2 Background material [03MQ]

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2 Background material

We now summarize the background material we will need to state our conjectures in §3. We discuss Calabi–Yau $m$ -folds, graded Lagrangians and special Lagrangians in §2.1 and Lagrangian mean curvature flow in §2.3, giving examples of SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ in §2.2 and solitons for Lagrangian MCF in §2.4. Section 2.5 explains Lagrangian Floer cohomology, obstructions to $HF^{*}$ , and derived Fukaya categories $D^{b}{\mathbin{\mathscr{F}}}(M)$ for embedded Lagrangians in Calabi–Yau $m$ -folds, and §2.6 considers the extension to immersed Lagrangians.

Some references are McDuff and Salamon [52] for symplectic geometry, the author [42] and Harvey and Lawson [27] for Calabi–Yau $m$ -folds and special Lagrangians, Mantegazza [50], Smoczyk [67] and Neves [56] for (Lagrangian) MCF, Fukaya [18, 19], Fukaya, Oh, Ohta and Ono [20] and Seidel [64] for Lagrangian Floer cohomology and Fukaya categories for embedded Lagrangians, and Akaho the author [2] for the extension to immersed Lagrangians.

2.1 Calabi–Yau $m$ -folds and special Lagrangians

We define Calabi–Yau $m$ -folds, graded Lagrangians, and special Lagrangians.

Definition 2.1.

A Calabi–Yau $m$ -fold is a quadruple $(M,J,g,\Omega)$ such that $(M,J)$ is an $m$ -dimensional complex manifold, $g$ is a Kähler metric on $(M,J)$ with Kähler form $\omega$ , and $\Omega$ is a holomorphic $(m,0)$ -form on $(M,J)$ satisfying

\omega^{m}/m!=(-1)^{m(m-1)/2}(i/2)^{m}\Omega\wedge\bar{\Omega}.

(2.1)

Then $g$ is Ricci-flat and its holonomy group is a subgroup of $\mathop{\rm SU}(m)$ . We do not require $M$ to be compact, or $g$ to have holonomy $\mathop{\rm SU}(m)$ , although many authors make these restrictions.

If $(M,J,g,\Omega)$ is a Calabi–Yau $m$ -fold with Kähler form $\omega$ , then $(M,\omega)$ is a symplectic manifold. A Lagrangian $L$ in $M$ is a real $m$ -dimensional submanifold (embedded or immersed) with $\omega|_{L}=0$ .

Let $L$ be a Lagrangian in $M$ . Then $\Omega|_{L}$ is a complex $m$ -form on $L$ . Equation (2.1) implies that $\big|\Omega|_{L}\big|=1$ , where $|\,.\,|$ is computed using the Riemannian metric $g|_{L}$ . Suppose $L$ is oriented. Then we have a volume form ${\rm d}V_{L}$ on $L$ defined using the metric $g|_{L}$ and orientation with $|{\rm d}V_{L}|=1$ , so $\Omega|_{L}=\Theta_{L}\cdot{\rm d}V_{L}$ , where $\Theta_{L}:L\rightarrow{\rm U}(1)$ is a unique smooth function, and ${\rm U}(1)=\{z\in{\mathbin{\mathbb{C}}}:|z|=1\}$ .

There is an induced morphism of cohomology groups $\Theta_{L}^{*}:H^{1}({\rm U}(1),{\mathbin{\mathbb{Z}}})\rightarrow H^{1}(L,{\mathbin{\mathbb{Z}}})$ . The Maslov class $\mu_{L}\in H^{1}(L;{\mathbin{\mathbb{Z}}})$ of $L$ is the image under $\Theta_{L}^{*}$ of the generator of $H^{1}({\rm U}(1),{\mathbin{\mathbb{Z}}})\cong{\mathbin{\mathbb{Z}}}$ . If $H^{1}(M,{\mathbin{\mathbb{R}}})=0$ then $\mu_{L}$ depends only on $(M,\omega),L$ and not on $g,J,\Omega$ . We call $L$ Maslov zero if $\mu_{L}=0$ .

A grading or phase function of an oriented Lagrangian $L$ is a smooth function $\theta_{L}:L\rightarrow{\mathbin{\mathbb{R}}}$ with $\Theta_{L}=\exp(i\theta_{L})$ , so that $\Omega|_{L}=e^{i\theta_{L}}{\rm d}V_{L}$ . That is, $i\theta_{L}$ is a continuous choice of logarithm for $\Theta_{L}$ . Gradings exist if and only if $L$ is Maslov zero. If $L$ is connected then gradings are unique up to addition of $2\pi n$ for $n\in{\mathbin{\mathbb{Z}}}$ . A graded Lagrangian $(L,\theta_{L})$ in $M$ is an oriented Lagrangian $L$ with a grading $\theta_{L}$ . Usually we refer to $L$ as the graded Lagrangian, leaving $\theta_{L}$ implicit.

An oriented Lagrangian $L$ in $M$ is called almost calibrated if $(\cos\phi\,\mathop{\rm Re}\Omega-\sin\phi\,\mathop{\rm Im}\Omega)|_{L}$ is a positive $m$ -form on $L$ for some $\phi\in{\mathbin{\mathbb{R}}}$ . Then $L$ admits a unique grading $\theta_{L}$ taking values in $(\phi-\frac{\pi}{2},\phi+\frac{\pi}{2})$ . If a graded Lagrangian $L$ has phase variation less than $\pi$ , then it is almost calibrated.

An oriented Lagrangian $L$ in $M$ is called special Lagrangian with phase $e^{i\phi}$ if $\Theta_{L}$ is constant with value $e^{i\phi}\in{\rm U}(1)$ . If we do not specify a phase, we usually mean phase 1. We will write SL for special Lagrangian, and SL $m$ -fold for special Lagrangian submanifold. SL $m$ -folds with phase $e^{i\phi}$ are Maslov zero, and graded with phase function $\theta_{L}=\phi$ . They are minimal submanifolds in $(M,g)$ . Compact SL $m$ -folds are volume-minimizing in their homology class.

Special Lagrangians were introduced by Harvey and Lawson [27, §III]. The deformation theory of SL $m$ -folds was studied by McLean [53, §3]:

Theorem 2.2.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and $L$ a compact SL $m$ -fold in $M$ . Then the moduli space ${\mathbin{\cal M}}_{\scriptscriptstyle L}$ of special Lagrangian deformations of $L$ is a smooth manifold of dimension $b^{1}(L),$ the first Betti number of $L$ .

2.2 Special Lagrangian $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$

Definition 2.3.

Let ${\mathbin{\mathbb{C}}}^{m}$ have coordinates $(z_{1},\dots,z_{m})$ and complex structure $J$ , and define a Kähler metric $g$ , Kähler form $\omega$ and $(m,0)$ -form $\Omega$ on ${\mathbin{\mathbb{C}}}^{m}$ by

\begin{split}g=|{\rm d}z_{1}|^{2}+\cdots+|{\rm d}z_{m}|^{2},\quad\omega&=\textstyle\frac{i}{2}({\rm d}z_{1}\wedge{\rm d}\bar{z}_{1}+\cdots+{\rm d}z_{m}\wedge{\rm d}\bar{z}_{m}),\\ \text{and}\quad\Omega&={\rm d}z_{1}\wedge\cdots\wedge{\rm d}z_{m}.\end{split}

(2.2)

Then $({\mathbin{\mathbb{C}}}^{m},J,g,\Omega)$ is the simplest example of a Calabi–Yau $m$ -fold.

Define a real 1-form $\lambda$ on ${\mathbin{\mathbb{C}}}^{m}$ called the Liouville form by

\lambda=-{\textstyle\frac{1}{2}}\mathop{\rm Im}(z_{1}{\rm d}\bar{z}_{1}+\cdots+z_{m}{\rm d}\bar{z}_{m}).

Then ${\rm d}\lambda=\omega$ . Thus, if $L$ is a Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ then ${\rm d}(\lambda|_{L})=0$ . We call $L$ an exact Lagrangian if $\lambda|_{L}={\rm d}f$ for some smooth $f:L\rightarrow{\mathbin{\mathbb{R}}}$ .

A (singular) Lagrangian $C$ in ${\mathbin{\mathbb{C}}}^{m}$ is called a cone if $C=tC$ for all $t>0$ , where $tC=\{t\,{\bf z}:{\bf z}\in C\}$ . Let $C$ be a closed Lagrangian cone in ${\mathbin{\mathbb{C}}}^{m}$ with an isolated singularity at 0. Then $\Sigma=C\cap{\cal S}^{2m-1}$ is a compact, nonsingular Legendrian $(m\!-\!1)$ -submanifold of ${\cal S}^{2m-1}$ , not necessarily connected. Let $g_{\smash{\scriptscriptstyle\Sigma}}$ be the metric on $\Sigma$ induced by the metric $g$ on ${\mathbin{\mathbb{C}}}^{m}$ in (2.2), and $r$ the radius function on ${\mathbin{\mathbb{C}}}^{m}$ . Define $\iota:\Sigma\times(0,\infty)\rightarrow{\mathbin{\mathbb{C}}}^{m}$ by $\iota(\sigma,r)=r\sigma$ . Then the image of $\iota$ is $C\setminus\{0\}$ , and $\iota^{*}(g)=r^{2}g_{\smash{\scriptscriptstyle\Sigma}}+{\rm d}r^{2}$ is the cone metric on $C\setminus\{0\}$ .

Let $L$ be a closed, nonsingular Lagrangian $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ , e.g. $L$ could be special Lagrangian, or a Lagrangian LMCF expander. We call $L$ asymptotically conical (AC) with rate $\rho<2$ and cone $C$ if there exists a compact subset $K\subset L$ and a diffeomorphism $\varphi:\Sigma\times(T,\infty)\rightarrow L\setminus K$ for some $T>0$ , such that

\big|\nabla^{k}(\varphi-\iota)\big|=O(r^{\rho-1-k})\quad\text{as $r\rightarrow\infty$, for all $k=0,1,2,\ldots.$}

Here $\nabla,|\,.\,|$ are computed using the cone metric $\iota^{*}(g)$ . Note that if $\rho<\sigma<2$ and $L$ is AC with rate $\rho$ , then $L$ is also AC with rate $\sigma$ .

