Learning Seminar: SYZ Mirror Symmetry

Introduction

After Kontsevich's famous proposal on Homological Mirror Symmetry [Kon94], the question of geometric construction of mirrors to a given variety has been an important question for mathematicians. One important proposal was made by Strominger-Yau-Zaslow [SYZ96], which stated that mirrors to a given Kähler manifold $X$ equipped with a Lagrangian torus fibration $X\to B$ should be a Kähler manifold $\check{X}\to B$ with a Lagrangian torus fibration over the same base $B$, and with "dual torus fiber".

This proposal has been successful firstly for elliptic curves whose corresponding Homological Mirror Symmetry was (partly) verified by Polishchuk-Zaslow [PZ98]. People quickly found finding a smooth Lagrangian torus fibration for any Kähler manifold $X$ is almost impossible: there can be toric singularities where the fibration has lower dimensional toric fibers, and there can be singular fibers. Cho and Oh [CO06] studied the effects of toric singularities for toric Fano varieties and showed that existence of toric singularities gives Landau-Ginzburg superpotentials $W$ of the mirror, via counting intersection numbers of holomorphic disks with the toric singularities. For toric varieties which admits canonical Lagrangian torus fibrations with only toric singularities, the corresponding Homological Mirror Symmetry statement was then proved by Fang-Liu-Treumann-Zaslow [FLTZ11] for smooth projective toric varieties and further generalized by Kuwagaki [Kuw20] to all toric stacks.

The simplest and the most generic case of singular fibers are non-degenerate singularities, similar to singular fibers of Lefschetz fibration, obtained by contracting a codimension $1$ subtorus to a point. This situation was studied by Auroux [Aur07] in the case when $X=\mathbb{C}^2\setminus\{xy=1\}$ and the Lagrangian torus fibration was constructed out of the Lefschetz fibration $\pi (x,y)=xy$.

In this case one can clearly see a wall-crossing phenomenon, where there's a single singular value for the torus fibration $X\to\mathbb{R}^2$ coming from singularities of the Lefschetz fibration, $\{xy=0\}\subseteq\mathbb{C}^2$, which corresponds to the wall $\{x=0\}\times\mathbb{R}\subseteq\mathbb{R}^2$. Auroux [Aur07] showed that for any $b=(b_1,b_2)$ with $b_1>1$ or $b_1<1$, disk countings computing mirror superpotential to $X\subseteq\mathbb{P}^2$ are different, so that the two moduli spaces of Lagrangian tori are glued together with non-trivial transition functions.

Although the phenomena seems mysterious, Fukaya proposed a possible explanation in [Fuk09] where we refine the previous picture by considering the Floer-theoretic deformation space of singular Lagrangian tori $L$ where the local systems play the role of deforming the Floer $A_{\infty}$-structure on $CF^{\ast} (L,L)$. For singular Lagrangian submanifolds $L\subseteq X$ the $A_{\infty}$-structure on $CF^{\ast} (L,L)$ is usually curved, meaning there exists a curvature term $\mu^0$ that prevents the identity $(\mu^1)^2=0$. However, there might exist an element $b\in H^1 (L,\lambda_0)$ where $\Lambda_0$ is the Novikov ring, such that the deformation $\mu_b$ satisfies $(\mu_b^1)^2=0$. The element $b$ must satisfy the Maurer-Cartan equation$$\sum_k\mathfrak{m}^k (b,b,\dotsb, b)=0,$$so we call elements satisfying this equation a Maurer-Cartan element, and use $\mathcal{MC} (L)$ for the space of all Maurer-Cartan elements (equivalently, the infinitesimal deformation space of $L$).

As for any choice of Maurer-Cartan element $b\in\mathcal{MC} (L)$, the pair $(L,b)$ defines an object in the Fukaya category of $X$, Fukaya proposed that the mirror to $X$ should be the quotient of a union $$\mathcal{MC} (\mathfrak{L}):=\bigcup_{u\in B}\mathcal{MC} (L_u)\big/H^1 (L_u;2\pi\sqrt{-1}\mathbb{Z})$$for a given family of Lagrangian submanifolds $\mathfrak{L}:=\{L_u\}_{u\in B}$ where $B$ is some smooth manifold with the same dimension as $H^1 (L;\mathbb{R})$. Moreover, for any given object $(L',b')\in\mathcal{F} (X)$, the Yoneda functor $$(u,b)\mapsto CF^{\ast}((L',b'),(L_u,b);\Lambda)$$should define a complex of coherent sheaves over $\mathcal{MC} (\mathfrak{L})$.

