Conifold transitions via affine geometry and mirror symmetry

96 Pages·2017·1.1 MB·English

Geometry & Topology18(2014)1769–1863
msp
Conifold transitions via affine geometry
and mirror symmetry
RICARDO CASTAÑO-BERNARD
DIEGO MATESSI
MirrorsymmetryofCalabi–YaumanifoldscanbeunderstoodviaaLegendreduality
betweenapairofcertainaffinemanifoldswithsingularitiescalledtropicalmanifolds.
In this article, we study conifold transitions from the point of view of Gross and
Siebert[11;12;13]. Weintroducethenotionsoftropicalnodalsingularity,tropical
conifolds,tropicalresolutionsandsmoothings. Weinterpretknownglobalobstruc-
tionstothecomplexsmoothingandsymplecticsmallresolutionofcompactnodal
Calabi–Yaumanifoldsintermsofcertaintropical 2–cyclescontainingthenodesin
theirassociatedtropicalconifolds. Weprovethattheexistenceofsuchcyclesimplies
thesimultaneousvanishingoftheobstructiontosmoothingtheoriginalCalabi–Yau
andtoresolvingitsmirror. Weformulateaconjecturesuggestingthattheexistence
ofthesecyclesshouldimplythatthetropicalconifoldcanberesolvedanditsmirror
can be smoothed, thus showing that the mirror of the resolution is a smoothing.
Wepartiallyprovetheconjectureforcertainconfigurationsofnodesandforsome
interestingexamples.
14J32;14J33,53D37
1 Introduction
Ageometrictransitionbetweenapairofsmoothvarietiesistheprocessofdeforming
thefirstvarietytoasingularoneandthenobtainingthesecondonebyresolvingthe
singularities. Thefirstvarietyiscalledasmoothingandthesecondonearesolution.
In [25], Morrison conjectures that, in certain circumstances, mirror symmetry should
map a pair of smooth Calabi–Yau manifolds, related by a geometric transition, to
anotherpair,alsorelatedbyageometrictransitionbutwiththerolesreversed,sothat
themirrorofasmoothingshouldbearesolutionandvice-versa. Thisideaissupported
byevidencesandexamples. Morrisonalsosuggeststhatanewunderstandingofthis
phenomenoncouldcomefromtheSYZinterpretationofmirrorsymmetryasaduality
of special Lagrangian torus fibrations. Building on ideas of Hitchin [18], Gross [7],
Gross and Wilson [14], Kontsevich and Soibelman [20; 21] and others on the SYZ
Published: 7July2014 DOI:10.2140/gt.2014.18.1769
1770 RicardoCastaño-BernardandDiegoMatessi
conjecture, Gross and Siebert show that mirror pairs of Calabi–Yau manifolds can
be constructed from a Legendre dual pair of affine manifolds with singularities and
polyhedral decompositions, also called tropical manifolds [11; 12; 13]. Inthis article
weconsiderthespecialcaseofconifoldtransitions. Weintroducethenotionoftropical
conifold,ieoftropicalmanifoldwithnodes,andweshowthatthesmoothing/resolution
processalsohasanaturaldescriptioninthiscontext(thiswasfirstobservedbyGross[7]
andRuan [27]). Indeed thesmoothingofa tropicalconifoldsimultaneouslyinduces
a resolution of its mirror, but in general the process is obstructed. To study global
obstructions we introduce the notion of tropical 2–cycle in a tropical conifold. Our
mainresultisthefollowing:
MainTheorem Theexistenceofatropical 2–cyclecontainingthenodesinatropical
conifoldimpliesthevanishingoftheobstructionstothesmoothingoftheassociated
Calabi–Yauvarietyandtotheresolutionofitsmirror.
SeeTheorem7.3fortheprecisestatement. Weformulateaconjectureclaimingthatthe
inversealsoholds,iethatthevanishingoftheseobstructionscanalwaysbedetectedby
tropical 2–cycles. Moreoverweexpectthattheexistenceofaresolution/smoothingof
asetofnodesinthetropicalconifoldshouldbeequivalenttosomepropertyexpressible
intermstropical 2–cyclescontainingthenodes. Thiswouldshowthatthesmoothing
andtheresolutionarethemselvesmirrorpairsinthesenseofGrossandSiebert. We
partially prove the conjecture for some special configurations of nodes and for an
interestingfamilyofexamples.
