Weinstein Lefschetz fibration data interface

A Weinstein domain is a compact smooth manifold $W$ with boundary, a smooth one-form ${\lambda }_W$ with ${\omega }_W=d{\lambda }_W$ nondegenerate, and a smooth Morse function $\varphi :W\to \mathbb{R}$ with $\mathrm{\partial }W={\varphi }^{-1}(0)$ a regular level and $\varphi <0$ in the interior. The vector field $Z$ defined by ${\iota }_Z{\omega }_W={\lambda }_W$ points outward at the boundary and is gradient-like for $\varphi$: its zeros are exactly the critical points of $\varphi$, $d\varphi (Z)>0$ elsewhere, and near each critical point $Z$ is the gradient of $\varphi$ for a smooth positive definite metric. An abstract Weinstein Lefschetz datum in dimension four consists of such a two-dimensional domain $\mathrm{\Sigma }$, and an ordered finite list of smooth embedded circles $L_i\subset \mathrm{int}\mathrm{\Sigma }$ satisfying ${\omega }_{\mathrm{\Sigma }}{\mathrm{\mid }}_{L_i}=0$ and ${\lambda }_{\mathrm{\Sigma }}{\mathrm{\mid }}_{L_i}=d{f}_i$ for specified smooth functions $f_i:L_i\to \mathbb{R}$.

A geometric realization is additional data, in the following compact-disc convention. Let $B=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\}$. Specify a compact smooth four-manifold $E$ with corners, a smooth one-form ${\lambda }_E$ with ${\omega }_E=d{\lambda }_E$ symplectic, and a smooth surjection $w:E\to B$. Smooth maps and forms extend across the boundary in local charts. Its boundary faces are ${\mathrm{\partial }}_{v}E=w^{-1}(\mathrm{\partial }B)$ and ${\mathrm{\partial }}_{h}E$, meeting in codimension-two corners. The critical points are finitely many interior points with distinct values in $\mathrm{int}B$. Near each critical point $p$ specify smooth real coordinates identifying a neighbourhood with a neighbourhood of $0\in {\mathbb{C}}^2$ in which $w(z)=w(p)+z_1^2+z_2^2$; ${\omega }_E$ tames the standard complex structure there, meaning ${\omega }_E(u,iu)>0$ for nonzero real tangent vectors $u$. Away from the critical values, $w$ is a smooth locally trivial bundle of compact surfaces with boundary, with ${\mathrm{\partial }}_{h}E$ its fibre-boundary bundle, and ${\omega }_E$ restricts to a symplectic form on each fibre. Require a collar of ${\mathrm{\partial }}_{h}E$ identified over $B$ with a product collar $[0,ϵ)\times \mathrm{\partial }\mathrm{\Sigma }\times B$, on which ${\lambda }_E$ is the sum of the fibre collar form and the pullback of a smooth base one-form. The fibre Liouville fields point outward along ${\mathrm{\partial }}_{h}E$. Specify a smoothing $M$ of the corners and Weinstein data on $M$ extending these exact data on the retained interior. The notation $(M,\lambda ,w)$ refers to this realization with its corner model retained when discussing boundary fibres.

On the regular locus define the horizontal planes to be $H=(\ker dw{)}^{{\perp }_{{\omega }_E}}$. Require their lifts along piecewise smooth paths avoiding critical values to exist throughout each finite path interval and preserve the fibre boundary. Choose a regular base point $b_0\in \mathrm{int}B$, an exact symplectic identification of $w^{-1}(b_0)$ with $\mathrm{\Sigma }$ (its pullback primitive differs from ${\lambda }_{\mathrm{\Sigma }}$ by an exact form), and embedded smooth paths ${\gamma }_i:[0,1]\to B$ from $b_0$ to the respective critical values, with disjoint interiors and no other critical values. Order their distinct initial tangent directions clockwise in the complex orientation of $B$. For each path require a smooth embedded Lagrangian disc in $E$ over that path, with centre the critical point and boundary in $w^{-1}(b_0)$, whose circles over $0\le t<1$ are carried into one another by this horizontal transport and collapse to the critical point as $t\to 1$. The identified boundary circle must be $L_i$. These discs define the vanishing cycles in this convention; exactness of each boundary is already part of the abstract data.

The abstract list and a geometric realization are separate data. This definition asserts neither existence of a realization for a proposed list nor any categorical comparison. Open-disc versions use the interior of this base and the corresponding completed or truncated total space with the chosen collar and transport data stated explicitly.