Asymptotically conical special Lagrangians are an important class of SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ . McLean’s Theorem, Theorem 2.2, was generalized to AC SL $m$ -folds by Marshall [51] and Pacini [60]. Here is a special case of their results:

Theorem 2.4.

Let $L$ be an asymptotically conical SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 3$ with cone $C$ and rate $\rho\in(2-m,0),$ and write ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ for the moduli space of deformations of $L$ as an AC SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ with cone $C$ and rate $\rho$ . Then ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ is a smooth manifold of dimension $b^{1}_{\rm cs}(L)=b^{m-1}(L)$ .

The next family of AC SL $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ was first found by Lawlor [45], and rewritten by Harvey [26, p. 139–140]. They are often called Lawlor necks.

Example 2.5.

Let $m>2$ and $a_{1},\ldots,a_{m}>0$ , and define polynomials $p,P$ by

p(x)=(1+a_{1}x^{2})\cdots(1+a_{m}x^{2})-1\quad\text{and}\quad P(x)=\frac{p(x)}{x^{2}}.

(2.3)

Define real numbers $\phi_{1},\ldots,\phi_{m}$ and $A$ by

\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\quad\text{and}\quad A=\int_{-\infty}^{\infty}\frac{{\rm d}x}{2\sqrt{P(x)}}\,.

Clearly $\phi_{k},A>0$ . But writing $\phi_{1}+\cdots+\phi_{m}$ as one integral gives

\phi_{1}+\cdots+\phi_{m}=\int_{0}^{\infty}\frac{p^{\prime}(x){\rm d}x}{(p(x)+1)\sqrt{p(x)}}=2\int_{0}^{\infty}\frac{{\rm d}w}{w^{2}+1}=\pi,

making the substitution $w=\sqrt{p(x)}$ . So $\phi_{k}\in(0,\pi)$ and $\phi_{1}+\cdots+\phi_{m}=\pi$ . This yields a 1-1 correspondence between $m$ -tuples $(a_{1},\ldots,a_{m})$ with $a_{k}>0$ , and $(m\!+\!1)$ -tuples $(\phi_{1},\ldots,\phi_{m},A)$ with $\phi_{k}\in(0,\pi)$ , $\phi_{1}+\cdots+\phi_{m}=\pi$ and $A>0$ .

For $k=1,\ldots,m$ , define a function $z_{k}:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{C}}}$ by

z_{k}(y)={\rm e}^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\quad\text{where}\quad\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,.

Now write ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$ , and define a submanifold $L_{{\boldsymbol{\phi}},A}$ in ${\mathbin{\mathbb{C}}}^{m}$ by

L_{{\boldsymbol{\phi}},A}=\bigl\{(z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m}):y\in{\mathbin{\mathbb{R}}},\;x_{k}\in{\mathbin{\mathbb{R}}},\;x_{1}^{2}+\cdots+x_{m}^{2}=1\bigr\}.

Then $L_{{\boldsymbol{\phi}},A}$ is closed, embedded, and diffeomorphic to ${\cal S}^{m-1}\times{\mathbin{\mathbb{R}}}$ , and Harvey [26, Th. 7.78] shows that $L_{{\boldsymbol{\phi}},A}$ is special Lagrangian. Also $L_{{\boldsymbol{\phi}},A}$ is asymptotically conical, with rate $\rho=2-m$ and cone $C$ the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two special Lagrangian $m$ -planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ given by

\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.

Apply Theorem 2.4 with $L=L_{{\boldsymbol{\phi}},A}$ and $\rho\in(2-m,0)$ . As $L\cong{\cal S}^{m-1}\times{\mathbin{\mathbb{R}}}$ we have $b^{1}_{\rm cs}(L)=1$ , so Theorem 2.4 shows that $\mathop{\rm dim}\nolimits{\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}=1$ . This is consistent with the fact that when $\boldsymbol{\phi}$ is fixed, $L_{{\boldsymbol{\phi}},A}$ depends on one real parameter $A>0$ . Here $\boldsymbol{\phi}$ is fixed in ${\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ as the cone $C=\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of $L$ depends on $\boldsymbol{\phi}$ , and all $\hat{L}\in{\mathbin{\cal M}}_{\scriptscriptstyle L}^{\rho}$ have the same cone $C$ , by definition.

Imagi, Oliveira dos Santos and the author [31, Th. 1.1] prove a uniqueness theorem for Lawlor necks. The proof involves Lagrangian Floer cohomology and Fukaya categories, and was motivated by the ideas of this paper.

Theorem 2.6.

Suppose $L$ is a closed, embedded, exact, asymptotically conical special Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 3,$ asymptotic at rate $\rho<0$ to a union $\Pi_{1}\cup\Pi_{2}$ of two transversely intersecting special Lagrangian planes $\Pi_{1},\Pi_{2}$ in ${\mathbin{\mathbb{C}}}^{m}$ . Then $L$ is equivalent under an $\mathop{\rm SU}(m)$ rotation to one of the ‘Lawlor necks’ $L_{\boldsymbol{\phi},A}$ found by Lawlor [45], and described in Example 2.5.

Here is an example based on Harvey and Lawson [27, §III.3.A]:

Example 2.7.

Define a special Lagrangian $T^{2}$ -cone $C$ in ${\mathbin{\mathbb{C}}}^{3}$ by

C=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:|z_{1}|=|z_{2}|=|z_{3}|,\;\>z_{1}z_{2}z_{3}\in[0,\infty)\bigr\}.

(2.4)

This will be important in §3.6 as it is a ‘stable’ special Lagrangian singularity in the sense of [33, Def. 3.6]. There are three families of explicit asymptotically conical SL 3-folds $L^{A}_{1},L^{A}_{2},L^{A}_{3}$ for $A>0$ in ${\mathbin{\mathbb{C}}}^{3},$ each diffeomorphic to ${\mathbin{\cal S}}^{1}\times{\mathbin{\mathbb{R}}}^{2}$ and asymptotic at rate $\rho=0$ to the cone $C$ , where

L^{A}_{1}=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:|z_{1}|^{2}-A=|z_{2}|^{2}=|z_{3}|^{2},\;\>z_{1}z_{2}z_{3}\in[0,\infty)\bigr\},

(2.5)

and $L^{A}_{2},L^{A}_{3}$ are obtained from $L^{A}_{1}$ by cyclic permutation of $z_{1},z_{2},z_{3}$ .

Example 2.8.

In [37, 38, 39] we study SL 3-folds in ${\mathbin{\mathbb{C}}}^{3}$ invariant under the ${\rm U}(1)$ -action

{\rm e}^{i\theta}:(z_{1},z_{2},z_{3})\longmapsto({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3})\quad\text{for ${\rm e}^{i\theta}\in{\rm U}(1)$.}

The three papers are surveyed in [40]. A ${\rm U}(1)$ -invariant SL 3-fold $N$ may locally be written in the form

\begin{split}N=\bigl\{(z_{1},z_{2},z_{3})\in{\mathbin{\mathbb{C}}}^{3}:\,&z_{1}z_{2}=v(x,y)+iy,\quad z_{3}=x+iu(x,y),\\ &|z_{1}|^{2}-|z_{2}|^{2}=2a,\quad(x,y)\in S\bigr\},\end{split}

(2.6)

where $S$ is a domain in ${\mathbin{\mathbb{R}}}^{2}$ , $a\in{\mathbin{\mathbb{R}}}$ and $u,v:S\rightarrow{\mathbin{\mathbb{R}}}$ satisfy (in a weak sense if $a=0$ ) the nonlinear Cauchy–Riemann equations

\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\quad\text{and}\quad\frac{\partial v}{\partial x}=-2\bigl(v^{2}+y^{2}+a^{2}\bigr)^{1/2}\frac{\partial u}{\partial y}.

(2.7)

If $S$ is simply-connected, as $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ there exists a potential $f$ for $u,v$ with $\frac{\partial f}{\partial y}=u$ , $\frac{\partial f}{\partial x}=v$ , satisfying

\Bigl(\Bigl(\frac{\partial f}{\partial x}\Bigr)^{2}+y^{2}+a^{2}\Bigr)^{-1/2}\frac{\partial^{2}f}{\partial x^{2}}+2\,\frac{\partial^{2}f}{\partial y^{2}}=0.

(2.8)

In [37, 38], for suitable strictly convex domains $S\subset{\mathbin{\mathbb{R}}}^{2}$ and boundary data $\phi:\partial S\rightarrow{\mathbin{\mathbb{R}}}$ , we prove the existence of a unique $f:S\rightarrow{\mathbin{\mathbb{R}}}$ satisfying (2.8) and $f|_{\partial S}=\phi$ , and then $u=\frac{\partial f}{\partial y}$ , $v=\frac{\partial f}{\partial x}$ satisfy (2.7) (possibly in a weak sense if $a=0$ ), and $N$ in (2.6) is special Lagrangian.

When $v=y=a=0$ , equations (2.7)–(2.8) become singular, and the SL 3-fold $N$ in (2.6) has a singularity at $(0,0,z_{3})=\bigl(0,0,x+iu(x,0)\bigr)$ in ${\mathbin{\mathbb{C}}}^{3}$ . In the simplest cases $N$ is locally modelled on the cone $C$ in (2.4) near $(0,0,z_{3})$ , but there are also infinitely many other topological types of singularities not locally modelled on cones. Note that the existence and uniqueness results for $N$ are entirely independent of the singularities appearing in the interior of $N$ .

The following will be important in §3.6. Using the results of [37, 38, 39, 40], by choosing a suitable family $\phi^{t}:t\in(-\epsilon,\epsilon)$ of boundary conditions for the potential $f^{t}$ , we can construct a family $N^{t}:t\in(-\epsilon,\epsilon)$ of exact ${\rm U}(1)$ -invariant SL 3-folds in ${\mathbin{\mathbb{C}}}^{3}$ of the form (2.6) with $a=0$ , with the following properties:

(i)

$N^{t}$ depends continuously on $t\in(-\epsilon,\epsilon)$ in a suitable sense, for instance as special Lagrangian integral currents in Geometric Measure Theory.
(ii)

$N^{t}$ is nonsingular for $t<0$ .
(iii)

$N^{0}$ has one singular point at $(0,0,0)\in{\mathbin{\mathbb{C}}}^{3}$ , which has tangent cone $\Pi_{1}\cup\Pi_{2}$ , where $\Pi_{1},\Pi_{2}$ are ${\rm U}(1)$ -invariant special Lagrangian planes in ${\mathbin{\mathbb{C}}}^{3}$ intersecting non-transversely with $\Pi_{1}\cap\Pi_{2}={\mathbin{\mathbb{R}}}$ .
(iv)

$N^{t}$ for $t>0$ has two singular points at $(0,0,\pm z(t))$ , where $z(t)$ depends smoothly on $t$ and $z(t)\rightarrow 0$ as $t\rightarrow 0$ . Each singular point is locally modelled on the special Lagrangian $T^{2}$ -cone $C$ in (2.4).

