SYZ mirror of Hirzebruch surface $\mathbb{F}_k$ and Morse homotopy

Technical Summary: SYZ Mirror of Hirzebruch Surfaces $F_k$ and Morse Homotopy

Problem Statement
This paper addresses the problem of establishing Homological Mirror Symmetry (HMS) for Hirzebruch surfaces $F_k$ as complex manifolds. While HMS for toric Fano surfaces (specifically $F_1$) was previously established by Futaki, Kajiura, and the author using Morse homotopy, the extension to non-Fano cases ($F_k$ for $k \geq 2$) presents significant challenges. Unlike the $F_1$ case, the dual coordinates of the Strominger-Yau-Zaslow (SYZ) fibrations for $F_k$ cannot be expressed explicitly in a manner that allows for direct computation of Lagrangian section intersections using moment polytope coordinates. Furthermore, the behavior of the Morse homotopy category differs between the Fano case ($k=1$) and the non-Fano cases ($k \geq 2$), particularly regarding the minimality of the category.

Methodology
The author employs the SYZ construction of mirror pairs via dual torus fibrations and utilizes the framework of Morse homotopy on the moment polytope, following the approach of Futaki and Kajiura [9, 10, 16].

1. SYZ Setup: The paper considers the Hirzebruch surface $F_k$ as a toric manifold equipped with a Kähler metric. It constructs the dual torus fibration $M \to B$ over the interior of the moment polytope $P$. The mirror manifold $\check{M}$ is identified with the torus bundle over $B$.
2. Complex Side (DG Category): The complex side is modeled by the derived category of coherent sheaves $Db(\text{Coh}(F_k))$. The author focuses on a full strongly exceptional collection of line bundles, denoted as $Ec = (\mathcal{O}, \mathcal{O}(1,0), \mathcal{O}(c,1), \mathcal{O}(c+1,1))$, which generates the derived category. A DG category $DG(F_k)$ is defined consisting of these holomorphic line bundles with specific connections.
3. Symplectic Side (Morse Homotopy): The symplectic side is modeled by the category $Mo(P)$ of weighted Morse homotopy. Objects in this category are Lagrangian sections of the dual torus fibration $M \to B$. The author constructs Lagrangian sections $L(a,b)$ corresponding to the line bundles in the exceptional collection.
4. Intersection Analysis: A critical technical step involves computing the intersections of these Lagrangian sections. Since explicit dual coordinates are difficult to handle for general $k$, the author computes intersections with respect to dual coordinates (as done in previous work [16]) and investigates the connected components of these intersections within the moment polytope $P$.
5. Morphisms and Differentials: The morphism spaces in $Mo(P)$ are spanned by connected components of the intersection of Lagrangian sections that satisfy specific stability manifold conditions (M1 and M2). The differential $m_1$ and composition $m_2$ are defined via gradient trees and their symplectic areas.

Key Contributions and Results

• Extension to Non-Fano Cases: The primary result is the proof that Homological Mirror Symmetry holds for Hirzebruch surfaces $F_k$ for any positive integer $k$. This extends the previous result for $F_1$ to the non-Fano regime (where $F_k$ is weak Fano for $k=2$ and not even weak Fano for $k \geq 3$).
• Equivalence of Categories: The paper establishes a DG-quasi-isomorphism between the full subcategory of the Morse homotopy category, $Mo_{Ec}(P)$, and the full subcategory of the DG category of holomorphic line bundles, $V'_{Ec}$ (which is quasi-isomorphic to $DGEc(F_k)$).
  $$Mo_{Ec}(P) \simeq V'_{Ec} \simeq DGEc(F_k)$$
  Consequently, the triangulated categories induced by these DG-categories are equivalent:
  $$Tr(Mo_{Ec}(P)) \simeq Db(\text{Coh}(F_k))$$
• Analysis of Minimality: The paper investigates the minimality of the Morse homotopy category.
  • For the full strongly exceptional collection $Ec$, the subcategory $Mo_{Ec}(P)$ is minimal (higher products $m_k$ for $k \geq 3$ vanish, and the differential is trivial on the generators of this specific subcollection).
  • However, the full category $Mo(P)$ is not minimal for $k \geq 2$. The author provides a concrete counterexample for $F_2$ (specifically the space of morphisms between $L(0,0)$ and $L(-7,3)$). In this case, the intersection of Lagrangian sections is non-connected, and a gradient trajectory exists between distinct connected components, resulting in a non-trivial differential $m_1$. This contrasts with the $F_1$ case, where $Mo(P)$ is minimal.
• Explicit Correspondence: The author constructs an explicit correspondence $\iota$ between the generators of the Morse homotopy category (connected components of intersections) and the basis elements of the cohomology of the complex side (holomorphic sections), showing compatibility with composition.

Significance
The paper claims significance in extending the SYZ-Morse homotopy approach to a broader class of toric surfaces beyond the Fano case. By demonstrating that HMS holds for $F_k$ ($k \geq 2$) using the moment polytope of the toric manifold itself (rather than a polytope derived from a tropical amoeba as in Abouzaid's approach [1]), the work validates the robustness of the Futaki-Kajiura framework. It highlights that while the full Morse homotopy category may exhibit non-minimal behavior in non-Fano cases, restricting to a full strongly exceptional collection recovers the necessary equivalence to the derived category of coherent sheaves. This provides a concrete example of homological mirror symmetry for non-Fano toric surfaces.