2.2 Principal evidence of Thomas-Yau [047R]

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2.2 Principal evidence of Thomas-Yau

Thomas and Yau offered a number of arguments in support of their picture. We now describe the evidence from first principles, and will later return to the evidence from analogies.

The following elementary observation is a first indication about how special Lagrangian objects resemble stable objects of a Bridgeland stability condition:

Proposition 2.4.

[66, section 5.2] Let $L$ , $L^{\prime}$ be unobstructed Lagrangian branes whose supports are compact special Lagrangian manifolds, with Lagrangian angles $\theta_{L}>\theta_{L^{\prime}}$ . Then $HF^{k}(L,L^{\prime})=0$ for $k\leq 0$ .

Proof.

After $C^{\infty}$ -small Hamiltonian perturbation we may assume any intersection point $p$ between $L$ and $L^{\prime}$ is transverse. We can write the tangent planes of $L,L^{\prime}$ inside $T_{p}X\simeq\mathbb{C}^{n}$ in the standard form

T_{p}L=\mathbb{R}^{n}\subset\mathbb{C}^{n},\quad T_{p}L^{\prime}=(e^{i\phi_{1}},\ldots e^{i\phi_{n}})\mathbb{R}^{n}\subset\mathbb{C}^{n},\quad 0<\phi_{i}<\pi.

The Floer degree of the intersection point $p$ is

\mu_{L,L^{\prime}}(p)=\frac{1}{\pi}(\theta_{L}-\theta_{L^{\prime}}+\sum_{1}^{n}\phi_{i})>0,

whence $HF^{k}(L,L^{\prime})=0$ for $k\leq 0$ . ∎

Remark 2.5.

As is clear from the proof, the special Lagrangian condition can be relaxed to $\inf_{L}\theta_{L}>\sup_{L^{\prime}}\theta_{L^{\prime}}$ .

Remark 2.6.

Our Floer degree convention follows Joyce [41], but is opposite to that of Seidel [69] and many other symplectic geometry texts. In our convention, shifting the Lagrangian phase by $k\pi$ for $k\in\mathbb{Z}$ is equivalent to considering the shifted object $L[k]$ . Thus a slight extension of the above is that if $\theta_{L}>\theta_{L^{\prime}}+l$ , then

HF^{k}(L,L^{\prime})=HF^{k-l}(L,L^{\prime}[l])=0,\quad k\leq l.

The most compelling evidence due to Thomas and Yau is

Theorem 2.5.

(Thomas-Yau uniqueness) [66] Let $L,L^{\prime}$ be unobstructed Lagrangian branes supported on embedded special Lagrangians of the same phase, which define isomorphic objects in $D^{b}Fuk(X)$ , then their supports coincide.

Remark 2.7.

The most general Thomas-Yau uniqueness, which includes immersed Lagrangians, seems to be due to Imagi [39]. The original Thomas-Yau theorem is phrased in terms of uniqueness in the Hamiltonian isotopy class, even though it can be cast in more general categorical terms. The categorical perspective is preferred, because it is closer to the spirit of homological mirror symmetry, and because one derived Fukaya category class may contain several Hamiltonian isotopy classes. If immersed Lagrangians are allowed, then isomorphism in $D^{b}Fuk(X)$ would also identify certain embedded Lagrangian objects with immersed objects of different topologies. When this happens, an interesting corollary of Thomas-Yau uniqueness is that at most one of these Hamiltonian classes contains special Lagrangian branes.

Proof.

(Sketch) Assume the suppports do not coincide. After small Hamiltonian perturbations, we can ensure the perturbed Lagrangians $\tilde{L},\tilde{L}^{\prime}$ have transverse intersections, and still define the same isomorphic objects in $D^{b}Fuk(X)$ . Under the special Lagrangian assumption, through judicious choice of the Hamiltonian via Morse theory as in Thomas-Yau [66, Thm 4.3], or by using genericity arguments based on real analyticity as in Joyce-Imagi-Santos [40, section 4.3], one can ensure there is no intersection point $\tilde{L}\cap\tilde{L}^{\prime}$ of degree $0,n$ , so in particular $HF^{0}(L,L^{\prime})\simeq HF^{0}(\tilde{L},\tilde{L}^{\prime})=0$ . However, this implies the cohomological unit of $HF^{*}(L,L)$ is zero, so the Floer cohomology ring of $L$ is zero, namely $L$ is a zero object in $D^{b}Fuk(X)$ , contradiction. ∎

The Thomas-Yau argument reveals the relevance of the Fukaya category to special Lagrangian geometry. The role of holomorphic curves, which are a central ingredient in the Fukaya category, is however rather opaque in this argument; their only appearance is to make the Floer cohomology defined.