Instructions for use
T itle
C onics on a generic hypersurface
A uthor(s )
J inzenji,Masao; Nakamura,Iku; S uzuki,Y asuki
C itation
Hokkaido University Preprint S eries in Mathematics, 710: 1-18
Is s ue D ate
2005
D O I
10.14943/83861
D oc UR L
http://hdl.handle.net/2115/69515
T ype
bulletin (article)
F ile Information
pre710.pdf
MASAOJINZENJI,IKUNAKAMURAANDYASUKISUZUKI
Abstract. Inthispap er,wecomputethecontributionsfromdoublecovermaps togenus0degree2Gromov-Witteninvariantsofgeneraltypeprojective hypersur-faces. Ourresultscorresp ondtoageneralizationofAspinwall-Morrisonformula togeneraltypehypersurfacesinsomesp ecial cases.
MSC-class: 14H99,14N35,32G20
1. Introduction
Inthispaper,wediscussageneralizationofthemultiplecoverformulaforrational Gromov-WitteninvariantsofCalabi-Yaumanifolds[AM],[M]todoublecovermaps ofaline LonadegreekhypersurfaceM
k N
inP N01
. Naively,foragivennitesetof
elements
j
2H
3 (M
k N
;Z),therationalGromov-WitteninvarianthO
1 O
2
111O n
i 0;d
of M
k N
counts the numb er of degree d (p ossibly singular and reducible) rational
curvesonM
k N
thatintersectrealsub-manifoldsofM k N
thatarePoincare-dualto j
. Recently, the mirror computation of rational Gromov-Witten invariants of M
k N withnegativerstchernclass(k0 N >0)wasestablishedin[CG],[Iri],[J]. Usingthe methodpresentedinthesearticles,wecancomputehO
e m
1 O
e m
2 111O
e m
n i
0;d
wheree is thegeneratorofH
1;1 (M
k N
;Z). Briey,mirrorcomputationof M k N
(k>N) in[J] goesas follows. We startfrom thefollowingODE:
(1)
(@ x
) N01
0k1exp(x)1(k @ x
+k01)(k @ x
+k02)111(k @ x
+1)
w (x)=0;
and construct thevirtual Gauss-Maninsystem asso ciated with(1):
@ x
~ N020m
(x) =
~ N010m
(x)+ 1 X
d=1
exp(dx)1 ~ L
N;k ;d m
1 ~
N010m0(N0k )d (x); (2)
where m runs through all the integers and
~ L
N;k ;d m
is non-zero only if 0 m
N01+(k0N)d. Fromthecompatibilityof(1)and(2), wecanderivetherecursive formulasthatdetermine all the
~ L
N;k ;d m
's: k 01
X
n=0 ~ L
N;k ;1 n
w n
=k1 k 01 Y
j=1
(jw+(k0j));
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z m
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l =2 (01)
l
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