1.1 Conifoldtransitions
Anodeisthe 3–foldsingularitywithlocalequation xy(cid:0)zwD0. Asmallresolution
ofanodehasa P1 asitsexceptionalcycle,withnormalbundle O .(cid:0)1/˚O .(cid:0)1/.
P1 P1
The smoothing of a node (ie xy (cid:0)zw D (cid:15)) produces a Lagrangian 3–sphere as a
vanishing cycle. A conifold transition is the geometric transition associated with a
3–fold with nodal singularities, ie a “conifold.” It was proved by Friedman [5] and
Tian [32] that a compact complex conifold can be smoothed to a complex manifold
ifandonlyif theexceptionalcyclesof asmallresolutionsatisfya“goodrelation” in
homology;seeEquation(20). Similarly,onthesymplecticside,itwasshownbySmith,
Thomas andYau[30] thata “symplectic conifold” has asymplectic (small)resolution,
withsymplecticexceptionalcycles,ifandonlyifthevanishingcyclesofasmoothing
satisfyagoodrelation. Thesetworesultsareamanifestationoftheideathatthemirror
ofacomplexsmoothingshouldbeasymplecticresolution.
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Conifoldtransitionsviaaffinegeometryandmirrorsymmetry 1771
1.2 SYZconjecture
Mirror symmetry is usually computed when the Calabi–Yau manifold is the generic
fibreofafamily W (cid:31)!C,wherethespecialfibre (cid:31) D (cid:0)1.0/ ishighlydegenerate
0
(eg one requires that (cid:31) has maximally unipotent monodromy). The Strominger–Yau–
0
{
Zaslow(SYZ) Conjecture[31] claimsthat mirrorCalabi–Yau pairs X and X should
admit“dual”specialLagrangianfibrations, fW X !B and f{W X{ !B. Thisoriginal
ideahasbeenrevisedbyGrossandWilson[7;14]andKontsevichandSoibelman[20],
whoclaimedthatspecialLagrangianfibrationsshouldexistonlyinsomelimitingsense
asthefibre (cid:31) D (cid:0)1.s/ approachesthesingularfibre (cid:31) (seealsothesurveypaper
s 0
byGross[10]). This“limitingfibration”canbedescribedintermsofacertainstructure
onthebase B ofthefibration. Here B isarealmanifoldandthestructureon B should
containinformationconcerningthecomplexandsymplecticstructureoftheCalabi–Yau
manifold. Moreover, this data contains intrinsically a duality given by a Legendre
transform. TheimportantfactisthatthestructureonB shouldallowthe“reconstruction”
oftheoriginalCalabi–Yau. Thisisknownasthereconstructionproblem. Therefore,
findingthe mirrorofagivenfamily W (cid:31)!C ofCalabi–Yaumanifolds becomesthe
processofconstructing B,withitsstructure,applyingtheLegendretransformtoobtain
{
the dual base B, with dual structure, and then reconstructing the mirror family via
somereconstructiontheorem. Forinstance,indimension 2,KontsevichandSoibelman
[21]constructarigidanalytic K3 fromanaffinestructureon S2 with 24 punctures.
1.3 Tropicalmanifoldsandmirrorsymmetry
In[11;12;13]GrossandSiebertcompletedthisprograminalldimensions. On B they
considerthe structureofanintegralaffinemanifoldwith singularitiesandpolyhedral
decompositions. Roughly this means B is obtainedby gluinga set of n–dimensional
integralconvexpolytopesin Rn byidentifyingfacesviaintegralaffinetransformations
(thisisthepolyhedraldecomposition,denoted P). Then,atthevertices v of P one
definesafanstructure,whichidentifiesthetangentwedgesofthepolytopesmeeting
at v with the cones of a fan †v in Rn. For a certain codimension-2 closed subset
(cid:129) (cid:26) B, this structure determines an atlas on B D B(cid:0)(cid:129) such that the transition
0
mapsareintegralaffinetransformations. Theset (cid:129),calledthediscriminantlocus,is
the set of singularities of the affine structure. An additional crucial piece of data is
apolarisation,consistingofaso-called“strictlyconvexmultivaluedpiecewiselinear
function” (cid:30) on B. Sucha (cid:30) isspecifiedbythedataofastrictlyconvexpiecewiselinear
function (cid:30)v definedoneveryfan †v,pluscompatibilityconditionsbetween (cid:30)v and (cid:30)w
for vertices v and w belonging to a common face. All these data, which we denote
by the triple .B;P;(cid:30)/, are also called a polarised tropical manifold. The “discrete
Geometry & Topology,Volume18(2014)
1772 RicardoCastaño-BernardandDiegoMatessi
Legendretransform”associatesto .B;P;(cid:30)/ anothertriple .B{;P{;(cid:30){/. Essentially,at
avertex v of B,thefan †v andfunction (cid:30)v providean n–dimensionalpolytope v{,
bythestandardconstructionintoricgeometry. Twopolytopes v{ and w{,associatedto
vertices v and w onacommonedgeof P,canbegluedtogetheralongafaceusing
thecompatibilities betweenthepairs .†v;(cid:30)v/ and .†w;(cid:30)w/. Thisgives B{ andthe
polyhedral decomposition P{. The fan structure and function (cid:30){ at the vertices of P{
comefromthe n–dimensionalpolytopesof P essentiallyusingtheinverseconstruction.
