We now explore Lagrangians \(K\subset X\), where we have a projection \(:X\to \CC\).

definition 0.0.1 [Sei08]

Let \((X, \omega , J)\) be a symplectic manifold equipped with compatible almost complex structure. A symplectic Lefschetz fibration is map \(\pi: X\to \CC\) satisfying the following properties:

\(\pi\) is \(J\)-holomorphic, in the sense that \(J\pi_*=\pi_*\jmath\), where \(\jmath\) is the standard complex structure on \(\CC\);
The map \(\pi\) has finitely many critical points;
The set of critical values \(\{\pi(x)\;|\;z\in \Crit(\pi)\}\) are disjoint and;
In a neighborhood of each critical point, there exists holomorphic coordinates \((z_1, \ldots, z_n)\) for \(X\) so that \(\pi=\sum_{i=1}^n z_i^2\).

If the map \(\pi\) is not proper (i.e. the fibers are allowed to be non-compact,) then we impose the additional requirement:

There exists a compact set \(X_0\subset X\) so that \(\pi:X_0\to \CC\) is a proper fibration and;
The fibration \(\pi:X\setminus X_0\to \CC\) is a trivial symplectic fibration, with split complex and symplectic structure.

Note that \(\pi: X\times \CC\to \CC\), the setting considered for Lagrangian cobordisms, trivially satisfies these criterion. A symplectic Lefschetz fibration is the symplectic geometry equivalent to a manifold equipped with a Morse function. The basic model that we consider is a function which models the neighborhood of a critical point.

example 0.0.2

The symplectic fibration which we will use as a running example through this section is \begin{align*} \pi:\CC^2\to& \CC\\ (z_1, z_2)\mapsto& z_1z_2. \end{align*} This has one critical value at \(z_1z_2=0\). The generic fiber \(\pi^{-1}(z)\) is symplectomorphic to \((\CC^*)^2\). At the origin, this degenerates to the union of two complex lines, \(\CC_{z_1=0}\cup \CC_{z_2=0}\). figure 0.0.3 is a drawing of this Lefschetz fibration.

figure 0.0.3:The model Lefschetz singularity

example 0.0.4

Another interesting piece of geometry comes from the cotangent bundle of the 2-sphere, which is a subvariety of \(\CC^{3}\), \[T^*S^2=\{(z_0, z_1, z_2)\;|\; z_0^2+z_1^2+z_2^2=1\}\] We check that this has the topology of the tangent bundle. Let \(S^2=\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2+x_2^2=1\}\subset \RR^3\). The tangent bundle is then described by the pairs \[T^*S^2:=\{(x_0, x_1, x_2, y_0, y_1, y_2)\;|\;x_0^2+x_1^2+x_2^2=1, \sum_{i=0}^2 x_iy_i=0 \}.\] These two constraints can be rephrased in terms of the real and imaginary components of \(z_0^2+z_1^2+z_2^2=1\). The complex structure on this cotangent bundle interchanges the \(x_i\) base directions with the \(y_i\) tangent bundle directions. The symplectic Lefschetz fibration that we consider for the cotangent bundle of the sphere sends \begin{align*} \pi: T^*S^2\to \CC\\ (z_0, z_1, z_2)\mapsto z_2 \end{align*} The fibers of this function are the conics \[\pi^{-1}(z)=\{(z_0, z_1, z_2)\;|\; z_2=z, z_0^1+z_1^2=1-z_2^2\}\] which are regular, provided that \(z_2\neq \pm 1\).

figure 0.0.5:Lefschetz fibration for the cotangent bundle of the sphere

One slightly confusing piece of notation in symplectic geometry is that the tangent bundle of \(\CP^1\) is usually equipped with a different symplectic form that the cotangent bundle \(T^*S^2\). This is because \(T\CP^1\) is usually considered with the symplectic form which makes \(\CP^1\) a symplectic, rather than Lagrangian submanifold. This can also be constructed as the symplectic blowup of \(\CC^2\) at the origin. This also comes with a projection to \(\CC\), by first considering the blowdown map \(T\CP^1\to \CC^2\), and then composing with the fibration considered in example 0.0.2. This is not a Lefschetz fibration, as the critical fiber above zero contains 2 critical points. The existence of a symplectic parallel transport map across the fibers of a Lefschetz fibration allows us to carry over many of the constructions which we considered for Lagrangian cobordisms to Lefschetz fibrations.