Thus, isolated singular points of SL $3$ -folds modelled on the $T^{2}$ -cone $C$ in (2.4) can appear or disappear in pairs under continuous deformation.

2.3 Lagrangian mean curvature flow

Next we discuss (Lagrangian) mean curvature flow. A book on mean curvature flow (MCF) for hypersurfaces in ${\mathbin{\mathbb{R}}}^{n}$ is Mantegazza [50]. Two useful surveys on Lagrangian MCF are Smoczyk [67] and Neves [56].

Let $(M,g)$ be a Riemannian manifold, and $N$ a compact manifold with $\mathop{\rm dim}\nolimits N<\mathop{\rm dim}\nolimits M$ , and consider embeddings or immersions $\iota:N\hookrightarrow M$ , so that $\iota(N)$ is a submanifold of $M$ . Mean curvature flow (MCF) is the study of smooth 1-parameter families $\iota_{t}$ , $t\in[0,T)$ of such $\iota_{t}:N\hookrightarrow M$ satisfying

\frac{{\rm d}\iota_{t}}{{\rm d}t}=H_{\iota_{t}},

where $H_{\iota_{t}}\in C^{\infty}(\iota_{t}^{*}(TM))$ is the mean curvature of the submanifold $\iota_{t}:N\hookrightarrow M$ . We usually write $N^{t}$ rather than $\iota_{t}:N\hookrightarrow M$ , suppressing the immersion, so that $\{N^{t}:t\in[0,T)\}$ is a family of submanifolds satisfying MCF.

Mean curvature flow is the gradient flow of the volume functional for compact submanifolds $N$ in $M$ . It has a unique short-time solution starting from any compact submanifold $N$ .

Now let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and $L$ a compact Lagrangian submanifold in $M$ . Then the mean curvature of $L$ is $H=J\nabla\Theta_{L}$ , where $\Theta_{L}:L\rightarrow{\rm U}(1)$ is the phase function from Definition 2.1. Thus $H$ is an infinitesimal deformation of $L$ as a Lagrangian. Smoczyk [66] shows that MCF starting from $L$ preserves the Lagrangian condition, yielding a 1-parameter family of Lagrangians $\{L^{t}:t\in[0,\epsilon)\}$ with $L^{0}=L$ , which are all in the same Hamiltonian isotopy class if $L$ is Maslov zero. This is Lagrangian mean curvature flow (LMCF). Special Lagrangians are stationary points of Lagrangian MCF.

We will be especially interested in Lagrangian MCF for graded Lagrangians. Suppose $\{L^{t}:t\in[0,T)\}$ is a family of compact, graded Lagrangians satisfying Lagrangian MCF. Then $L^{t}$ are all Hamiltonian isotopic, that is, graded Lagrangian MCF stays within a fixed Hamiltonian isotopy class. Also, if the phase function $\theta_{L^{0}}$ takes values in an interval $[a,b]$ or $(a,b)$ , then so does $\theta_{L^{t}}$ for $t\in[0,T)$ . Thus, Lagrangian MCF preserves the almost calibrated condition.

It is an important problem to understand the singularities which arise in Lagrangian mean curvature flow. Singularities in Lagrangian MCF are often locally modelled on soliton solutions, Lagrangians in ${\mathbin{\mathbb{C}}}^{m}$ which move by rescaling or translation under Lagrangian MCF.

Definition 2.9.

A closed Lagrangian $L$ in ${\mathbin{\mathbb{C}}}^{m}$ is called an LMCF expander if $H=\alpha F^{\perp}$ in $C^{\infty}(T{\mathbin{\mathbb{C}}}^{m}|_{L})$ , where $H$ is the mean curvature of $L$ and $F^{\perp}$ is the orthogonal projection of the position vector $F$ (that is, the inclusion $F:L\hookrightarrow{\mathbin{\mathbb{C}}}^{m}$ ) to the normal bundle $TL^{\perp}\subset T{\mathbin{\mathbb{C}}}^{m}|_{L}$ , and $\alpha>0$ is constant.

This implies that (after reparametrizing by diffeomorphisms of $L$ ) the family of Lagrangians $L^{t}:=\sqrt{2\alpha t}\,L$ for $t\in(0,\infty)$ satisfy Lagrangian mean curvature flow. That is, Lagrangian MCF expands $L$ by dilations.

Similarly, we call $L$ an LMCF shrinker if $H=\alpha F^{\perp}$ for $\alpha<0$ , and then $L^{t}:=\sqrt{2\alpha t}\,L$ for $t\in(-\infty,0)$ satisfy LMCF, so LMCF shrinks $L$ by dilations.

We call $L$ an LMCF translator if $H=v^{\perp}$ , where $v\in{\mathbin{\mathbb{C}}}^{m}$ is the translating vector of $L$ , and $v^{\perp}$ the orthogonal projection of $v$ to $TL^{\perp}$ . Then $L^{t}:=L+tv$ for $t\in{\mathbin{\mathbb{R}}}$ satisfy LMCF, so Lagrangian MCF translates $L$ in ${\mathbin{\mathbb{C}}}^{m}$ .

Finite time singularities of MCF have a fundamental division into ‘type I’ and ‘type II’ singularities:

Definition 2.10.

Let $(M,g)$ be a compact Riemannian manifold (e.g. a Calabi–Yau $m$ -fold) and $\{L^{t}:t\in[0,T)\}$ a family of compact immersed submanifolds in $M$ (e.g. Lagrangians) satisfying mean curvature flow. We say that the family has a finite time singularity at $t=T$ if the flow cannot be smoothly continued to $[0,T+\epsilon)$ for any $\epsilon>0$ . As in Wang [71, Lem. 5.1] this implies that $\mathop{\rm lim\,sup}_{t\rightarrow T}\|A^{t}\|_{C^{0}}\rightarrow\infty$ , where $A^{t}$ is the second fundamental form of $L^{t}$ .

We call such a finite time singularity of type I if $\|A^{t}\|_{C^{0}}^{2}\leqslant\penalty C/(T-t)$ for some $C>0$ and all $t\in[0,T)$ . Otherwise we call the singularity of type II.

We call $x\in M$ a singular point of the flow if $\mathop{\rm lim\,sup}_{t\rightarrow T}\|A^{t}|_{U\cap L^{t}}\|_{C^{0}}=\infty$ for all open neighbourhoods $U$ of $x$ in $M$ .

Huisken [29] showed that type I singularities developing a singularity at $x\in M$ are locally modelled in a strong sense on MCF shrinkers in ${\mathbin{\mathbb{R}}}^{n}=T_{x}M$ , through a process known as ‘type I blow up’, as in Smoczyk [67, Prop. 3.17] or Mantegazza [50, §3].

However, we are interested in MCF of graded Lagrangians in Calabi–Yau $m$ -folds, and it turns out that type I singularities do not occur in graded Lagrangian MCF, as was proved by Wang [71, Rem. 5.1] and Chen and Li [13, Cor. 6.7] in the almost calibrated case (i.e. Lagrangians $L^{t}$ with phase variation less than $\pi$ ) and by Neves [55, Th. A] in the graded (or Maslov zero) case.

Theorem 2.11.

Let $(M,J,g,\Omega)$ be a compact Calabi–Yau $m$ -fold and $\{L^{t}:t\in[0,T)\}$ a family of compact, immersed, graded Lagrangians in $M$ satisfying Lagrangian MCF. Then the flow cannot develop a type I singularity.

A parallel result of Neves [56, Cor. 3.5] says that there exist no nontrivial, immersed, graded Lagrangian MCF shrinkers in ${\mathbin{\mathbb{C}}}^{m}$ (satisfying a few extra conditions such as closed in ${\mathbin{\mathbb{C}}}^{m}$ and of bounded Lagrangian angle), so there are no possible local models for type I blow ups of graded Lagrangian MCF. Examples of Lagrangian MCF shrinkers in ${\mathbin{\mathbb{C}}}^{m}$ can be found in Abresch and Langer [1] for $m=1$ and in Anciaux [3] and Joyce, Lee and Tsui [43, Th. F] in higher dimensions, but none of them are graded.

So, for graded Lagrangian MCF, all finite time singularities are of type II. It is a well known ‘folklore’ theorem that type II singularities of MCF admit ‘type II blow ups’, eternal smooth solutions of MCF in ${\mathbin{\mathbb{R}}}^{n}$ modelling the formation of the singularity in the small region where the second fundamental form $A^{t}$ is largest as $t\rightarrow T$ . The idea of type II blow ups is due to Hamilton, and explanations can be found in Smoczyk [67, §3.4] and Mantegazza [50, §4.1], and for Lagrangian MCF in Han and Li [24, §2]. We state it for graded LMCF:

Theorem 2.12.

Let $(M,J,g,\Omega)$ be a compact Calabi–Yau $m$ -fold and $\{L^{t}:t\in[0,T)\}$ a family of compact, immersed, graded Lagrangians in $M$ satisfying Lagrangian MCF, with a finite time singularity at $t=T$ . Then at some singular point $x\in M$ of the flow there exists a type II blow up.