In order to have satisfactory reconstruction theorems it is necessary to put further
technical restrictions on .B;P;(cid:30)/. Gross and Siebert define such conditions and
call them “positivity and simplicity.” For convenience, we will say that a polarised
tropical manifold is smooth if it satisfies these nice conditions. In particular, in the
3–dimensional case smoothness of B amounts to the fact that (cid:129) is a 3–valent
graph and the vertices can be of two types: “positive” or “negative”, depending
on the local monodromy of the affine structure. The Gross–Siebert reconstruction
theorem [13] ensures that given a smooth polarised tropical manifold .B;P;(cid:30)/, it
is possible to construct a toric degeneration W (cid:31) ! C of Calabi–Yau varieties,
such that B is the dual intersection complex of the singular fibre (cid:31) . The mirror
0
family {W (cid:31){ ! C is obtained by applying the reconstruction theorem to the Le-
gendredual .B{;P{;(cid:30){/.
The integral affine structure on B DB(cid:0)(cid:129) implies the existence of a local system
0
ƒ(cid:3)(cid:26)T(cid:3)B ,whosefibres ƒ ŠZn aremaximallatticesin T(cid:3)B . Thenonecanform
0 b b 0
the n–torusbundle X DT(cid:3)B =ƒ(cid:3) over B . Thestandardsymplecticformon T(cid:3)B
B0 0 0 0
descends to X and the projection f W X !B is a Lagrangian torus fibration.
B0 0 B0 0
In[4],weprovedthatif B isa 3–dimensionalsmoothtropicalmanifoldthenonecan
formasymplecticcompactificationof X . Thisisasymplecticmanifold X ,con-
B0 B
taining X asadenseopensubset,togetherwithaLagrangianfibration fW X !B
B0 B
whichextends f . ThisisdonebyinsertingsuitablesingularLagrangianfibresover
0
points of (cid:129). Topologically the compactification X is based on the one found by
B
Grossin[8]. Itisexpectedthat X shouldbediffeomorphictoasmoothfibre (cid:31) ofthe
B s
family W (cid:31)!C intheGross–Siebertreconstructiontheorem,whosedualintersection
complexis .B{;P{;(cid:30){/. ThisresulthasbeenannouncedinGross[9,Theorem0.1]. A
completeproofforthequintic 3–foldin P4 isfoundin[8]. Wealsoexpectthat X
B
shouldbesymplectomorphicto (cid:31) withasuitableKählerform,althoughthereisno
s
proofofthisyet.
1.4 Summaryoftheresults
Indimension 3,smoothnessof B ensuresthegeneralfibre (cid:31) of W (cid:31)!C issmooth.