proposition 0.0.6

The regular fibers \(X_z:=\pi^{-1}(z)\) of \(\pi\) are symplectic submanifolds, and there exists a connection on \(TX\) whose parallel transport is a symplectomorphism of the fibers. We first show that \(\ker(\pi_*)\) is a symplectic subspace. Let \(0\neq v\in \ker(\pi_*)\) be any tangent vector. Since \(\pi_*\) is \(J\)-holomorphic, \(Jv\in \ker(\pi_*)\). We conclude that \(\omega(v, Jv)=g(v, v)\neq 0\), and so \(\omega\) is non-degenerate on the subspace \(\ker(\pi_*)=T_zX\). It remains to show that the form is closed. Let \(i:X_z\into X\) be and inclusion of a fiber of the Lefschetz fibration. Then \(di^*\omega= i^*d\omega =0\), so the \(\omega|_{X_z}\) is a symplectic form on the fiber. We can construct a connection by picking a horizontal complement to the kernel of the projection. As \(\ker(\pi_*)\) is a symplectic subspace, \((\ker(\pi_*))^{\omega\bot}\), its symplectic complement, is a complementary subspace in the sense that \(\ker(\pi_*)\oplus (\ker(\pi_*))^{\omega\bot}=TX\). This is a choice of horizontal complement, defining a connection on this fiber bundle. In fact, the splitting of the tangent bundle locally splits the symplectic form.

proposition 0.0.7

In the local splitting \(T_xX=\ker(\pi_*)\oplus(\ker(\pi_*))^{\omega^\bot}\), the symplectic form \(\omega_X\) can be written as \[\omega_X=\omega|_{X_z}\oplus f\omega_\CC.\] for some smooth function \(f:X\to \RR\). Pick \(p\in X\) a point, and tangent vectors \((v_1, w_1), (v_2, w_2)\in T_pX=\ker(\pi_*)\oplus(\ker(\pi_*))^{\omega^\bot}.\) Because this these two spaces are symplectic orthogonal, \(\omega_X(v_1, w_2)=0\) and \(\omega_X(v_2, w_1)=0\). Since \(v_1, v_2\) are tangent vectors to \(X_p\), and symplectic forms on \(\CC\) are all scalar multiples, there exists a function \(f: X\to \RR\) yielding the decomposition \begin{align*} \omega_X((v_1, w_1), (v_2, w_2))=&\omega_X((v_1, 0), (v_2, 0))+ \omega_X((0, w_1), (0, w_2)). =&\omega_{X_p}(v_1, v_2)+f(p) \pi^*\omega_\CC(w_1, w_2) \end{align*} We now show that parallel transport is a symplectomorphism of the fibers. This is done by computing the vertical component of the Lie derivative of \(\omega\). Let \(v\in \ker(\pi_*)^{\omega \bot}\) be a horizontal vector. Then \[\mathcal L_{v \omega}=d\iota_v \omega\] We note that \(\iota_v\omega\) vanishes on vertical vectors. Therefore, \(d\iota_v\omega\) vanishes on pairs of vertical vectors, and so the vertical component of the Lie derivative of \(\omega\) is zero. This allows us to make the following definition.

