Non abelian theta functions and the theta map
95 pages
English

Non abelian theta functions and the theta map

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95 pages
English
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En savoir plus

Description

Non-abelian theta functions and the theta map Arnaud Beauville Universite de Nice Berkeley, April 2009 Arnaud Beauville Non-abelian theta functions and the theta map

  • poles ≤

  • abelian theta

  • theta functions

  • bundles

  • riemann surface

  • functions

  • complex torus

  • universite de nice

  • line bundles trivial

  • line bundles


Sujets

Informations

Publié par
Nombre de lectures 23
Langue English

Exrait

andurAllviauBebe-aoneNtehtnailoitcnufahttemapasnnatdeh
Berkeley, April 2009
Non-abelian
theta functions and
the
theta
map
Arnaud Beauville
Universit´edeNice
Tin,llyalicogolopissabdeehtyrgedunebesdlCaonclremoc.alsssefoileneedegLZ.Jd:={is0J=}CnoocaJeht=solendbuedreegfdCs/goturliflWΓweofCbianplex=com)6(L}h=0L={H0J|gJ=:Θ.1suco:Jnosor).DehetadivicaieJnt(pyreusfr:=k}errdfosoonticnufateht{noitinctiocfunrphiromo{=emΘk))O(J0HJ(nCloiaivafetthgsnoitcnuufottfilJwitnsoneskhpolnibe}Θl;sertnuldΓ.nctionsonCg,quas-iepirdoci.w.r.t
C
g .
genus
of
curve
(=
Riemann
surface)
map
theta
the
Non-abelian
theta
functions
and
Abelian theta functions
Arnaud
Beauville
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C curve (= Riemann surface)
of genus g .
tama
deg L
degree
the
by
Topologically,
line
Z .
bundles
on
C
are
classified
oisnnogCq,auisp-eriodicw.r.t.Γ.AuanraeBdlivuoNelabn-iaelhentfutalibethanActunafetnsiolcsaesoslfnibenuJd:={isom.focuwill/ΓWeusCgtxropmel=oconCfiaobaceJth0==J}Cnodeergedfoseld).Denitadivisorni(Jhttesrruafec0}6=pehy|HJL)0(Θ.1L{=:JnosgJ=:
nsiocthttefanuAebilnaobaceJth=fCnoia=0JllfoWewinJ:=cusoeltxocpmgCΓ/rosuoromicph)=Θ)er{mJnoshtiwcnufnoitionsoforetafunct(0JJOk(edkr:}H=isivadetthJ(inceht{noitineD.)ro|H0({LJ.Θ:=Jg1ruafepsr}0yh)L=6vuaeelli-noNlebantiatahencfuontigCq,auisp-reoiidcw.r.t.Γ.ArnaudBufatehtlsnoitcnunofttifonnsioctsΘkopelenub;}ilstrindleonCgvial
:=
{ isom.
classes
of line bundles of degree d
on C }
=
J d
bundles on C
are classified by the degree deg L Z .
C curve (= Riemann surface) of genus g . Topologically, line
pmataheetthndsa
}:=H0(Joforderknutcoisn{nhttefaontinJsoicphncfurem{romok(JO=))Θvialstrindlenebu;}ilsΘkopeliwhtioctunofttifslonitcnufatehtgCnoiweWofllosuc)L=6H|(0epsr}0yhJg1nJ:={LJ.Θ:=.)rosivioitineDincefauradetthJ(onns,qCgsiuaer-pidoir.wcΓ.t.nrA.audBeauvilleNon-baleaitnehatufcnndsaontitaheetth
C g / Γ
J d := { isom. classes of line bundles of degree d on C } J 0 = the Jacobian of C = complex torus
bundles on C are classified by the degree deg L Z .
C curve (= Riemann surface) of genus g . Topologically, line
pam
=
/ Γ
= the Jacobian of C
C
ilnahtteAeboitcnufasn
Θ:tahencfutini{ton)roseD.atehividaceinJ(typersurfL(6)0=h}{=L|J0H.t.r.wcidoirep-iΓ.
Abelian theta functions C curve (= Riemann surface) of genus g . Topologically, line bundles on C are classified by the degree deg L Z . J d := { isom. classes of line bundles of degree d on C } ∼ = J 0 = the Jacobian of C = complex torus C g / Γ We will focus on J : J . = g 1 Arnaud Beauville Non-abelian theta functions and the theta map
rphiromoctiocfunwJtisnnosekphlofosoonti:=k}errd(JOJ(0Hem{=))ΘkunctionslifttofucnitnoosCn,guqsa;lΘ}ebindluntresaiviCnolhtgfate
snoitcnufaetthanlibeAeldnubenlaivirtshetCgontincfutafittnolstcoifonuCg,qnson-peruasirofokredH=:}J(0(kOJ)=Θ)er{moromhpciufcnitnoosJnwithpoleskΘ};liitioDenoisnnutctefa{nht
Θ := { L J | H 0 ( L ) 6 = 0 }
map
C curve (= Riemann surface) of genus g . Topologically, line bundles on C are classified by the degree deg L Z .
J d := { isom. classes of line bundles of degree d on C } = J 0 = the Jacobian of C = complex torus C g / Γ
We will focus on J := J g 1 .
hypersurface in J ( theta divisor ) .
atehtehtdnasnoitncfutahentiaelaboN-nlielaevuuaBd.Arn.t.Γcw.riodi
lophtiwJl;}Θksedlunebiniaivtresorom{=emcfunrphinsonctioasqupei-soong,nC.t.r.Γdoir.wcietafunctlonCgthotufcnitoisniltf
Abelian theta functions C curve (= Riemann surface) of genus g . Topologically, line bundles on C are classified by the degree deg L Z . J d := { isom. classes of line bundles of degree d on C } ∼ = J 0 = the Jacobian of C = complex torus C g / Γ We will focus on J := J g 1 . Θ := { L J | H 0 ( L ) 6 = 0 } hypersurface in J ( theta divisor ) . Definition { theta functions of order k } := H 0 ( J O J ( k Θ)) Arnaud Beauville Non-abelian theta functions and the theta map
aitnbAleinlunebesdlivtrctionslifttofuncaiolCnghttefanu.Γ.t.r.wcidoirepi-asqug,nCsoontitefanahtebilnoa-lleNauviudBeArna
= { meromorphic functions on J with poles k Θ } ;
Definition { theta functions of order k } := H 0 ( J O J ( k Θ))
Θ := { L J | H 0 ( L ) 6 = 0 } hypersurface in J ( theta divisor ) .
We will focus on J := J g 1 .
mate
C curve (= Riemann surface) of genus g . Topologically, line bundles on C are classified by the degree deg L Z . J d := { isom. classes of line bundles of degree d on C } = J 0 = the Jacobian of C = complex torus C g / Γ
padtanthhectunnsioufcnehatsitno
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