s
Weintroducethenotionof(polarised)tropicalconifold,inwhichthediscriminantlocus
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Conifoldtransitionsviaaffinegeometryandmirrorsymmetry 1773
isallowedtohave 4–valentvertices. Suchvertices,whichwecall(tropical)nodes,are
of two types: negative and positive. Away from these nodes, a tropical conifold is a
smoothtropicalmanifold. WebelievetheGross–Siebertreconstructiontheoremcanbe
extended alsoto tropical conifolds, butthe generalfibre (cid:31) shouldbe avarietywith
s
nodes. Thisishintedbythefactthatthelocalconifold
xy(cid:0)wzD0
has a pair of torus fibrations which induce on the base B the same structure as in a
neighbourhoodofpositiveornegativenodes. Infact,inCorollary6.7weshowif B
is a tropical conifold, then X can be topologically compactified to a topological
B0
conifoldX (ieasingulartopologicalmanifoldwithnodalsingularities). Aninteresting
B
observationisthattheLegendretransformofapositivenodeisthenegativenode. In
particular we also have the mirror conifold X{. This extends topological mirror
B
symmetryof[8]toconifolds. Thenwegivealocaldescriptionofthesmoothingand
resolution of a node in a tropical conifold (see Figures 9 and 10). It turns out that
the Legendre dual of a resolution is indeed a smoothing. At the topological level
this was already observed by Gross [7] and Ruan [27], who also discusses a global
example. Theinterestingquestion isglobal: givenacompact tropicalconifold,canwe
simultaneously resolve or smooth its nodes? We give a precise procedure to do this.
Itturnsout thatthesmoothingofnodes inatropicalconifold simultaneously induces
the resolution of the nodes in the mirror. What are the obstructions to the tropical
resolution/smoothing? Forthispurposewedefinethenotionoftropical 2–cycleinside
a tropical conifold. These objects resemble the usual notion of a tropical surface as
definedforinstancebyMikhalkinin[23]. Atropical 2–cycleisgivenbyaspace S
andanembedding jW S !B withsomeadditionalstructure. Thespace S hasvarious
types of interior and boundary points. For instance at generic points, S is locally
Euclidean, at the codimension-1 points S is modelled on the tropical line times an
intervalandatcodimension-2 points S ismodelledonthetropicalplane(seeFigure13)
andsoon. InTheorem7.3weprovethatif j.S/ containstropicalnodes,thenboththe
vanishingcyclesassociatedtothenodesin X andtheexceptionalcurvesassociatedto
B
thenodesin X{ satisfyagoodrelation. Theideaisthattropical 2–cyclescanbeused
B
toconstructeither 4–dimensionalobjectsin XB or 3–dimensionalonesin XB{ (see
also Aspinwall, Bridgeland, Craw, Douglas, Gross, Kapustin, Moore, Segal, Szendro˝i,
andWilson[1,Chapter6],wherethelocaldualitybetween A–branesand B–branesis
explained.) Thus obstructions vanish on both sides of mirror symmetry. The results
of Friedman, Tian, and Smith, Thomas and Yau then lead us to Conjecture 8.3. It
statesanygoodrelationamongthevanishingcyclesofasetofnodesin X isalinear
B
combinationofgoodrelationscomingfromtropical 2–cyclesin B. Moreoverthere
shouldexistsome propertyofthesetropical 2–cycleswhichis equivalentto thefact
Geometry & Topology,Volume18(2014)
1774 RicardoCastaño-BernardandDiegoMatessi
that B canbetropicallyresolved. Asapartialconfirmationofthisconjecture,weprove
thenodescontainedinsomespecialconfigurationsoftropical 2–cyclescanalwaysbe
tropicallyresolved(Theorems8.5,8.7,8.9andCorollary8.6).
Finallyweapplytheseresultstospecificexamples. WeconsiderthecaseofSchoen’s
Calabi–Yau[29],whichisafibredproductoftworationalellipticsurfaces. Acorre-
spondingtropicalmanifoldhasbeendescribedbyGrossin[9]. Itispossibletomodify
the example in many ways so that we obtain a tropical conifold with various nodes.
We show how these nodes can be resolved/smoothed and thus obtain new tropical
manifolds. Theinterestingfactisthatthisprocedureautomaticallyproducesthemirror
families via discrete Legendre transform and the reconstruction theorems. For this
classofexampleswealsopartiallyproveConjecture8.3.
Notation
We denote the convex hull of a set of points q ;:::;q in Rn by Conv.q ;:::;q /.
1 r 1 n
Givenasetofvectors v ;:::;v 2Rn theconespannedbythesevectorsistheset
1 r
(cid:26)Xr ˇ (cid:27)
Cone.v ;:::;v /D t v ˇt (cid:21)0;j D1;:::;r :
1 r j j ˇ j
jD1
2 Affine manifolds with polyhedral decompositions
Wegiveaninformalintroductiontoaffinemanifoldswithsingularitiesandpolyhedral
decompositions. Wereferto[11]forprecisedefinitionsandproofs.
