Bridgeland stability, Joyce’s proposal [047J]

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Bridgeland stability, Joyce’s proposal

To trace the subsequent development, and move beyond the almost calibrated setting, we need to recall an important piece of homological algebra known as Bridgeland stability conditions on a triangulated category [13].

Definition 2.3.

A Bridgeland stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$ consists of a group homomorphism (‘central charge’) $Z$ from the (numerical) Grothendieck group $K(\mathcal{D})$ to $\mathbb{C}$ , and full additive subcategories $\mathcal{P}(\phi)\subset\mathcal{D}$ for each $\phi\in\mathbb{R}$ , whose objects are called ‘semistable objects of phase angle $\pi\phi$ ’,⁸⁸ 8 In the convention of Bridgeland, the phase is $\phi$ . We have instead opted to call $\pi\phi$ the phase, which is more naturally identified with the phase angle of special Lagrangians. satisfying the following axioms:

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(Phase) If $L\in\mathcal{P}(\phi)$ then $Z(L)=m(L)\exp(i\pi\phi)$ for some $m(L)>0$ ,
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(Shift) For all $\phi\in\mathbb{R}$ , $\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]$ ,
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(Monotonicity) If $\phi_{1}>\phi_{2}$ and $L_{j}\in\mathcal{P}(\phi_{j})$ , $j=1,2$ then $Hom_{\mathcal{D}}(L_{1},L_{2})=0$ ,
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For each nonzero object $L\in\mathcal{D}$ there are a finite sequence of real numbers $\phi_{1}>\phi_{2}>\ldots>\phi_{N}$ and a Harder-Narasimhan decomposition

$0=\mathcal{E}_{0}\to\mathcal{E}_{1}\to\ldots\to\mathcal{E}_{N}=L,$ (4)

with distinguished triangles

$\mathcal{E}_{i-1}\to\mathcal{E}_{i}\to L_{i}\to\mathcal{E}_{i-1}[1]$

such that $L_{j}\in\mathcal{P}(\phi_{j})$ .
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(Calibration) For any fixed norm on the finite dimensional vector space $K(\mathcal{D})\otimes_{\mathbb{Z}}\mathbb{R}$ , we have a uniform constant $C$ , such that any semistable object $L$ satisfies $\left\lVert L\right\rVert\leq C|Z(L)|.$

Remark 2.2.

For the heuristic but naïve geometric meaning, one can imagine that $\mathcal{D}$ is the derived Fukaya category, $K(\mathcal{D})$ is the sublattice of $H_{n}(X)$ generated by the Lagrangians, the central charge is $Z(L)=\int_{L}\Omega$ , the subcategory $\mathcal{P}(\phi)$ is generated by the special Lagrangians of constant phase angle $\theta=\pi\phi$ , the Harder-Narasimhan decomposition means a multiple Lagrangian connected sum with decreasing phase angles

L\simeq L_{1}\#L_{2}\#\ldots\#L_{N},

and the calibration property comes from the fact that the total mass of a special Lagrangian is equal to $|Z(L)|$ , and the mass bounds any norm of $[L]\in H_{n}(X)$ using Poincaré duality, assuming $H_{n}(X)$ is finite dimensional.

Remark 2.3.

We write the subcategory generated by all the $\mathcal{P}(\phi)$ within the interval $\phi\in(\phi_{0},\phi_{1}]$ as $\mathcal{P}(\phi_{0}<\phi\leq\phi_{1})$ . The important property is that $\mathcal{P}(\phi_{0}<\phi\leq\phi_{0}+1)$ is an abelian category, known as the heart of a bounded $t$ -structure. The advantage is that while in general triangulated categories only distinguished triangles make sense, for abelian categories we can talk about exact sequences, subobjects and quotients.

With the hindsight of Bridgeland stability condition, and more than one decade of progress on the mean curvature flow as well as symplectic geometry, Joyce [41] produced a major update of the Thomas-Yau picture. We shall discuss more about the LMCF considerations in section 4.1, but it suffices here to give away its main punchline (cf. [41, Conj 3.34]):

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Let $(X,\omega,\Omega)$ be a Calabi-Yau manifold ⁹⁹ 9 The complex Monge-Ampère equation is convenient but not indispensable, see section 4.1., either compact or Stein. There should be an enlarged version of the derived Fukaya category $D^{b}Fuk(X)$ , including classes of immersed or mildly singular Lagrangians, and a Bridgeland stability condition $(Z,\mathcal{P})$ on $D^{b}Fuk(X)$ , whose central charge is

$Z(L)=\int_{L}\Omega.$ (5)
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Given a Lagrangian brane $L$ (with grading, orientation, relative spin structure, local system and bounding cochain data) such that the Floer cohomology is unobstructed, and suppose $L$ is generic in its Hamiltonian isotopy class. Then the LMCF $(L_{t})_{t>0}$ (with brane structure) starting from $L$ exists for all time with surgeries at a finite series of singular times. The nature of these surgeries have conjectural descriptions. The Lagrangian can change its Hamiltonian class at these surgeries, but maintains its derived category class. At $t\to\infty$ , the Lagrangians $L_{t}$ converges in the geometric measure theory sense to a union of graded special Lagrangian currents $L_{1},\ldots L_{N}$ of phase angle $\hat{\theta}_{1}>\hat{\theta}_{2}>\ldots>\hat{\theta}_{N}$ with multiplicities counted:

$\lim_{t\to\infty}L_{t}=L_{1}+\ldots L_{N}.$ (6)
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If there is only one constant phase angle $\hat{\theta}$ appearing in the infinite time limit, then $L$ defines a semistable object of phase $\hat{\theta}$ in $D^{b}Fuk(X)$ . Otherwise $L$ is not semistable with respect to any phase angle, and the infinite time limit supposedly give rise to the Harder-Narasimhan decomposition.

Joyce’s program is much more ambitious in the sense that it attempts to capture the entire triangulated category $D^{b}Fuk(X)$ . The original Thomas-Yau proposal, which is concerned only with almost calibrated Lagrangians, fits into the Joyce picture as an abelian subcategory $\mathcal{P}(-\pi/2<\theta\leq\pi/2)\subset D^{b}Fuk(X)$ ,¹⁰¹⁰ 10 Here we ignore the ‘small’ difference between $<$ and $\leq$ . This would be justified if $\Omega$ is suitably generic so that there is no special Lagrangian of phase $\pi/2$ , namely that the countably many numbers $\text{Arg}\int_{L}\Omega$ for $[L]\in H_{n}(X,\mathbb{Z})$ miss the number $\pi/2$ . the heart of a certain $t$ -structure. The main appeal of Joyce’s mean curvature flow perspective, is that if the infinite time limiting currents $L_{1},\ldots L_{N}$ can be given suitable brane structures to be admitted as objects of $D^{b}Fuk(X)$ , then the program gives a conjectural dynamical mechanism to explain the Harder-Narasimhan decomposition, which is the most nontrivial aspect of the Bridgeland stability. Its principal drawback is that Joyce offers no a priori information on the Bridgeland stability beyond the central charge, other than letting off the mean curvature flow to find its own destiny. This limits its practical applicability, for instance to existence questions of special Lagrangians in prescribed classes in $D^{b}Fuk(X)$ . See section 3.6 for more discussions.