The main goal of this learning seminar is to understand the current state of research concerning SYZ mirror constructions, in particular the work of Abouzaid [Abo21] on SYZ and Homological Mirror Symmetry for smooth Lagrangian torus fibrations (and hence for all abelian varieties) and Abouzaid-Auroux-Katzarkov [AEK16] on blow-ups of toric varieties along hypersurfaces near tropical limits.

Plan of the Seminar.

References

[Abo14] Abouzaid, M. (2014). Family Floer cohomology and mirror symmetry. Proceedings of the International Congress of Mathematicians---Seoul 2014. Vol. II, 813–836. Retrieved from http://www.ams.org/mathscinet-getitem?mr=3728639

[Abo21] Abouzaid, M. (2021). Homological mirror symmetry without correction. Journal of the American Mathematical Society, 34(4), 1059–1173. doi:10.1090/jams/973

[AEK16] Abouzaid, M., Auroux, D., & Katzarkov, L. (2016). Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publications Mathématiques. Institut de Hautes Études Scientifiques, 123, 199–282. doi:10.1007/s10240-016-0081-9

[AS21] Abouzaid, M., & Sylvan, Z. (2021). Homological Mirror Symmetry for local SYZ singularities. arXiv [Math.SG]. Retrieved from http://arxiv.org/abs/2107.05068

[Aur07] Auroux, D. (2007). Mirror symmetry and T-duality in the complement of an anticanonical divisor. Journal of Gökova Geometry Topology. GGT, 1, 51–91. Retrieved from http://www.ams.org/mathscinet-getitem?mr=2386535

[Aur25] Auroux, D. (2025). Holomorphic discs of negative Maslov index and extended deformations in mirror symmetry. arXiv [Math.SG]. Retrieved from http://arxiv.org/abs/2309.13010

[CO06] Cho, C.-H., & Oh, Y.-G. (2006). Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math., 10(4), 773–814. doi: 10.4310/AJM.2006.v10.n4.a10

[Eva23] Evans, J. (2023). Lectures on Lagrangian torus fibrations (p. xiii+225). doi:10.1017/9781009372671

[Fuk09] Fukaya, K., (2009). Lagrangian surgery and Rigid analytic family of Floer homologies. https://www.math.kyoto-u.ac.jp/~fukaya/Berkeley.pdf

[FLTZ11] Fang, B., Liu, C.-C. M., Treumann, D., & Zaslow, E. (2011). A categorification of Morelli’s theorem. Invent. Math., 186(1), 79–114. doi: 10.1007/s00222-011-0315-x

[Kon94] Kontsevich, M. (1994). Homological Algebra of Mirror Symmetry. arXiv, alg-geom/9411018. Retrieved from https://arxiv.org/abs/alg-geom/9411018v1.

[Kuw20] Kuwagaki, T. (2020). The nonequivariant coherent-constructible correspondence for toric stacks. Duke Math. J., 169(11), 2125–2197. doi: 10.1215/00127094-2020-0011

[LZ20] Lau, S.-C., & Zheng, X. (2020). SYZ mirror symmetry for hypertoric varieties. Communications in Mathematical Physics, 373(3), 1133–1166. doi:10.1007/s00220-019-03535-z

[Jin25] Jing, H. (2025). On SYZ mirrors of Hirzebruch surfaces. arXiv [Math.SG]. Retrieved from http://arxiv.org/abs/2504.01889

[PZ98] Polishchuk, A., & Zaslow, E. (1998). Categorical mirror symmetry: The elliptic curve. Adv. Theor. Math. Phys., 2(2), 443–470. doi: 10.4310/ATMP.1998.v2.n2.a9

[Sym03] Symington, M. (2003). Four dimensions from two in symplectic topology. In Proc. Sympos. Pure Math.: Vol. 71. Topology and geometry of manifolds (Athens, GA, 2001) (pp. 153–208). doi:10.1090/pspum/071/2024634

[SYZ96] Strominger, A., Yau, S. T., & Zaslow, E. (1996). Mirror symmetry is T-duality. Nuclear Physics B, 479(1-2), 243-259.