We consider the suspension operation on Lefschetz fibrations, which takes
p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant,
and changes the category of the fibre (or more precisely, the subcategory
consisting of a basis of vanishing cycles) in a specific way. As an
application, we prove part of Homological Mirror Symmetry for the total spaces
of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio
Let I(t)=∮δ(t)ω be an Abelian integral, where
H=y2−xn+1+P(x) is a hyperelliptic polynomial of Morse type, δ(t) a
horizontal family of cycles in the curves {H=t}, and ω a polynomial
1-form in the variables x and y. We provide an upper bound on the
multiplicity of I(t), away from the critical values of H. Namely: $ord\
I(t) \leq n-1+\frac{n(n-1)}{2}if\deg \omega <\deg H=n+1.Thereasoninggoesasfollows:weconsidertheanalyticcurveparameterizedbytheintegralsalong\delta(t)ofthen‘‘Petrov′′formsofH(polynomial1−formsthatfreelygeneratethemoduleofrelativecohomologyofH),andinterpretthemultiplicityofI(t)astheorderofcontactof\gamma(t)andalinearhyperplaneof\textbf C^ n.UsingthePicard−Fuchssystemsatisfiedby\gamma(t),weestablishanalgebraicidentityinvolvingthewronskiandeterminantoftheintegralsoftheoriginalform\omegaalongabasisofthehomologyofthegenericfiberofH.Thelatterwronskianisanalyzedthroughthisidentity,whichyieldstheestimateonthemultiplicityofI(t).Still,insomecases,relatedtothegeometryatinfinityofthecurves\{H=t\}
\subseteq \textbf C^2,thewronskianoccurstobezeroidentically.Inthisalternativeweshowhowtoadapttheargumenttoasystemofsmallerrank,andgetanontrivialwronskian.Foraform\omegaofarbitrarydegree,weareledtoestimatingtheorderofcontactbetween\gamma(t)andasuitablealgebraichypersurfacein\textbf C^{n+1}.Weobservethatord I(t)growslikeanaffinefunctionwithrespectto\deg \omega$.Comment: 18 page