That is, identifying $M$ near $x$ with $T_{x}M\cong{\mathbin{\mathbb{C}}}^{m}$ near $0,$ there exist sequences $(t_{i})_{i=1}^{\infty}$ in $[0,T),$ $(x_{i})_{i=1}^{\infty}$ in $M$ and $(\lambda_{i})_{i=1}^{\infty}$ in $(0,\infty),$ such that $t_{i}\rightarrow T,$ $x_{i}\rightarrow x,$ $\lambda_{i}\rightarrow\infty$ and $\lambda_{i}^{2}(T-t_{i})\rightarrow 0$ as $i\rightarrow\infty,$ and for each $s\in{\mathbin{\mathbb{R}}}$ the limit

\tilde{L}^{s}=\lim_{i\rightarrow\infty}\lambda_{i}\cdot(L^{t_{i}+\lambda_{i}^{-2}s}-x_{i})

exists as a nonempty, noncompact, smooth, closed, immersed, exact, graded Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ whose mean curvature $\tilde{A}^{s}$ is nonzero (so that $\tilde{L}^{s}$ is not a union of Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$ ). All derivatives of $\tilde{A}^{s},$ and the phase function $\theta_{\smash{\tilde{L}^{s}}},$ are uniformly bounded independently of $s\in{\mathbin{\mathbb{R}}}$ . Also $\tilde{L}^{s}$ depends smoothly on $s\in{\mathbin{\mathbb{R}}},$ and $\{\tilde{L}^{s}:s\in{\mathbin{\mathbb{R}}}\}$ satisfies Lagrangian MCF in ${\mathbin{\mathbb{C}}}^{m}$ .

A solution $\{\tilde{L}^{s}:s\in{\mathbin{\mathbb{R}}}\}$ of MCF for all $s\in{\mathbin{\mathbb{R}}}$ is called an eternal solution. Two obvious classes of eternal solutions of Lagrangian MCF in ${\mathbin{\mathbb{C}}}^{m}$ are

(a)

$\tilde{L}^{s}=L$ is independent of $s\in{\mathbin{\mathbb{R}}}$ , and is an SL $m$ -fold in ${\mathbin{\mathbb{C}}}^{m}$ .
(b)

$\tilde{L}^{s}=L+sv$ for $s\in{\mathbin{\mathbb{R}}}$ , where $L$ is a Lagrangian MCF translator in ${\mathbin{\mathbb{C}}}^{m}$ with translating vector $v\in{\mathbin{\mathbb{C}}}^{m}$ .

Many examples of special Lagrangian $m$ -folds in ${\mathbin{\mathbb{C}}}^{m}$ are known suitable for use in (a), but for (b) there are few, as we explain in §2.4.

2.4 Examples of solitons for Lagrangian MCF

We now give examples of solitons for Lagrangian MCF. We are interested in graded Lagrangians, and as in §2.3 there are no graded Lagrangian MCF shrinkers. The next example describes a family of LMCF expanders from Joyce, Lee and Tsui [43, Th.s C & D], generalizing the ‘Lawlor necks’ of Example 2.5.

Example 2.13.

Let $m>2$ , $\alpha\geqslant 0$ and $a_{1},\ldots,a_{m}>0$ , and define a smooth function $P:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{R}}}$ by $P(0)=\alpha+a_{1}+\cdots+a_{m}$ and

P(x)=\textstyle\frac{1}{x^{2}}\bigl(e^{\alpha x^{2}}\prod_{k=1}^{m}(1+a_{k}x^{2})-1\bigl),\quad x\neq 0.

(2.9)

Define real numbers $\phi_{1},\ldots,\phi_{m}$ by

\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,,

For $k=1,\ldots,m$ define a function $z_{k}:{\mathbin{\mathbb{R}}}\rightarrow{\mathbin{\mathbb{C}}}$ by

z_{k}(y)={\rm e}^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\;\>\text{where}\;\>\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{{\rm d}x}{(1+a_{k}x^{2})\sqrt{P(x)}}\,.

Now write ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$ , and define a submanifold $L_{\boldsymbol{\phi}}^{\alpha}$ in ${\mathbin{\mathbb{C}}}^{m}$ by

L_{\boldsymbol{\phi}}^{\alpha}=\bigl\{(z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m}):y\in{\mathbin{\mathbb{R}}},\;x_{k}\in{\mathbin{\mathbb{R}}},\;x_{1}^{2}+\cdots+x_{m}^{2}=1\bigr\}.

Then $L_{\boldsymbol{\phi}}^{\alpha}$ is a closed, embedded Lagrangian diffeomorphic to ${\mathbin{\cal S}}^{m-1}\times{\mathbin{\mathbb{R}}}$ and satisfying $H=\alpha F^{\perp}$ . If $\alpha>0$ it is an LMCF expander, and if $\alpha=0$ it is one of the Lawlor necks $L_{{\boldsymbol{\phi}},A}$ from Example 2.5. It is graded, with Lagrangian angle

\theta_{L_{\boldsymbol{\phi}}^{\alpha}}\bigl((z_{1}(y)x_{1},\ldots,z_{m}(y)x_{m})\bigr)=\textstyle\sum_{k=1}^{m}\psi_{k}(y)+\arg\bigl(-y-iP(y)^{-1/2}\bigr).

Note that the only difference between the constructions of $L_{{\boldsymbol{\phi}},A}$ in Example 2.5 and $L_{\boldsymbol{\phi}}^{\alpha}$ above is the term $e^{\alpha x^{2}}$ in (2.9), which does not appear in (2.3). If $\alpha=0$ then $e^{\alpha x^{2}}=1$ , and the two constructions agree.

As in [43, Th. D], $L_{\boldsymbol{\phi}}^{\alpha}$ is asymptotically conical, with cone $C$ the union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ of two Lagrangian $m$ -planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ given by

\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.

But in contrast to Example 2.5, for $\alpha>0$ we do not have $\phi_{1}+\cdots+\phi_{m}=\pi$ , so $\Pi_{\boldsymbol{\phi}}$ and $C$ are not special Lagrangian.

In [43, Th. D] we prove that for fixed $\alpha>0$ , the map $\Phi^{\alpha}:(a_{1},\ldots,a_{m})\mapsto(\phi_{1},\ldots,\phi_{m})$ gives a diffeomorphism

\Phi^{\alpha}:(0,\infty)^{m}\longrightarrow\bigl\{(\phi_{1},\ldots,\phi_{m})\in(0,\pi)^{m}:0<\phi_{1}+\cdots+\phi_{m}<\pi\bigr\}.

That is, for all $\alpha>0$ and ${\boldsymbol{\phi}}=(\phi_{1},\ldots,\phi_{m})$ with $0<\phi_{1},\ldots,\phi_{m}<\pi$ and $0<\phi_{1}+\cdots+\phi_{m}<\pi$ , the above construction gives a unique LMCF expander $L_{\boldsymbol{\phi}}^{\alpha}$ asymptotic to $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}}$ .

Motivated by the ideas of this paper, Imagi, Oliveira dos Santos and the author [31, Th. 1.1] prove a uniqueness theorem for these LMCF expanders when $m\geqslant 3$ . The case $m=2$ was already proved by Lotay and Neves [47].

Theorem 2.14.

Suppose $L$ is a closed, embedded, exact, asymptotically conical Lagrangian MCF expander in ${\mathbin{\mathbb{C}}}^{m}$ for $m\geqslant 2,$ satisfying the expander equation $H=\alpha F^{\perp}$ for $\alpha>0,$ and asymptotic at rate $\rho<2$ to a union $\Pi_{1}\cup\Pi_{2}$ of two transversely intersecting Lagrangian planes $\Pi_{1},\Pi_{2}$ in ${\mathbin{\mathbb{C}}}^{m}$ . Then $L$ is equivalent under a ${\rm U}(m)$ rotation to one of the LMCF expanders $L_{\boldsymbol{\phi}}^{\alpha}$ found by Joyce, Lee and Tsui [43, Th.s C & D], and described in Example 2.13.

Example 2.15.

In dimension $m=1$ , the unique connected Lagrangian MCF translator in ${\mathbin{\mathbb{C}}}$ , up to rigid motions and rescalings, is the ‘grim reaper’

\bigl\{x+iy\in{\mathbin{\mathbb{C}}}:y\in(-\pi/2,\pi/2),\quad x=-\log\cos y\bigr\},

with translating vector $v=1\in{\mathbin{\mathbb{C}}}$ , which is sketched in Figure 2.1.

Figure 2.1: ‘Grim reaper’ Lagrangian MCF translating soliton in

{\mathbin{\mathbb{C}}}

Here is a family of LMCF translators from Joyce, Lee and Tsui [43, Cor. I]:

Example 2.16.

For given constants $\alpha>0$ and $a_{1},\ldots,a_{m-1}>0,$ define

\psi_{j}(y)=\int_{-\infty}^{y}\frac{{\rm d}t}{(\frac{1}{a_{j}}+t^{2})\sqrt{P(t)}}\,,\;\>\text{where}\;\>P(t)=\frac{1}{t^{2}}\bigg(\prod_{k=1}^{m-1}(1+a_{k}t^{2})e^{\alpha t^{2}}-1\bigg),

for $j=1,\ldots,m-1$ and $y\in{\mathbin{\mathbb{R}}}$ . Then

	$\displaystyle L=$	$\displaystyle\bigl\{\bigl(x_{1}\textstyle\sqrt{\frac{1}{a_{1}}\!+\!y^{2}}\,e^{i\psi_{1}(y)},\ldots,x_{m-1}\sqrt{\frac{1}{a_{m-1}}\!+\!y^{2}}\,e^{i\psi_{m-1}(y)},\textstyle{\textstyle\frac{1}{2}}y^{2}\!-\!{\textstyle\frac{1}{2}}\sum_{j=1}^{m-1}x_{j}^{2}$
		$\displaystyle-\textstyle\frac{i}{\alpha}\sum_{j=1}^{m-1}\psi_{j}(y)-\textstyle\frac{i}{\alpha}\arg(y+iP(y)^{-1/2})\bigr):x_{1},\ldots,x_{m-1},y\in{\mathbin{\mathbb{R}}}\bigr\}$		(2.10)

is a closed, embedded Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ diffeomorphic to ${\mathbin{\mathbb{R}}}^{m},$ which is a Lagrangian MCF translator with translating vector $(0,\ldots,0,\alpha)\in{\mathbin{\mathbb{C}}}^{m}$ .

Define $\phi_{1},\ldots,\phi_{m-1}\in{\mathbin{\mathbb{R}}}$ by

\phi_{j}=\int_{-\infty}^{\infty}\frac{{\rm d}t}{(\frac{1}{a_{j}}+t^{2})\sqrt{P(t)}}\,.