2.1 Affinemanifoldswithsingularities
Let M ŠZn bealatticeanddefine M DM ˝ R andlet
R Z
Aff.M/DM ÌGl.Z;n/
0
bethegroupofintegralaffinetransformationsof M . If M and M aretwolattices,
R
then Aff.M;M0/ is the Z–module of integral affine maps between M and M0 .
R R
Recall that an integral affine structure A on an n–manifold B is given by an open
cover fU g and an atlas of charts (cid:30) W U ! M whose transition maps (cid:30) ı(cid:30)(cid:0)1
i i i R j i
are in Aff.M/. An integral affine manifold is a manifold B with an integral affine
structure A. Acontinuousmap fW B !B0 betweentwointegralaffinemanifoldsis
integralaffineif,locally, f isgivenbyelementsof Aff.M;M0/.
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Conifoldtransitionsviaaffinegeometryandmirrorsymmetry 1775
An affine manifold with singularities is a triple .B;(cid:129);A/, where the B is an n–
manifold, (cid:129) (cid:18) B is a closed subset such that B D B (cid:0)(cid:129) is dense in B and A
0
is an integral affine structure on B . The set (cid:129) is called the discriminant locus. A
0
continuousmap fW B !B0 ofintegralaffinemanifoldswithsingularitiesisintegral
affine if f(cid:0)1.B00/\B0 is dense in B and fjf(cid:0)1.B00/\B0W f(cid:0)1.B00/\B0 ! B00 is
integral affine. Furthermore f is an isomorphism of integral affine manifolds with
singularitiesif fW .B;(cid:129)/!.B0;(cid:129)0/ isahomeomorphismofpairs.
2.2 Paralleltransportandmonodromy
Given an affine manifold B, let .U;(cid:30)/ 2 A be an affine chart with coordinates
u ;:::;u . Then the tangent bundle TB (resp. cotangent bundle T(cid:3)B) has a flat
1 n
connection r definedby
r@ D0 .resp. rdu D0/
uj j
for all j D1;:::;n and all charts .U;(cid:30)/2A. Then parallel transport along loops
basedat b2B givesthemonodromyrepresentation (cid:26)zW (cid:25) .B;p/!Gl.T B/. Gross
1 b
andSiebertalsointroducethenotionofholonomyrepresentation,whichisdenoted (cid:26)
and has values in Aff.T B/, and (cid:26)z coincides with the linear part of (cid:26). In the case
b
ofanaffinemanifoldwithsingularities .B;(cid:129);A/,themonodromyrepresentationis
(cid:26)zW (cid:25) .B ;p/!Gl.T B /.
1 0 b 0
Integralityimpliestheexistenceofamaximalintegrallatticeƒ(cid:26)TB (resp.ƒ(cid:3)(cid:26)T(cid:3)B )
0 0
definedby
(1) ƒj Dspan h@ ;:::;@ i .resp. ƒ(cid:3)j Dspan hdu ;:::;du i/:
U Z u1 un U Z 1 n
Wecanthereforeassumethat (cid:26)z hasvaluesin Gl.Z;n/.
2.3 Polyhedraldecompositions
Rather than recalling here the precise definition of an integral affine manifold with
singularitiesandpolyhedraldecompositions (ie [11, Definition 1.22]), it is better to
recall the standard procedure to construct them; see [11, Construction 1.26]. We
0
start with a finite collection P of n–dimensional integral convex polytopes in M .
R
0
Themanifold B isformedbygluing together thepolytopesof P viaintegralaffine
identificationsof theirproper faces. Then B hasa celldecomposition whosecells are
0
theimagesoffacesofthepolytopesof P . Denoteby P thissetofcells. Weassume
that B isacompactmanifoldwithoutboundary. Wenowconstructtheintegralaffine
atlas A on B. Firstofall,theinteriorofeachmaximalcellof P canberegardedasthe
domainofanintegralaffinechart,sinceitcomesfromtheinteriorofapolytopein M .