definition 0.0.8

Let \(\gamma:[0,1]_t\to X\setminus \Crit(\pi)\). Define the symplectomorphism \(\phi_\gamma^t: X_{\gamma(0)}\to X_{\gamma(t)}\) to be the symplectic parallel transport along the path \(\gamma\). Define the symplectic inclusion \(i_\gamma^t: X_{\gamma(0)}\to X\) to be the symplectic parallel transport map composed with the inclusion of the fiber into the total space \(X\). Finally, we prove that the autosymplectomorphism of the fiber taken by small loops are Hamiltonian isotopies. . Suppose that we have a family of loops \(\gamma_s:[0,1]_s\times [0,1]_t\to \CC\) with \(\gamma_s(0)=\gamma_s(1)=\gamma_1(t)=z\). This gives us a family of maps \(i_\gamma^t: X_z\into X\) and a symplectomorphisms \(\phi_{s, 1}: X_z\to X_z\). Let \(c:S^1\to X_{z}\) represent a class in \(H_1(X_z)\), and consider the flux \(\int_{\phi_{s, 1}\circ c}\omega|_{X_z}\). We can write this chain as the boundary of a 3 chain, \[c\times [0, 1]_s\times\{t=1\}\cup c\times [0, 1]_s\times \{t=0\}\cup c\times \{0\}_s \times [0, 1]_t\cup c\times \{1\}_s\times [0, 1]_t=\partial (c\times [0, 1_s]\times [0, 1]_t)\] This allows us to replace the flux integral with \begin{align*} 0=&\int_{i_\gamma^t\circ c} d\omega = \int_{\partial(i_\gamma^t\circ c)}\omega\\ =& \int_{i{s, 1}\circ c}\omega-\int_{i{s, 0}\circ c}\omega+\int_{i{1, t}\circ c}\omega-\int_{i_{0, t}\circ c}\omega \end{align*}The first term is an integral restricted to the fiber \(X_z\), so we may replace \(\omega\) with \(\omega|_{X_z}\). The second term and fourth term are zero as the map is constant. \begin{align*} =& \int_{i{s, 1}\circ c}\omega|_{X_z}+\int_{i{1, t}\circ c}\omega\\ \end{align*}In the last term \(\frac{d}{dt}i_{1, t}\circ c\) lies in the horizontal tangent space, and \(\frac{d}{d\theta} i_{1,t}\circ c\) lies in the vertical tangent space. Therefore \(\omega\) vanishes on \(T(i_{1, t}\circ c)\) \begin{align*} =&\int_{i{s, 1}\circ c}\omega|_{X_z}= \Flux_{i_{s, 1}}(c). \end{align*}

proposition 0.0.9

Consider a symplectic Lefschetz fibration \(\pi: X\to \CC\). Let \(\gamma:\RR\to \CC\) be a path avoiding the critical values of a symplectic Lefschetz fibration. Additionally, pick a Lagrangian submanifold of a fiber \(L\subset X_{\gamma(0)}\), parameterized by \(\li: L\to X\). Consider the submanifold \(K\) parameterized by \begin{align*}\li_t: L\times \RR\to& X\\ (x, t) \mapsto& (i\gamma^t\circ \li(x)) \end{align*} where \(i_\gamma^t: X_{\gamma_0}\to X\) is given by parallel transport along \(\gamma\). \(K\) is a Lagrangian submanifold of \(X\). Let \(\li_t:L\times \RR\to X\) be a parameterization of the submanifold \(K\). By reparameterization, it suffices to check that this is a Lagrangian submanifold at points \((p, 0)\). The tangent space \(T_{(p, 0)}(L\times \RR)\) is spanned by vectors \(\{\partial_{x_1}, \ldots, \partial_{x_{n-1}}, \partial_t\}\). Because \(L\) is a Lagrangian of the fiber, and the fiber is a symplectic submanifold, \[\li_t^*\omega(\partial_{x_i}, \partial_{x_j})=0.\] Since \(K\) was constructed via parallel transport, \((\li_t)_*\partial_t\in (\ker(\pi_*))^{\omega\bot}\). Since \(L\) is a Lagrangian of the fiber, \((\li_t)_*\partial_{x_i}\in \ker(\pi_*)\). Therefore, \begin{align*}\li_t^*\omega(\partial_{x_i}, \partial_{t})=0 \end{align*}

figure 0.0.10:Product Torus constructed via Lefschetz fibration

figure 0.0.11:Chekanov Torus constructed via Lefschetz fibration

When we have a curve \(\gamma: \RR\to \CC\) which avoids the critical values, and a Lagrangian submanifold \(L\subset X_{\gamma(0)}\), we will abuse notation and denote the parallel transport of \(L\) along \(\gamma\) by \(L\times \gamma\).