Then $\phi_{1},\ldots,\phi_{m-1}\in(0,\pi)$ with $\phi_{1}+\cdots+\phi_{m-1}<\pi$ , and $\psi_{j}(y)\rightarrow\phi_{j}$ as $y\rightarrow\infty$ , and $\psi_{j}(y)\rightarrow 0$ as $y\rightarrow-\infty$ . For fixed $\alpha>0,$ the map $(a_{1},\ldots,a_{m-1})\mapsto(\phi_{1},\ldots,\phi_{m-1})$ is a 1-1 correspondence from $(0,\infty)^{m-1}$ to $\bigl\{(\phi_{1},\ldots,\phi_{m-1})\in(0,\pi)^{m-1}:\phi_{1}+\cdots+\phi_{m-1}<\pi\bigr\}$ .

The phase function $\theta_{L}$ of $L$ in (2.10) is a monotone decreasing function of $y$ only, with limits $\pi$ as $y\rightarrow-\infty$ and $\sum_{j=1}^{m-1}\phi_{j}$ as $y\rightarrow+\infty$ . Thus, by choosing $\sum_{j=1}^{m-1}\phi_{j}$ close to $\pi,$ the phase variation of $L$ can be made arbitrarily small.

We can give the following heuristic description of $L$ in (2.10). If $y\gg 0$ then $\psi_{j}(y)\approx\phi_{j}$ and $\sqrt{\frac{1}{a_{j}}+y^{2}}\approx y$ , and the terms $-\frac{i}{\alpha}\sum_{j=1}^{n}\psi_{j}(y)-\frac{i}{\alpha}\arg(y+iP(y)^{-1/2})$ are negligible compared to ${\textstyle\frac{1}{2}}y^{2}$ in the last coordinate. Thus, the region of $L$ with $y\!\gg\!0$ is in a weak sense approximate to

\bigl\{\bigl(x_{1}ye^{i\phi_{1}},\ldots,x_{m-1}ye^{i\phi_{m-1}},\textstyle{\textstyle\frac{1}{2}}y^{2}-{\textstyle\frac{1}{2}}\sum_{j=1}^{m-1}x_{j}^{2}\bigr):x_{1},\ldots,x_{m-1}\in{\mathbin{\mathbb{R}}},\;y>0\bigr\}.

But this is just an unusual way of parametrizing

\Pi_{\boldsymbol{\phi}}=\bigl\{\bigl(y_{1}e^{i\phi_{1}},\ldots,y_{m-1}e^{i\phi_{m-1}},y_{m}\bigr):y_{j}\in{\mathbin{\mathbb{R}}}\bigr\}\setminus\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\},

the complement of a ray in a Lagrangian plane. Similarly, the region of $L$ with $y\ll 0$ is in a weak sense approximate to

\Pi_{0}=\bigl\{(y_{1},\ldots,y_{m-1},y_{m}):y_{j}\in{\mathbin{\mathbb{R}}}\bigr\}\setminus\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\}.

So, $L$ can be roughly described as asymptotic to the union of two Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}\cong{\mathbin{\mathbb{R}}}^{m}$ which intersect in an ${\mathbin{\mathbb{R}}}$ in ${\mathbin{\mathbb{C}}}^{m}$ , the $y_{m}$ -axis $\bigl\{(0,\ldots,0,y_{m}):y_{m}\in{\mathbin{\mathbb{R}}}\bigr\}$ . To make $L$ , we glue these Lagrangian planes by a kind of ‘connect sum’ along the negative $y_{m}$ -axis $\bigl\{(0,\ldots,0,y_{m}):y_{m}\leqslant\penalty 0\bigr\}$ . Under Lagrangian mean curvature flow, $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ remain fixed, but the gluing region translates in the positive $y_{m}$ direction, as though $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ are being ‘zipped together’.

A slightly more accurate description of the ends of $L$ for large $y$ is that $L$ approximates $\tilde{\Pi}_{\boldsymbol{\phi}}$ when $y\gg 0$ and $\tilde{\Pi}_{0}$ when $y\ll 0$ , where $\tilde{\Pi}_{\boldsymbol{\phi}}$ and $\tilde{\Pi}_{0}$ are the non-intersecting affine Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$

\begin{split}\tilde{\Pi}_{\boldsymbol{\phi}}&=\bigl\{\bigl(y_{1}e^{i\phi_{1}},\ldots,y_{m-1}e^{i\phi_{m-1}},y_{m}\!-\!\textstyle\frac{i}{\alpha}(\phi_{1}\!+\!\cdots\!+\!\phi_{m-1})\bigr):y_{j}\!\in\!{\mathbin{\mathbb{R}}}\bigr\},\\ \tilde{\Pi}_{0}&=\bigl\{\bigl(y_{1},\ldots,y_{m-1},y_{m}-\textstyle\frac{i\pi}{\alpha}\bigr):y_{j}\in{\mathbin{\mathbb{R}}}\bigr\}.\end{split}

(2.11)

We will discuss these Lagrangian MCF translators further in Example 3.32.

Castro and Lerma [12] give more examples of Lagrangian MCF translators in ${\mathbin{\mathbb{C}}}^{2}$ . Neves and Tian [58] prove some nonexistence results.

2.5 Lagrangian Floer cohomology and Fukaya categories

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, which may be compact or noncompact, with Kähler form $\omega$ . We now explain a little about (embedded) Lagrangian branes $(L,E)$ in $(M,\omega)$ , bounding cochains $b$ for $(L,E)$ and obstructions to $HF^{*}$ , Lagrangian Floer cohomology $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ , the Fukaya category ${\mathbin{\mathscr{F}}}(M)$ , and the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ . Section 2.6 discusses the extension of all this to immersed Lagrangians.

The construction of $D^{b}{\mathbin{\mathscr{F}}}(M)$ in the generality we need may not yet be available in the literature. As this paper is wholly conjecture anyway, and clearly the theory will eventually work, this does not matter very much.

The version of bounding cochains, obstructions to $HF^{*}$ , and Lagrangian Floer cohomology we need is in Fukaya, Oh, Ohta and Ono [20]. An early explanation of how to define the (derived) Fukaya category ${\mathbin{\mathscr{F}}}(M),D^{b}{\mathbin{\mathscr{F}}}(M)$ is Fukaya [18], and a more recent survey is Fukaya [19]. Floer [17] originally introduced Lagrangian Floer cohomology.

For exact Lagrangians in Liouville manifolds (a class of noncompact, exact symplectic manifolds), a simpler, more complete, and more satisfactory theory of Lagrangian Floer cohomology and Fukaya categories is given in Seidel [64], which we used in [31] to prove Theorems 2.6 and 2.14. In Seidel’s theory there are no bounding cochains or obstructions to $HF^{*}$ .

However, for our purposes Seidel’s theory will not do: we need to extend the theory to immersed Lagrangians, and even for exact Lagrangians, bounding cochains and obstructions to $HF^{*}$ will then appear. Also, we wish to stress the idea that Lagrangian MCF is better behaved for Lagrangians with $HF^{*}$ unobstructed, and in Seidel’s framework this issue is hidden by restricting to exact, embedded Lagrangians, for which $HF^{*}$ is automatically unobstructed.

Definition 2.17.

Fix a field ${\mathbin{\mathbb{F}}}$ , in which we will do ‘counting’ of $J$ -holomorphic curves. If nontrivial $J$ -holomorphic ${\mathbin{\mathbb{CP}}}^{1}$ ’s can exist in the symplectic manifold $(M,\omega)$ we are interested in, the virtual counts can be rational, so ${\mathbin{\mathbb{F}}}$ must have characteristic zero, and ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{Q}}},{\mathbin{\mathbb{R}}}$ or ${\mathbin{\mathbb{C}}}$ are the obvious possibilities. If $M$ has no $J$ -holomorphic ${\mathbin{\mathbb{CP}}}^{1}$ ’s (for example, if $\omega$ is exact, or if $\pi_{2}(M)=0$ ) then ${\mathbin{\mathbb{F}}}$ can be arbitrary, so we can take ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{Z}}}_{2}$ , for instance, which means we do not have to worry about orientations on moduli spaces of $J$ -holomorphic curves.

The Novikov ring $\Lambda_{\rm nov}$ is the field of formal power series $\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ for $a_{i}\in{\mathbin{\mathbb{F}}}$ and $\lambda_{i}\in{\mathbin{\mathbb{R}}}$ with $\lambda_{i}\rightarrow+\infty$ as $i\rightarrow\infty$ , for $P$ a formal variable. Write $\Lambda_{\rm nov}^{\geqslant 0}$ for the subring of $\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ in $\Lambda_{\rm nov}$ with all $\lambda_{i}\geqslant 0$ , and $\Lambda_{\rm nov}^{+}$ for the ideal of $\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ in $\Lambda_{\rm nov}$ with all $\lambda_{i}>0$ .

Definition 2.18.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold. A Lagrangian brane in $M$ is a pair $(L,E)$ , where $L$ is a compact, spin, graded Lagrangian in $M$ , and $E\rightarrow L$ is a rank one ${\mathbin{\mathbb{F}}}$ -local system on $L$ , for ${\mathbin{\mathbb{F}}}$ as in Definition 2.17. That is, $E$ is a locally constant rank one ${\mathbin{\mathbb{F}}}$ -vector bundle over $L$ , so that if $p\in L$ then $E|_{p}$ is a dimension one ${\mathbin{\mathbb{F}}}$ -vector space, which is locally independent of $p$ .

In this section we take $L$ to be embedded, but in §2.6 $L$ can be immersed, and in §3 we will (conjecturally) allow $L$ to have certain kinds of singularities.

Remark 2.19.

‘Lagrangian branes’ are the objects for which we will define Lagrangian Floer cohomology and Fukaya categories; the term is used in the same way by Seidel [64, §12a] and Haug [28, §3.1], for instance, although with different definitions. Our definition is designed to try to make the programme of §3 work. The precise details of Definition 2.18 will be important in Remark 3.7 and §3.4, and are discussed in Remark 3.13.