R
Geometry & Topology,Volume18(2014)
1776 RicardoCastaño-BernardandDiegoMatessi
To define a full atlas we need charts around points belonging to lower-dimensional
cells. In factthis willbe possibleonly afterremovingfrom B aset (cid:129)0 whichwe now
define. Let Bar.P/ bethefirstbarycentricsubdivisionof P. Thendefine (cid:129)0 tobethe
unionofallsimplicesof Bar.P/ notcontainingavertexof P (ie 0–dimensionalcells)
or the barycenter of a maximal cell. For a vertex v 2P, let Wv be the union of the
interiorsof allsimplicesof Bar.P/ containing v. Then Wv isan openneighbourhood
of v and
fWv jv isavertexofPg[fInt.(cid:27)/j(cid:27) 2Pmaxg
forms a covering of B(cid:0)(cid:129)0. A chart on the open set Wv is given by a fanstructure
atthevertex v;see[11,Construction1.26]. Thisconstructiongivesanintegralaffine
atlason B(cid:0)(cid:129)0. Inmanycasestheset (cid:129)0 istoocrudeandtheaffinestructurecanbe
extendedtoalargersetthan B(cid:0)(cid:129)0. Thiscanbedoneasfollows. Noticethat (cid:129)0 isa
unionofcodimension-2 simplices. Thenlet (cid:129) betheunionofthosesimplicesaround
which local monodromy is not trivial. In [11, Proposition 1.27] it is proved that the
affinestructureon B(cid:0)(cid:129)0 canbeextendedto B(cid:0)(cid:129).
GrossandSiebertalsointroducethecrucialnotionoftoricpolyhedraldecomposition.
Essentiallythis conditionestablishes certaincompatibilities between fans †v and †w
atvertices v and w lyingonsomecommoncell. Wewillcomebacktothisinthenext
twoparagraphs.
2.4 Localpropertiesofmonodromy
Themonodromyrepresentationofaffinemanifoldswithpolyhedraldecompositions
has someuseful distinguishedproperties, which wenowdescribe. Firstof allnotice
that (cid:129) iscontainedinthecodimension-1 skeleton. Let (cid:28) beacellof P ofcodimension
at least 1, then it is shown in [11, Proposition 1.29] that the tangent space to (cid:28) is
monodromyinvariantwithrespecttothelocalmonodromynear(cid:28). Moreprecisely,there
existsaneighbourhood U(cid:28) of Int.(cid:28)/ suchthat,if b2(cid:28)(cid:0)(cid:129) and (cid:13) 2(cid:25)1.U(cid:28)(cid:0)(cid:129);b/,then
(cid:26)z.(cid:13)/.w/Dw forevery w tangentto (cid:28) in b. Moreover(see[11,Proposition1.32])the
polyhedralsubdivisionistoricifandonlyifforevery (cid:28) thereexistsaneighbourhood U(cid:28)
of Int.(cid:28)/ suchthat,if b2(cid:28)(cid:0)(cid:129) and (cid:13) 2(cid:25)1.U(cid:28)(cid:0)(cid:129);b/,then (cid:26)z.(cid:13)/.w/(cid:0)w istangent
to (cid:28) forevery w2T B .
b 0
2.5 Quotientfans
If the polyhedral decomposition is toric (see above), then to every cell (cid:28) 2 P one
can associate a complete fan †(cid:28), called the quotient fan of (cid:28), whose dimension is
equal to the codimension of (cid:28). It is defined as follows. Let b 2Int.(cid:28)/(cid:0)(cid:129), then to
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Conifoldtransitionsviaaffinegeometryandmirrorsymmetry 1777
every (cid:27) such that (cid:28) (cid:26)(cid:27), one can associate the tangent wedge of (cid:27) at b. This can
be viewed as a convex rational polyhedral cone inside T B (with lattice structure
b 0
given by ƒ). The union of all such cones forms a complete fan in T B , which is
b 0
the pullback of a complete fan in the quotient TbB0=Tb(cid:28). Let †(cid:28) be such a fan.
The toric condition ensures that the quotient spaces TbB0=Tb(cid:28) and the fan †(cid:28) are
independentof b2(cid:28)(cid:0)(cid:129),infacttheycanbeall identified viaparalleltransportalong
pathscontainedinasuitablysmallneighbourhood U(cid:28) of Int.(cid:28)/. Thelocalpropertiesof
monodromy, assuming thetoric condition, implythat thisidentification is independent
ofthechosenpath.