example 0.0.12

Recall our running example \(\pi:\CC^2\to \CC\) from example 0.0.2. We will prove that the symplectic parallel transport map preserves a class of Lagrangian submanifolds of the fiber. Consider the function \(H(z_1, z_2)= \frac{1}{2}\left(|z_1|^2-|z_2|^2\right)=\frac{1}{2}\left( x_1^2+y_1^2-x_2^2-y_2^2\right)\). The exterior derivative of this function, in local coordinates, is given by \[dH= x_1dx_1 +y_1dy_1 -x_2dx_2- y_2dy_2.\] We prove that \(H\) is invariant under the action of symplectic parallel transport along the fibration \(\pi:\CC^2\to \CC\). In this example, we can explicitly compute that \(H\) is invariant under vectors contained in \(\ker(d\pi)^{\omega_\bot}\). The kernel of \(d\pi=z_2dz_1+z_1dz_2\) at a point \((z_1, z_2)\) is the complex subspace generated by the vector \begin{align*} \ker_{(z_1, z_2)}(d\pi)=&\Span_\CC(\langle z_1, -z_2\rangle)\\ =&\Span_\RR(\langle x_1, y_1, -x_2, -y_2\rangle, \langle -y_1, x_1, y_2, -x_2\rangle ). \end{align*} In this setting, the symplectic complement is described by the orthogonal complement, and so \begin{align*} (\ker_{(z_1, z_2)}(d\pi))^{\omega\bot}=&\Span_\CC(\langle \bar z_2, \bar z_1\rangle)\\ =&\Span_\RR(\langle x_2, -y_2, x_1, -y_1\rangle, \langle y_2, x_2, y_1, x_1\rangle ). \end{align*} One then checks that \(dH\) vanishes on this by computing \(dH(v)=0\) for \(v\in (\ker_{(z_1, z_2)}(d\pi))^{\omega\bot}\) \begin{align*} \langle x_1, y_1, -x_2,- y_2\rangle\cdot \langle x_2, -y_2, x_1, -y_1\rangle=&0\\ \langle x_1, y_1, -x_2,-y_2\rangle\cdot \langle y_2, x_2, y_1, x_1 \rangle=&0 \end{align*} This means that the level sets of \(H\) are preserved under parallel transport. We use to this to describe some Lagrangian submanifolds of \(\CC^2\). If we take a level set of \(H\) and restrict to a fiber above the point \(re^{\jmath c}\), the level set \(H^{-1}(\lambda)\cap \pi^{-1}(re^{\jmath c})\) can be explicitly parameterized by \(S^1:=\theta\mapsto re^{\jmath c}\cdot(s e^{\jmath\theta}, s^{-1} e^{-\jmath\theta})\), where \(s\) is determined by \(r^2(s^2-s^{-2})=\lambda\). Simply because every curve is a Lagrangian submanifold of a \(\CC^*\), the level set of \(H\) restricted to a fiber of \(\pi\) is a Lagrangian submanifold. We can now apply proposition 0.0.9 to obtain some new Lagrangian submanifolds of \(\CC^2\) from parallel transport of these level sets. Let \(\gamma:[0, 1]\to \CC\setminus 0\) be a closed curve, and \(\lambda\in \RR\) some value. Define the Lagrangian \(L_{\gamma, \lambda}\) to be the parallel transport of the \(\lambda\)-level set along the curve \(\gamma\). This already gives several interesting examples of Lagrangian submanifolds inside of \(\CC^2\). These Lagrangian submanifolds can also be characterized in the following way: \[L_{\gamma, \lambda}:=\{(z_1, z_2)\;|\; H(z_1, z_2)=\lambda, \pi(z_1, z_2)\in \Im(\gamma)\}.\] A good example of one of these Lagrangians is the product torus. Let \(\gamma_r=re^{\jmath\theta}\). Let \(s\) be the real value so that \(r^2(s^2-s^{-2})=\lambda\). Then the Lagrangian \(L_{\gamma_r, \lambda}\) is explicitly parameterized by: \[L_{\gamma_r, \lambda}=\{r(se^{\jmath\theta_1}, se^{\jmath\theta_2})\}\] This agrees with the definition of the product torus from (product torus). A Lefschetz fibration \(\pi: X\to \CC\) equips \(X\) with two Hamiltonian flows: the real and imaginary coordinates of the complex parameter.