If we take ${\mathbin{\mathbb{F}}}={\mathbin{\mathbb{C}}}$ then $E\rightarrow L$ is a complex line bundle on $L$ with a flat connection $\nabla_{E}$ , which is determined up to isomorphism by its holonomy $\mathop{\rm Hol}\nolimits(\nabla_{E}):\pi_{1}(L)\rightarrow{\mathbin{\mathbb{C}}}^{*}$ . In String Theory and Mirror Symmetry it is natural to suppose that $\nabla_{E}$ preserves a unitary metric on $E$ , so that $\mathop{\rm Hol}\nolimits(\nabla_{E})$ takes values in ${\rm U}(1)\subset{\mathbin{\mathbb{C}}}^{*}$ . One can also allow $E$ to be an ${\mathbin{\mathbb{F}}}$ -local system of higher rank. Kontsevich [44] and Fukaya [18, §2.1] include a unitary local system $E\rightarrow L$ of arbitrary rank in objects of their Fukaya categories.

We need to restrict to $E$ of rank one, and not to impose the unitary condition.

Much of the literature on Lagrangian Floer cohomology and Fukaya categories including [20, 64] omits the local system $E\rightarrow L$ , which is equivalent to taking $E$ to be trivial, $E={\mathbin{\mathbb{F}}}\times L\rightarrow L$ . As in §3.4, we cannot do this, since in the programme of §3.2 involving families $(L^{t},E^{t},b^{t})$ for $t\in[0,\infty)$ , starting with $E^{0}$ trivial, after a surgery at $t=T_{i}$ , we can have $E^{t}$ nontrivial for $t>T_{i}$ .

Definition 2.20.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and $L,L^{\prime}$ graded Lagrangians in $M$ , with phase functions $\theta_{L},\theta_{L^{\prime}}$ , which intersect transversely at $p\in M$ . By a kind of simultaneous diagonalization, we may choose an isomorphism $T_{p}M\cong{\mathbin{\mathbb{C}}}^{m}$ which identifies $J|_{p},g|_{p},\omega|_{p}$ on $T_{p}M$ with the standard versions (2.2) on ${\mathbin{\mathbb{C}}}^{m}$ , and identifies $T_{p}L,T_{p}L^{\prime}$ with the Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ respectively, where

\Pi_{0}=\bigl\{(x_{1},\ldots,x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},\;\>\Pi_{\boldsymbol{\phi}}=\bigl\{({\rm e}^{i\phi_{1}}x_{1},\ldots,{\rm e}^{i\phi_{m}}x_{m}):x_{j}\in{\mathbin{\mathbb{R}}}\bigr\},

(2.12)

for $\phi_{1},\ldots,\phi_{m}\in(0,\pi)$ . Then $\phi_{1},\ldots,\phi_{m}$ are independent of choices up to order. Define the degree $\mu_{L,L^{\prime}}(p)\in{\mathbin{\mathbb{Z}}}$ of $p$ by

\mu_{L,L^{\prime}}(p)=(\phi_{1}+\cdots+\phi_{m}+\theta_{L}(p)-\theta_{L^{\prime}}(p))/\pi.

This an integer as $\theta_{L^{\prime}}(p)=\theta_{L}(p)+\phi_{1}+\cdots+\phi_{m}\mod\pi{\mathbin{\mathbb{Z}}}$ . Exchanging $L,L^{\prime}$ replaces $\phi_{1},\ldots,\phi_{m}$ by $\pi-\phi_{1},\ldots,\pi-\phi_{m}$ , so that $\mu_{L,L^{\prime}}(p)+\mu_{L^{\prime},L}(p)=m$ . Since $\phi_{1},\ldots,\phi_{m}\in(0,\pi)$ , we see that

(\theta_{L}(p)-\theta_{L^{\prime}}(p))/\pi<\mu_{L,L^{\prime}}(p)<(\theta_{L}(p)-\theta_{L^{\prime}}(p))/\pi+m.

(2.13)

Here is the basic idea of Lagrangian Floer cohomology. Let $(L,E),(L^{\prime},E^{\prime})$ be Lagrangian branes in a Calabi–Yau $m$ -fold $(M,J,g,\Omega)$ , and suppose $L,L^{\prime}$ intersect transversely. The aim is to define a $\Lambda_{\rm nov}$ -module $HF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ called the Lagrangian Floer cohomology, which is the cohomology of a complex of $\Lambda_{\rm nov}$ -modules $\bigl(CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),{\rm d}\bigr)$ called the Floer complex.

Figure 2.2: Holomorphic disc

\Sigma

with boundary in

L\cup L^{\prime}

Define a free, graded $\Lambda_{\rm nov}$ -module $CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ by

CF^{k}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)=\bigoplus_{p\in L\cap L^{\prime}:\mu_{L,L^{\prime}}(p)=k}\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E|_{p},E^{\prime}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}.

Initially we define ${\rm d}:CF^{k}(L,L^{\prime})\rightarrow CF^{k+1}(L,L^{\prime})$ by

{\rm d}\alpha_{p}=\!\!\!\!\!\!\!\!\!\bigoplus_{\begin{subarray}{l}q\in L\cap L^{\prime}:\\ \mu_{L,L^{\prime}}(q)=k+1\end{subarray}}\!\!\!\sum_{A>0}\bigl(\#_{\rm virt}{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}\bigr)\,P^{A}\cdot\mathop{PT}\limits_{\begin{subarray}{l}\text{$p\rightarrow q$ in}\\ \partial\Sigma\cap L^{\prime}\end{subarray}}(E^{\prime})\circ\alpha_{p}\circ\mathop{PT}\limits_{\begin{subarray}{l}\text{$q\rightarrow p$ in}\\ \partial\Sigma\cap L\end{subarray}}(E),

(2.14)

for $p\in L\cap L^{\prime}$ with $\mu_{L,L^{\prime}}(p)=k$ and $\alpha_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E|_{p},E^{\prime}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$ , where ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ is the moduli space of stable $J$ -holomorphic discs $\Sigma$ in $M$ with boundary in $L\cup L^{\prime}$ , corners at $p,q$ and area $A$ , of the form shown in Figure 2.2, where $\#_{\rm virt}{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}\in{\mathbin{\mathbb{Q}}}$ is the ‘virtual number of points’ in ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ , and the sum is weighted by composition with the parallel transport maps $PT_{\cdots}(E)\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E|_{q},E|_{p}\bigr)$ and $PT_{\cdots}(E^{\prime})\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E^{\prime}|_{p},E^{\prime}|_{q}\bigr)$ in the ${\mathbin{\mathbb{F}}}$ -local systems $E,E^{\prime}$ along the two segments of $\partial\Sigma$ . These $PT_{\cdots}(E),PT_{\cdots}(E^{\prime})$ are locally constant on ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ .

Constructing an appropriate geometric structure (‘Kuranishi space’ or ‘polyfold’) on ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ , and defining the virtual count $\#_{\rm virt}{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ , raise many complicated issues which we will not go into.

For exact Lagrangians in an exact symplectic manifold, as in Seidel [64], the differential ${\rm d}$ in (2.14) has ${\rm d}^{2}=0$ , so $HF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ is well-defined. However, in the non-exact case we may have ${\rm d}^{2}\neq 0$ , because of contributions to the boundaries $\partial{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ from holomorphic discs with boundary in $L$ or in $L^{\prime}$ .

To get round this, Fukaya, Oh, Ohta and Ono [20, §3.6] introduce the notion of a bounding cochain $b$ for $(L,E)$ , an element $b$ of the singular $(m-1)$ -chains $C_{m-1}(L,\Lambda_{\rm nov}^{+})$ of $L$ with coefficients in $\Lambda_{\rm nov}^{+}\subset\Lambda_{\rm nov}$ , satisfying an equation in $C_{m-2}(L,\Lambda_{\rm nov}^{+})$ which is (very roughly, and oversimplified) of the form

\partial b+\sum_{k\geqslant 0}\sum_{A>0}P^{A}\cdot\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}\times_{L^{k}}b^{k}\bigr]_{\rm virt}\cdot\mathop{\rm Hol}\nolimits_{\partial\Sigma}(E)=0,

(2.15)

where ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}$ is the moduli space (as a Kuranishi space or polyfold, of virtual dimension $m+k-2$ ) of isomorphism classes $[\Sigma,\vec{z}]$ where $\Sigma$ is a stable $J$ -holomorphic disc of area $A>0$ in $M$ with boundary in $L$ , and $\vec{z}=(z_{0},z_{1},\ldots,z_{k})$ are cyclically ordered marked points in $\partial\Sigma$ . Also ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}\times_{L^{k}}b^{k}$ is the moduli space of such $[\Sigma,\vec{z}]$ in which $z_{1},\ldots,z_{k}$ intersect the chain $b$ in $L$ , and $\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}\times_{L^{k}}b^{k}\bigr]{}_{\rm virt}$ is a virtual chain for this. The sum is weighted by the holonomy $\mathop{\rm Hol}\nolimits_{\partial\Sigma}(E)\in{\mathbin{\mathbb{F}}}^{*}$ of the rank one ${\mathbin{\mathbb{F}}}$ -local system $E\rightarrow L$ around $\partial\Sigma\subset L$ , which depends only on $[\partial\Sigma]\in H_{1}(L,{\mathbin{\mathbb{Z}}})$ , and is locally constant on ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}$ .

If a bounding cochain $b$ exists for $(L,E)$ , we say that $(L,E)$ has $HF^{*}$ unobstructed, otherwise we say that $(L,E)$ has $HF^{*}$ obstructed. Implicitly we will always consider bounding cochains $b$ up to the appropriate notion of equivalence.

To oversimplify even further, suppose that the terms for $k\geqslant 1$ in (2.15) are zero, and $\partial{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}=\emptyset$ for all $A>0$ when $k=0$ , so that $\partial\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}\bigr]{}_{\rm virt}=0$ , and write $b=\sum_{A>0}P^{A}\cdot b_{A}$ for $b_{A}\in C_{m-1}(L,{\mathbin{\mathbb{Q}}})$ . Then (2.15) becomes $\partial b_{A}=\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}\bigr]{}_{\rm virt}\cdot\mathop{\rm Hol}\nolimits_{\partial\Sigma}(E)$ for all $A>0$ . So a bounding cochain $b$ exists if $\bigl[\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}\bigr]{}_{\rm virt}\bigr]=0$ in $H_{m-2}(L,{\mathbin{\mathbb{Q}}})$ for all $A>0$ . In particular, if $H_{m-2}(L,{\mathbin{\mathbb{Q}}})=0$ then a bounding cochain exists.

