
## 1. Problem list

1. ### Finite-dimensional interpretation of Bridgeland stability

#### Problem 1.05.

Find or describe a finite-dimensional analogue of the dHYM equation and its relation to stability, similar to how finite-dimensional geometric invariant theory is formally similar to traditional Hermitian Yang–Mills and slope stability, which may be seen as an infinite-dimensional Kempf–Ness theorem. Furthermore, go back up to the mirror side and investigate a statement of the special Lagrangian problem or Thomas–Yau in the language of this finite-dimensional analogue. Investigate under what conditions the Kähler metric in the finite-dimensional analogue is definite, and see if this provides any insights into the necessary positivity conditions arising from subvarieties in the study of dHYM.
• ### Dimensional reduction of deformed Hermitian Yang–Mills

#### Problem 1.1.

Understand the solvability of the dHYM equation on higher rank bundles in any examples. In particular:
• On complex surfaces with (special) Lagrangian torus fibrations and semi-flat metrics, such as elliptically fibered K3 surfaces.
• By dimensionally reducing from 4 (real) dimensions to two dimensions using similar techniques to Hitchin for Yang–Mills to Higgs bundles. For example even on $\mathbb{R}^4$.
• By dimensionally reducing from 4 to 3 or 4 to 1 or 4 to 0 dimensions, similarly to monopoloes, Nahm’s equations, and ADHM respectively.
In particular to answer the following questions in any of the above cases:
• What does the equation dimensionally reduce to, even locally?
• Existence of solutions.
• Classification of solutions.
• ### Toric mirror symmetry and asymptotically admissible Lagrangians

#### Problem 1.15.

Provide an algebro-geometric or categorical interpretation of the central charge-type term $$\int_{L_\mathcal{X}^\delta} \tilde \Omega$$ appearing in the study of deformations of special Lagrangians in toric mirror symmetry for mirrors to pushforwards of $\mathcal{O}(-\delta E)$ coming from birational models $\mathcal{X}$ of $X\times \mathbb{P}^1$.
• How does this term change under deformation of the special Lagrangian to an asymptotically admissible Lagrangian? Can the correction term be understood or interpreted.
• Is there a canonical deformation of $L_\mathcal{X}^\delta$ to an asymptotically admissible Lagrangian?
• Is there a generalisation of the construction of $L_\mathcal{X}^\delta$ from $L$ for an arbitrary SYZ fibration using similar techniques of mirrors to pushfowards of exceptional divisors on birational models.
• Is there a notion of stability with generalises or is an alternative to Bridgeland stability which corresponds to special Lagrangians and accounts for the example of Collins–Shi on $Bl_p \mathbb{CP}^2$.
• ### Lagrangian mean curvature flow for Kähler–Einstein manifolds with definite first Chern class.

#### Problem 1.2.

What does Lmcf look like for Kähler–Einstein manifolds with definite first Chern class. In particular:
• What does Lmcf look like for $c_1(X)>0$. For example Evans work on $\mathbb{CP}^2$ shows possible oscillating behaviour for Lmcf where a (non minimal) Clifford torus can flow to a Chekanov torus and back to Clifford any number of times before eventually flowing down to the minimal Clifford torus. What happens for example in the case of $\mathbb{CP}^n$?
• What does Lmcf look like for $c_1(X)<0$? Is it better behaved than the $c_1(X)=0$ case? For example consider a graph $$\Gamma \subset \Sigma_g \times \Sigma_g$$ of some symplectomorphism $f: \Sigma_g \to \Sigma_g$ for higher genus Riemann surfaces $\Sigma_g, g>1$. See work of Smoczyk for the special almost-calibrated case.
Furthermore:
• Is there a notion of a Bridgeland-type stability condition for $\mathbb{Z}/2$-graded Fukaya categories like those appearing in mirror symmetry for Fanos? Can this be understood using the differential geometry of the mirror similarly to dHYM?
• What happens for embedded curve-shortening flow for $\mathbb{P}^1$ and its mirror. In particular what are the mirror objects in the Landau–Ginzburg model to minimal Lagrangians (the Fano/general type analogues of dHYM as it relates to special Lagrangians on Calabi–Yaus).
• ### Lagrangian mean curvature flow with dilations

#### Problem 1.25.

What happens under Lmcf to Lagrangians in, for example, $\mathbb{R}^{2n} \cong T^* \mathbb{R}^n$ where one also allows dilations of the fibres (see work of Neves who shows that singularities are generic). In particular if you squish the initial Lagrangian very close to the zero section before running Lmcf are singularities still generic? What about if you allow dynamic dilations as you progress along the flow? Related, is there some natural geometric flow which produces the Eliashberg–Polterovich isotopy?
• ### Definition of dHYM in higher rank for complexes of vector bundles

#### Problem 1.3.

What is the right definition of the higher rank dHYM equation for complexes of vector bundles? Such a definition would need to satisfy certain desirable properties:
• It should be mirror to the special Lagrangian condition, possibly for multisection special Lagrangians in a semi-flat SYZ model.
• Existence of solutions to the equation must be invariant under quasi-isomorphism of complexes.
Can one develop even a notion of Hermitian Yang–Mills metrics on a complex of vector bundles. How does this differ from previous work on HYM for quiver bundles?
• ### Deriving higher rank dHYM from Lagrangian multisections using mirror symmetry

#### Problem 1.35.

How can one derive mathematically the higher rank dHYM equation from Lagrangian multisections using mirror symmetry. For example consider the case $T\mathbb{CP}^2$ relating to a multisection with one ramification point.
• ### Thomas–Yau–Joyce conjecture for the flat 2-torus