lemma 0.0.13

Let \(\pi: X\to \CC\) be a symplectic Lefschetz fibration. Let \(g_X=\omega_X\circ (J_X\tensor \id)\) be the compatible metric on \(X\). Then we have the following relations between gradient and Hamiltonian flows.

The gradient flow of \(\pi_\RR\) is the Hamiltonian flow of \(\pi_{i\RR}\).
The gradient flow of \(\pi_{i\RR}\) is the Hamiltonian flow of \(-\pi_\RR\).

This follows from (symplecticGradient).

proposition 0.0.14

Let \(p\in \Crit(\pi)\) be a critical point of a symplectic Lefschetz fibration. Consider the function \(\Re(\pi):X\to \RR\). The point \(p\) is also critical point of \(\Re(\pi)\) and \(W^-_p\), the downward flow space of \(p\), is a Lagrangian submanifold. In the local model at a critical point of \(\pi\), projection to the real coordinate can be written in local holomorphic coordinates as \[\Re(\pi)=x_1^2+\ldots x_n^2-y_1^2-\ldots y_n^2.\] The downward flow space is parameterized by \[W^-_p=\{x_1=y_1, \ldots, x_n=y_n\}\] which is a Lagrangian submanifold for the standard symplectic structure.

example 0.0.15

Once again we consider the Lefschetz fibration \(\pi: \CC^2\to \CC\) from example 0.0.2. The only critical value of this function is \(0\). Given a path \(\gamma:[0, 1]\to \CC\) with \(\gamma(0)=0\), the thimble can be described by the construction of example 0.0.12, \[D^n_\gamma=L_{\gamma, 0}.\] In particular case of \(\gamma\) being the real positive \(\RR_{\geq0}\subset \CC\), \begin{align*} L_{\gamma, 0}=&\{(z_1, z_2)\;|\;z_1z_2\in \RR_{\geq 0}, |z_1|^2-|z_2|^2=0\}\\ =&\{(z, \bar z)\;|\; z\in \CC\}. \end{align*} Note that \(\pi(W^-_p)\) is a ray emerging from \(\pi(p)\) and heading in the positive real direction. Let \(z\) be in the interior of \(\pi(W^-_p)\). Then \(\pi^{-1}(z)\cap W^-_p\) is a Lagrangian sphere \(S^{n-1}\subset \pi^{-1}(z)\). More generally, one can take any path \(\gamma\subset \CC\) with left end on a critical endpoint to determine a Lagrangian sphere in the fiber. Note that the map \((\phi_\gamma^{t_0})^{-1}: X_{\gamma(t_0)}\to X_{\gamma(0)}\) is still a well defined map. One can check that \((\phi_{\gamma}^{t_0})^{-1}(z)\subset X_{\gamma(0)}\) is a Lagrangian sphere in the fiber.