In the general case, if $H_{m-2}(L,{\mathbin{\mathbb{Q}}})=0$ then (2.15) may be solved for $b=\sum_{A>0}P^{A}\cdot b_{A}$ by an inductive procedure in increasing $A$ , yielding:

Lemma 2.21.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold and $(L,E)$ an embedded Lagrangian brane in $M$ . If $b_{m-2}(L)=0$ then $(L,E)$ has $HF^{*}$ unobstructed.

Suppose $b,b^{\prime}$ are bounding cochains for $(L,E),(L^{\prime},E^{\prime})$ . Then Fukaya et al. [20] define a modification ${\rm d}^{b,b^{\prime}}$ of ${\rm d}$ in (2.14) involving $b,b^{\prime}$ and satisfying $({\rm d}^{b,b^{\prime}})^{2}=0$ . The Lagrangian Floer cohomology $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is the cohomology of $\bigl(CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),{\rm d}^{b,b^{\prime}}\bigr)$ , which may depend on $b,b^{\prime}$ . Here are some properties of Lagrangian Floer cohomology in the theory of Fukaya, Oh, Ohta and Ono [20]:

(a)

The Lagrangian Floer cohomology $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is independent of the choice of almost complex structure $J$ up to canonical isomorphism, although $\bigl(CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),{\rm d}^{b,b^{\prime}}\bigr)$ does depend on $J$ .
(b)

Let $(L^{t},E^{t}):t\in[0,1]$ be a smooth family of Lagrangian branes, with the $L^{t}$ Hamiltonian isotopic and the $E^{t}$ locally constant in $t$ , and let $b^{0}$ be a bounding cochain for $L^{0}$ . By a kind of ‘parallel transport’ we can extend $b^{0}$ to a family of bounding cochains $b^{t}$ for $(L^{t},E^{t})$ for $t\in[0,1]$ . If $(L^{\prime},E^{\prime})$ is another Lagrangian brane with bounding cochain $b^{\prime}$ then $HF^{*}\bigl((L^{t},E^{t},b^{t}),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is independent of $t\in[0,1]$ up to canonical isomorphism. Thus $HF^{*}\bigl((L,E,b),\allowbreak(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is an invariant of Lagrangian branes up to Hamiltonian isotopy.

Remark 2.22.

We need $M$ to be (symplectic) Calabi–Yau and $L,L^{\prime}$ to be graded to define the degree $\mu_{L,L^{\prime}}(p)\in{\mathbin{\mathbb{Z}}}$ , which determines the grading of $CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ and $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ . If we took $M$ symplectic and $L,L^{\prime}$ oriented, then $CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ would only be graded over ${\mathbin{\mathbb{Z}}}_{2}$ rather than ${\mathbin{\mathbb{Z}}}$ .

Lagrangian Floer cohomology is only the beginning of a more general theory of Fukaya categories, which may be still incomplete in the general case. Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold. The idea is to define the Fukaya category ${\mathbin{\mathscr{F}}}(M)$ of $M$ , an $A_{\infty}$ -category whose objects are triples $(L,E,b)$ of a Lagrangian brane $(L,E)$ in $M$ with $HF^{*}$ unobstructed, and a bounding cochain $b$ for $(L,E)$ , such that the morphisms $\mathop{\rm Hom}\nolimits\bigl((L_{0},E_{0},b_{0}),(L_{1},E_{1},b_{1})\bigr)$ in ${\mathbin{\mathscr{F}}}(M)$ are the graded $\Lambda_{\rm nov}$ -modules $CF^{*}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)$ from above, with $A_{\infty}$ -operations

\begin{split}\mu^{k}:CF^{a_{k}}\bigl((L_{k-1},E_{k-1}),(L_{k},E_{k})\bigr)\times\cdots\times CF^{a_{1}}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)&\\ \longrightarrow CF^{a_{1}+\cdots+a_{k}+2-k}\bigl((L_{0},E_{0}),(L_{k},E_{k})\bigr)&\end{split}

(2.16)

for $k\geqslant 1$ , with $\mu^{1}:CF^{a_{1}}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)\allowbreak\rightarrow CF^{a_{1}+1}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)$ the differential ${\rm d}^{b_{0},b_{1}}$ in the Floer complex. The coefficients in the $\Lambda_{\rm nov}$ -multilinear map $\mu^{k}$ in (2.16) are obtained by ‘counting’ $J$ -holomorphic $(k\!+\!1)$ -gons in $M$ with boundary in $L_{0}\cup\cdots\cup L_{k}$ , weighted by parallel transport maps in $E_{0},\ldots,E_{k}$ .

By a category theory construction, one then defines the derived Fukaya category, a triangulated category. There are two versions, which we will write $D^{b}{\mathbin{\mathscr{F}}}(M)$ and $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ . For $D^{b}{\mathbin{\mathscr{F}}}(M)$ , the objects are twisted complexes, as in Seidel [64, §3l]. Roughly speaking, a twisted complex consists of objects $(L_{1},E_{1},b_{1}),\ldots,(L_{n},E_{n},b_{n})$ in ${\mathbin{\mathscr{F}}}(M)$ together with Floer cochains $b_{ij}\in CF^{*}\bigl((L_{i},E_{i}),(L_{j},E_{j})\bigr)$ for $1\leqslant\penalty i<j\leqslant\penalty n$ satisfying an equation related to the bounding cochain equation. In particular, objects $(L,E,b)$ in ${\mathbin{\mathscr{F}}}(M)$ are also objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ .

The translation functor $[1]$ in the triangulated category $D^{b}{\mathbin{\mathscr{F}}}(M)$ acts on objects $(L,E,b)$ by reversing the orientation of $L$ and changing the grading $\theta_{L}$ to $\theta_{L}+\pi$ . The (graded) morphisms of objects $(L,E,b),(L^{\prime},E^{\prime},b^{\prime})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ are $\mathop{\rm Hom}\nolimits^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)=HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ .

The second version $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ , called the idempotent completion, Karoubi completion, or split closure of $D^{b}{\mathbin{\mathscr{F}}}(M)$ , is obtained by applying a further category theory construction to $D^{b}{\mathbin{\mathscr{F}}}(M)$ , which adds direct summands (idempotents) of objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ as extra objects, as in Seidel [64, §4].

Kontsevich’s Homological Mirror Symmetry Conjecture [44], motivated by String Theory, says (very roughly) that if $M,\check{M}$ are ‘mirror’ Calabi–Yau $m$ -folds then there should be an equivalence of triangulated categories

D^{\pi}{\mathbin{\mathscr{F}}}(M)\simeq D^{b}\mathop{\rm coh}(\check{M}).

This has driven much research in the area.

For Mirror Symmetry, one must use $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ rather than $D^{b}{\mathbin{\mathscr{F}}}(M)$ , as the mirror category $D^{b}\mathop{\rm coh}(\check{M})$ is automatically idempotent complete. In §3.1 we will conjecture that in the situation we are interested in, our enlarged version of $D^{b}{\mathbin{\mathscr{F}}}(M)$ should be idempotent complete, so that $D^{\pi}{\mathbin{\mathscr{F}}}(M)\simeq D^{b}{\mathbin{\mathscr{F}}}(M)$ .

2.6 $HF^{*}$ and $D^{b}{\mathbin{\mathscr{F}}}(M)$ for immersed Lagrangians

For the programme of §3, it will be necessary to enlarge the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ of a Calabi–Yau $m$ -fold $(M,J,g,\Omega)$ to include immersed Lagrangians. As a first step in doing this, Akaho and the author [2] explain how to generalize the Lagrangian Floer cohomology of Fukaya, Oh, Ohta and Ono [20] from embedded Lagrangians to immersed Lagrangians with transverse self-intersections. We now explain some of the main ideas in [2].

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and $(L,E)$ a Lagrangian brane in $M$ . As in §2.5, in the embedded case [20], a bounding cochain for $(L,E)$ is a singular $(m\!-\!1)$ -chain $b\in C_{m-1}(L,\Lambda_{\rm nov}^{+})$ (or equivalence class of such chains), satisfying an equation (2.15) involving virtual chains for moduli spaces ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{\rm main}(J,\beta)$ of $J$ -holomorphic discs in $M$ with boundary in $L$ .

In the immersed case [2], if $L$ has transverse self-intersections, a bounding cochain $b$ for $(L,E)$ consists of two pieces of data: a chain $b_{\rm ch}$ in $C_{m-1}(L,\Lambda_{\rm nov}^{+})$ as above, and also, for each point $p\in M$ at which two local sheets $L_{+},L_{-}$ of $L$ intersect transversely with $\mu_{L_{+},L_{-}}(p)=1$ , an element

b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}|_{p},E_{-}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0},

(2.17)

where we write $E_{\pm}$ for the restriction of $E\rightarrow L$ to the local sheets $L_{\pm}$ . These $b_{\rm ch},b_{p}$ must satisfy equations involving virtual chains for moduli spaces of $J$ -holomorphic discs in $M$ with boundary in $L$ , but now these $J$ -holomorphic discs can be polygons with ‘corners’ at self-intersection points of $L$ .

For example, suppose $(L_{1},E_{1}),(L_{2},E_{2})$ are embedded, transversely intersecting Lagrangian branes in $M$ . Then $(L,E)=(L_{1},E_{1})\cup(L_{2},E_{2})$ is an immersed Lagrangian brane in $M$ . A bounding cochain $b$ for $(L,E)$ could consist of $b_{\rm ch}=b_{1}\oplus b_{2}$ , where $b_{i}\in C_{m-1}(L_{i},\Lambda_{\rm nov}^{+})$ for $i=1,2$ are embedded bounding cochains for $(L_{1},E_{1}),(L_{2},E_{2})$ , together with elements $b_{p}$ in (2.17) for $p\in L_{1}\cap L_{2}$ with $\mu_{L_{1},L_{2}}(p)=1$ or $\mu_{L_{2},L_{1}}(p)=1$ which encode how the objects $(L_{1},E_{1},b_{1}),(L_{2},E_{2},b_{2})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ are glued together to make $(L,E,b)$ . For instance, if we have a distinguished triangle in $D^{b}{\mathbin{\mathscr{F}}}(M)$

then the $b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ for $p\in L_{1}\cap L_{2}$ with $\mu_{L_{2},L_{1}}(p)=1$ form a chain in $CF^{1}\bigl((L_{2},E_{2}),(L_{1},E_{1})\bigr)$ representing $\beta$ , and $b_{p}=0$ otherwise.