2.6 Examples
Inthefollowingexamples B willbeallowedtohaveboundaryoreventobeconstructed
usingunboundedpolytopes. Theconstructionabovecanbeeasilyadaptedtothesecases.
Example2.1 (Thefocus-focussingularity) Herethedimensionis nD2. Theset P0
isgivenbytwopolytopes: astandardsimplexandasquare Œ0;1(cid:141)(cid:2)Œ0;1(cid:141). Gluethem
along one edge to form B (see Figure 1). Let e be the common edge, and let v
1
and v betheverticesof e. Thediscriminantlocus (cid:129) consistsofthebarycenter of e.
2
Consider the fan in R2 whose 2–dimensional cones are two adjacent quadrants, ie
Cone.e ;e / and Cone.e ;(cid:0)e /,where fe ;e g isthestandardbasisof R2. Thenthe
1 2 1 2 1 2
fanstructureat v , j D1;2,identifiesthetangentwedgesofthetwopolytopeswith
j
thesetwocones, insuchawaythat theprimitivetangentvectorto e at v ismapped
j
to e (seeFigure1).
1
v (cid:0)e
2 2
e
2 e
1
e
1
(cid:0)e v e
2 1 2
Figure1
Consider a loop (cid:13) which starts at v , goes into the square, passes through v and
1 2
comesbackto v whilepassinginsidethetriangle. Onecaneasilycalculatethat (cid:26)z.(cid:13)/,
1
computedwithrespecttothebasis fe ;e g,asdepictedinFigure1,isthematrix
1 2
(cid:18) (cid:19)
1 1
:
0 1
Thesingularpoint (cid:129) inthisexampleiscalledthefocus-focussingularity.
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1778 RicardoCastaño-BernardandDiegoMatessi
Example2.2 (Genericsingularity) Thisa 3–dimensionalexampleand itisjust the
productofthepreviousexampleby Œ0;1(cid:141). Here (cid:129) consistsofasegment.
v v
2 2
e
2 e3 e2 (cid:0)e
3
v3 e1 v1 v1 e1 v3
Figure2
Example 2.3 (The negative vertex) Here nD3. Let Pn be the standard simplex
in Rn. The set P0 consists of two polytopes: P3 and P2(cid:2)P1 which we glue by
identifyingthetriangularface P2(cid:2)f0g withafaceof P3. InFigure2wehavelabelled
theverticesofthesetwofacesby v ;v ;v andtheidentificationis donebymatching
1 2 3
the vertices with the same labelling. Now consider the fan in R3 whose cones are
two adjacent octants (ie Cone.e ;e ;e / and Cone.e ;e ;(cid:0)e /, where fe ;e ;e g
1 2 3 1 2 3 1 2 3
is the standard basis of R3). At every vertex v identify the tangent wedges of the
j
two polytopes with these two cones, in such a way that the tangent wedge to the
commonfaceismappedto Cone.e ;e /. Thereismorethanonewaytodothis(since
1 2
Cone.e ;e / hasnontrivialautomorphisms),butanychoiceisagoodchartoftheaffine
1 2
structure. If one fixes an orientation then a choice can be made so that the chart is
oriented. The discriminant locus (cid:129) is the Y-shaped figure depicted in (red) dashed
linesinFigure2. Nowlet (cid:13) bethepathgoingfrom v to v bypassinginto P3 and
j 3 j
thencomingbackto v bypassinginto P2(cid:2)P1. Itcanbeeasilyshownthat (cid:26)z.(cid:13) /
3 1
and (cid:26)z.(cid:13) / aregivenrespectivelybythematrices
2
0 1 0 1
1 0 1 1 0 0
@0 1 0A; @0 1 1A:
0 0 1 0 0 1
Thevertexof (cid:129) inthisexampleiscalledthenegativevertex.
Example 2.4 (The positive vertex) In this case B D R2(cid:2)Œ0;1(cid:141) with polyhedral
decompositiongivenbythefollowingunboundedpolytopes(seeFigure3):
Q Dfx(cid:21)maxfy;0g;z2Œ0;1(cid:141)g
1
Q Dfy (cid:21)maxfx;0g;z2Œ0;1(cid:141)g
2
Q Dfx(cid:20)0;y (cid:20)0;z2Œ0;1(cid:141)g
3
Geometry & Topology,Volume18(2014)