely,givenϕ,leta =ϕ(v )foreachρ∈Σ(1).
ρ ρ
Theclaimonintegralityisstraightforwardtoverify. ThisfinishestheproofoftheProposition. (cid:3)