definition 0.0.16

Let \(\pi: X\to \CC\) be a symplectic fibration. Let \(\gamma:[0, 1]\to \CC\) be path with \(\gamma(0)\) a critical value. Let \(p\) be the critical point above this critical value. Suppose \(\gamma(t)\) avoids critical values for \(t\neq 0\). Let \(W^{-1}(\gamma)\) be the collection of fibers above the path \(\gamma\). Consider the map \((\phi_{\gamma}^{t})^{-1}:W^{-1}(\gamma)\mapsto X_{\gamma(0)}\) given by parallel transport. Then the thimble of \(\gamma\) from \(p\) is a Lagrangian disk \(D^n_\gamma:= (\phi_{\gamma}^{t})^{-1}(z)\subset W^{-1}(\gamma)\subset X\). The vanishing cycle of \(\gamma\) is a Lagrangian sphere \(S^{n-1}_\gamma (\phi_{\gamma}^{1})^{-1}(z)\subset X_{\gamma(0)}\), which may also be identified with \( D^n_\gamma\cap \pi^{-1}(\gamma(1))\). Sometimes, a sphere in the fiber is the vanishing cycle of more than one critical point. If \(\gamma_1, \gamma_2\) are two paths so that \(S^{n-1}_{\gamma_1}=S^{n-1}_{\gamma_2}\), whose concatenation \(\gamma=\gamma_1\cdot \gamma_2^{-1}\) is a smooth path, we call \(\gamma\) a matching path for the critical values \(\gamma_1(0)\) and \(\gamma_2(0)\). When we have a matching path, we can construct a Lagrangian sphere by gluing the corresponding thimbles together at their mutual vanishing cycle boundary.

example 0.0.17

We continue our discussion of the sphere from example 0.0.4. The Lefschetz fibration \(\pi: T^*S^2\to S^2\) has two critical values, \(\{-1, 1\}\), whose critical points corresponding to the north and south pole of the sphere. We now look at the thimbles drawn in figure 0.0.18. The first example we consider is the Lagrangian thimble constructed from \(\gamma(t)=-1-t\), the real negative ray with endpoint on the critical value of the south pole. The symplectic parallel transport along \(\gamma(t)\) is the negative gradient flow of the imaginary coordinate of \(\pi(z_0, z_1, z_2)=z_2\) from the critical point \((0,0,1)\). The gradient flow of the imaginary coordinate is \begin{align*} \grad_{T^*S^2}(\Im(z_2))=&\text{proj}_{T(T^*S^2)}\grad_{\CC^3}(\Im(z_2))\\ =&\langle 0, 0, 1 \rangle- \cdot\frac{ \langle 0, 0, 1 \rangle\cdot \langle 2z_0, 2z_1, 2z_2 \rangle }{(2z_0)^2+(2z_1)^2+(2z_2)^2}\langle 2z_0, 2z_1, 2_2\rangle\\ =& h(z_0, z_1, z_2)\langle 0, 0, 1\rangle \end{align*} For some function \(h(z_0, z_1, z_2)\). The space \(\{(ix_0, ix_1, 1+x_0^2+x_1^2)\}\) is a 2-dimensional Lagrangian subspace which contains \((0,0, 1)\) and is parallel to the \(\grad_{T^*S^2}(\Im(z_2))\), and therefore the Lagrangian thimble over \(\gamma(t)\). This also corresponds to the cotangent fiber above the south pole, \(T^*_{sp}S^2\). In this example, we can also consider the path \(\gamma(t)=1-2t\), which starts at the critical value for the south pole and ends at the critical value of the north pole. This is a matching path, and therefore there is a Lagrangian \(S^2_{\gamma}\subset T^*S^2\) which lives above this path. The zero section of the sphere, given by \(\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2+x_2^2=1, x_i\in \RR\}\) is a 2 dimensional submanifold of \(T^*S^2\) which lies parallel to \(\grad_{T^*S^2}(\Im(z_2))\). The image of the zero section under \(\pi\) is the curve \(\gamma\), and the \(S^2\) zero section clearly contains the north and south pole. Therefore, the Lagrangian sphere associated to the mapping path \(\gamma\) is exactly the zero section.

figure 0.0.18:The matching math in the example of \(\pi: T^*S^2\to \CC\) gives a Lagrangian sphere. The vanishing cycle above \(0\) corresponds to the equator of the sphere.

These collections are useful, because Lagrangian spheres can be used to construct symplectomorphisms.

References

[Sei08]

Paul Seidel. Fukaya categories and Picard-Lefschetz theory, volume 10. European Mathematical Society, 2008.