#### Problem 1.4.

Prove a version of the Thomas–Yau–Joyce conjecture for curves in the flat 2-torus, or just on $\mathbb{R}^2$. Technicalities that occur include:
• One must restrict to the immersed case, as the embedded case is already understood.
• Finite time singularities will occur, due to for example figure-eights.
• It is necessary to assume genericity of singularities, including shrinkers, translators (“grim reapers"). For each type one needs to perform a surgery where weak unobstructedness is preserved.
• ### Geometric structures on the space of complexified Kahler classes and of stability conditions

#### Problem 1.45.

What are natural Weil–Peterson-type metrics on moduli of complexified Kähler classes, and what can be said about the geometry of the moduli space using them. For example what are geodesics? (This requires a search of the Physics and Bridgeland stability literature beforehand, where such questions have been investigated) Similarly what can be said about the Bridgeland stability moduli space $Stab(X)$ and its geometry (see recent work of Bridgeland on metrics on $Stab(X))$. For example consider the case of a blow up of $\mathbb{CP}^n$.
• ### Examples of Joyce–Lee–Tsui translators in explicit cases

#### Problem 1.5.

Study whether an almost-calibrated translator arises in Lmcf, for example in the simplest case ($\mathbb{C}^2$). The grim reaper translator is not allowed as this is not strictly almost-calibrated (having phase change exactly $\pi$). The Joyce program predicts the existence of such translators in the immersed case in dimension larger than 1. For example, does the Joyce–Lee–Tsui translator occur in some example as some immersed Lagrangian. This should be related to J-holomorphic teardrops. In particular can one find a suitable structure at infinity to glue to such a translator to produce a well-defined immersed Lagrangian.
• ### Study mirror symmetry in model Calabi-Yau metrics arising out of real Monge–Ampere equations near large volume limit.

#### Problem 1.55.

Study the existence and flows for dHYM and special Lagrangians in semi-flat SYZ model Calabi–Yau metrics arising from real Monge–Ampere equations near the large volume limit.
• ### Finite time singularities in dHYM flows and graphical Lagrangian mean curvature flow.

#### Problem 1.6.

Study the possibility of finite-time singularities for dHYM flows (any which seem applicable to the problem, or new flows). Note that finite-time singularity formation does occur in Lmcf which should be mirror to dHYM flow, but that dHYM arises as mirror to the study of Lagrangian sections and currently it is not known whether finite-time singularity formation occurs for such Lagrangian graphs.
• ### Subsolutions of higher rank dHYM and Z-critical equation

#### Problem 1.65.

Is a solution to the higher rank dHYM equation necessarily a subsolution in the sense of Dervan–McCarthy–Sektnan. The analogous condition for dHYM in rank 1 is an equivalence by work of Gao Chen, and it is easily seen that in higher rank not every subsolution is a solution, but the converse implication is not known. It is conjectured that the subsolution condition should be equivalent to an algebro-geometric stability condition coming from subvarieties (just as occurs for rank 1 or for the J-equation). Can such an algebro-geometric stability condition be written down, perhaps assuming a higher rank analogue of the super or hypercritical phase condition? Since it is conjectured that solutions correspond to stable objects, every solution should therefore satisfy the subsolution condition. In particular this would imply the ellipticity of the higher rank dHYM equation at a solution as well as the positivity of the natural Kähler metric on the space of connections which turns the dHYM equation into a moment map problem.
• ### Generating Lagrangians in the Fukaya category of P^2 relative to an elliptic curve

#### Problem 1.7.

Do special Lagrangians generate the Fukaya–Seidel category $\mathrm{FS}(W)$ for a superpotential $W$ mirror to $\mathbb{CP}^2$ relative to an elliptic curve $E$, $\mathbb{CP}^2/E$.
• ### Chern number inequalities in higher rank dHYM

#### Problem 1.75.

Does existence of solutions to the higher rank dHYM equation imply any interesting Chern number inequalities, for example a generalised Bogomolov–Gieseker inequality. For example consider the case of rank 2 on a surface or threefold. The corresponding result is known for dHYM on line bundles in dimensions up to 4. The analogous result for HYM follows from a straightforward local calculation using the HYM condition; can this argument be adjusted to the dHYM equation, perhaps in the large volume limit where the HYM term dominates? Is there an interpretation of these Chern inequalities in terms of bounds on some functional when solutions exist, allowing general arguments which may apply to existence of dHYM on line bundles in dimension larger than 4, or to higher rank dHYM simultaneously? Current methods are largely adhoc based on the local expression for dHYM on line bundles which are known (and which do not easily generalise to the higher rank dHYM equation).
• ### Isotopy to holomorphic Landau–Ginzburg model

#### Problem 1.8.

Suppose $W:X\to \mathbb{C}$ is a symplectic Landau–Ginzburg model. That is, there exists a compact $K\subset \mathbb{C}$ such that on $X\backslash W^{-1}(K)$ the fibers are symplectic, and symplectic parallel transport is well-defined. Are there simple obstructions to $W$ being isotopic to a holomorphic Landau–Ginzburg model? This is mirror to asking when the B-model is Kähler/projective, so for instance, one wants a mirror obstruction to $h^2, h^{1,1} \ne 0$. Is there an interpretation in terms of (non-)existence of a stability condition?

Cite this as: AimPL: Stability in mirror symmetry, available at http://aimpl.org/stabmirror.