Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

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

On the multiplicity of the hyperelliptic integrals

Let $I(t)= \oint_{\delta(t)} \omega$ be an Abelian integral, where $H=y^2-x^{n+1}+P(x)$ is a hyperelliptic polynomial of Morse type, $\delta(t)$ a horizontal family of cycles in the curves $\{H=t\}$, and $\omega$ 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$\deg \omega <\deg H=n+1$. The reasoning goes as follows: we consider the analytic curve parameterized by the integrals along$\delta(t)$of the$n$``Petrov'' forms of$H$(polynomial 1-forms that freely generate the module of relative cohomology of$H$), and interpret the multiplicity of$I(t)$as the order of contact of$\gamma(t)$and a linear hyperplane of$\textbf C^ n$. Using the Picard-Fuchs system satisfied by$\gamma(t)$, we establish an algebraic identity involving the wronskian determinant of the integrals of the original form$\omega$along a basis of the homology of the generic fiber of$H$. The latter wronskian is analyzed through this identity, which yields the estimate on the multiplicity of$I(t)$. Still, in some cases, related to the geometry at infinity of the curves$\{H=t\} \subseteq \textbf C^2$, the wronskian occurs to be zero identically. In this alternative we show how to adapt the argument to a system of smaller rank, and get a nontrivial wronskian. For a form$\omega$of arbitrary degree, we are led to estimating the order of contact between$\gamma(t)$and a suitable algebraic hypersurface in$\textbf C^{n+1}$. We observe that$ord I(t)$grows like an affine function with respect to$\deg \omega$.Comment: 18 page