Note that $\beta$ is represented by $(b_{p})$ with $b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$ , but to define a bounding cochain we require that $b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ . This can be achieved by multiplying $\beta,b_{p}$ by $P^{\lambda}$ for $\lambda\gg 0$ , which does not change $(L,E,b)$ up to isomorphism in $D^{b}{\mathbin{\mathscr{F}}}(M)$ .

Figure 2.3:

J

-holomorphic ‘teardrop’ making immersed

HF^{*}

obstructed

The new cause of obstructions to $HF^{*}$ for immersed Lagrangians $L$ with transverse self-intersections is ‘teardrop-shaped’ $J$ -holomorphic discs $\Sigma$ of the form shown in Figure 2.3, with one corner at $q\in M$ , and with $\mu_{L_{+},L_{-}}(q)=2$ , where $L_{\pm}$ are the local sheets of $L$ intersecting at $q$ . As $\mu_{L_{+},L_{-}}(q)=2$ the moduli space of such discs has virtual dimension 0. Such $\Sigma$ only obstruct $HF^{*}$ if they have ‘small area’ (that is, $\mathop{\rm area}(\Sigma)$ is smaller than the areas of other relevant curves with boundary in $L$ ). Thus we deduce an analogue of Lemma 2.21:

Lemma 2.23.

Suppose $(M,J,g,\Omega)$ is a Calabi–Yau $m$ -fold and $(L,E)$ is an immersed Lagrangian brane in $M$ with only transverse self-intersections. If $b_{m-2}(L)=0$ and $L$ has no self-intersection points $p$ with $\mu_{L^{+},L^{-}}(p)=2$ or $m-2,$ where $L_{\pm}$ are the local sheets of $L$ at $p,$ then $(L,E)$ has $HF^{*}$ unobstructed.

In §2.5 we explained that if $(L^{t},E^{t}):t\in[0,1]$ is a smooth family of embedded Lagrangian branes with the $L^{t}$ Hamiltonian isotopic and the $E^{t}$ locally constant in $t$ , and $b^{0}$ is a bounding cochain for $(L^{0},E^{0})$ , then $b^{0}$ extends to bounding cochains $b^{t}:t\in[0,1]$ for $(L^{t},E^{t})$ by a kind of ‘parallel transport’, and the isomorphism class of $(L^{t},E^{t},b^{t})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ is independent of $t\in[0,1]$ .

In the immersed case, things are more complicated. Firstly, there are two notions of Hamiltonian isotopy. Let $\iota^{t}:L\rightarrow M$ for $t\in[0,1]$ be a smooth family of compact, immersed Lagrangians in $M$ , where we also write $\iota^{t}:L\rightarrow M$ as $L^{t}$ . We call the family globally Hamiltonian isotopic if $\frac{{\rm d}}{{\rm d}t}\iota^{t}$ for $t\in[0,1]$ is Hamiltonian flow by $H^{t}\circ\iota^{t}$ for some smooth $H^{t}:M\rightarrow{\mathbin{\mathbb{R}}}$ . We call the family locally Hamiltonian isotopic if $\frac{{\rm d}}{{\rm d}t}\iota^{t}$ for $t\in[0,1]$ is Hamiltonian flow by some smooth $H^{t}:L\rightarrow{\mathbin{\mathbb{R}}}$ , where there may exist $p_{+},p_{-}\in L$ with $\iota^{t}(p_{+})=\iota^{t}(p_{-})$ but $H^{t}(p_{+})\neq H^{t}(p_{-})$ , so that $H^{t}$ does not descend from $L$ to $M$ .

There is a notion of ‘parallel transport’ for bounding cochains $b^{t}$ along such local Hamiltonian isotopies, but it does not work all the time. Suppose for simplicity that $L^{t}$ has only transverse self-intersections for all $t\in[0,1]$ . Then the self-intersection points of $L^{t}$ in $M$ depend smoothly on $t\in[0,1]$ , so we can write $p^{t}=\iota^{t}(p^{t}_{+})=\iota^{t}(p^{t}_{-})$ for the intersection of local sheets $L^{t}_{+}\ni p^{t}_{+}$ , $L^{t}_{-}\ni p^{t}_{-}$ of $L^{t}$ for $t\in[0,1]$ , where $p^{t}_{\pm},L^{t}_{\pm}$ depend smoothly on $t$ . Then $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})$ is independent of $t$ .

Let $b^{t}$ be a bounding cochain for $(L^{t},E^{t})$ depending smoothly on $t$ , with $(L^{t},E^{t},b^{t})\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ . Then $b^{t}$ evolves in time by a kind of ‘parallel transport’. Let $p^{t},p^{t}_{\pm},L^{t}_{\pm}$ be as above with $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})=1$ . As above, $b^{t}$ includes an element $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ . Since the local systems $E^{t}$ are locally constant in $t$ , we can identify the fibres $E_{+}^{t}|_{p^{t}}$ for $t\in[0,1]$ , and the fibres $E_{-}^{t}|_{p^{t}}$ for $t\in[0,1]$ , and so regard $\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ as being independent of $t$ . Then $b^{t}_{p^{t}}$ is not constant, but evolves by

\frac{{\rm d}}{{\rm d}t}b^{t}_{p^{t}}=\bigl(H^{t}(p^{t}_{+})-H^{t}(p^{t}_{-})\bigr)\cdot\log P\cdot b^{t}_{p^{t}}.

(2.18)

Integrating this over $[0,t]$ gives

b^{t}_{p^{t}}=P^{\textstyle\int_{0}^{t}(H^{t}(p^{s}_{+})-H^{t}(p^{s}_{-}){\rm d}s}\cdot b^{0}_{p^{0}}.

(2.19)

Suppose $b^{0}_{p^{0}}\neq 0$ , and write $b^{0}_{p^{0}}=\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ with $a_{i}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}(E_{+}^{0}|_{p^{0}},E_{-}^{0}|_{p^{0}})$ , $a_{0}\neq 0$ , and $0\leqslant\penalty\lambda_{0}<\lambda_{1}<\lambda_{2}<\cdots$ . Then

b^{t}_{p^{t}}=\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}+\textstyle\int_{0}^{t}\bigl(H^{t}(p^{s}_{+})-H^{t}(p^{s}_{-})\bigr){\rm d}s}.

(2.20)

Thus $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}\subset\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$ , required for $b^{t}$ to be a bounding cochain by (2.17), if and only if

\lambda_{0}+\int_{0}^{t}\bigl(H^{t}(p^{s}_{+})-H^{t}(p^{s}_{-})\bigr){\rm d}s\geqslant 0.

(2.21)

Hence we have the following situation, which will be important in §3.4. Let $(L^{t},E^{t}),$ $t\in[0,1]$ be a local Hamiltonian isotopy of Lagrangian branes in $M$ , and $b^{0}$ a bounding cochain for $(L^{0},E^{0})$ . We may extend $b^{0}$ to a family of bounding cochains $b^{t}:t\in[0,T]$ for $(L^{t},E^{t}),t\in[0,T]$ for some $T\in[0,1]$ , so that $(L^{t},E^{t},b^{t})\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ . But at time $t=T$ we may cross a ‘wall’ when the l.h.s. of (2.21) becomes zero, and we cannot define $b^{t}$ for $t>T$ . Either $(L^{t},E^{t})$ for $t>T$ may have $HF^{*}$ obstructed, or a bounding cochain $\tilde{b}^{t}$ may exist but $(L^{t},E^{t},\tilde{b}^{t})\not\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ .

The Lagrangian $h$ -principle, due to Gromov [23, p. 60-61] and Lees [46], says that two Lagrangians $L,L^{\prime}$ are locally Hamiltonian isotopic in $(M,\omega)$ if and only if they are homotopic in a weak sense, which can be well understood using homotopy theory, and is weaker than isomorphism in $D^{b}{\mathbin{\mathscr{F}}}(M)$ . So we should expect local Hamiltonian isotopies to connect Lagrangians with $HF^{*}$ unobstructed and with $HF^{*}$ obstructed, or to connect non-isomorphic Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M)$ .

Remark 2.24.

As in §2.5, in the embedded case, the Fukaya category ${\mathbin{\mathscr{F}}}(M)$ has objects $(L,E,b)$ for $(L,E)$ an embedded Lagrangian brane and $b$ a bounding cochain, but the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ has objects twisted complexes, consisting of objects $(L_{1},E_{1},b_{1}),\ldots,(L_{n},E_{n},b_{n})$ in ${\mathbin{\mathscr{F}}}(M)$ together with $b_{ij}\in CF^{*}\bigl((L_{i},E_{i}),(L_{j},E_{j})\bigr)$ for $1\leqslant\penalty i<j\leqslant\penalty n$ satisfying an equation.

In the immersed case, we can regard such a twisted complex as a single object $(L,E,b)$ in ${\mathbin{\mathscr{F}}}(M)$ , where $L$ is the disjoint union $L_{1}\amalg\cdots\amalg L_{n}$ , considered as a single immersed Lagrangian, $E|_{L_{i}}=E_{i}$ , and $b$ is a bounding cochain for $(L,E)$ built from $b_{1},\ldots,b_{n}$ and $b_{ij}$ for $i<j$ . Thus there is no need to add twisted complexes, and we can suppose all objects of $D^{b}{\mathbin{\mathscr{F}}}(M)$ are of the form $(L,E,b)$ .

The idempotent completion $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ of $D^{b}{\mathbin{\mathscr{F}}}(M)$ as in §2.5 could still include objects which are direct summands of some $(L,E,b)$ , but do not have a good geometric interpretation. However, in §3.1 we will conjecture that in the situation we are interested in, $D^{b}{\mathbin{\mathscr{F}}}(M)$ is already idempotent complete, so that we can take all objects of $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ to be of the form $(L,E,b)$ .