Lagrangian multi-sections and their toric equivariant mirror

Abstract

The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called $N$-generic condition with $N\geq 3$. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section.

Figures (7)

• Figure 1: The linking disk $D_{m,-\xi_0}$ agrees with $F_{-\xi_0}$ on $T^*U_{m,-\xi_0}$.
• Figure 2: The function $(r,\theta)\mapsto\rho(r,\theta)$. The curved arrows indicate the value of $\rho(r,\theta)$ at different regions. The $r$-derivative decay at a rate $r^{-3}$ (resp. $r^{-2}$) after $r\geq R+\varepsilon$. We extend $\rho$ to the whole $\mathbb{R}$ by declaring that $\rho(r,\theta+\pi)=\rho(r,\theta)+\pi$ in the odd case and $\rho(r,\theta+2\pi)=\rho(r,\theta)+2\pi$ in the even case.
• Figure 3: We modifying $L_{\geq R}$ by $\rho$ so that the angles $\theta_i^d(r)$'s match with $\theta_i$'s.
• Figure 4: This is Case (O), where $N=3$. As $\varphi_{\geq R}$ is obtained by rounding codimension 1 corners of $\varphi^{\mathrm{trop}}$, we may assume, by changing the branch of $\varphi_{f_d}$ if needed, that is, by adding $\pi$ to $\theta$, the differences $\varphi_{f_d}^{\rho}(r,\theta)-\varphi_{f_d}^{\rho}(r,\theta+\pi)$ and $\varphi_{\geq R}(r,\theta)-\varphi_{\geq R}(r,\theta+\pi)$ have same sign everywhere on $[0,\pi)$.
• Figure 5: We glue $\mathbb{L}_{\varphi_{\geq R}}$ and $L_{f_d}^{\rho}$ along the cylinder $C_{[R+\varepsilon,R+1]}$ by shrinking $a_d>0$ to ensure embeddedness on $C_{[R+\varepsilon,R+1]}$.

Theorems & Definitions (78)

• Theorem 1.2: =Theorem 1.9 in Morse_theory_TVB
• Theorem 1.3: =Theorem \ref{['thm:unobs']}
• Theorem 1.4: =Theorem \ref{['thm:degree0']}
• Corollary 1.5: =Corollary \ref{['cor:MS_gives_bundle']}
• Theorem 1.7: =Theorem \ref{['thm:unobs_immersed_Lag']}
• Remark 1.8
• Corollary 1.9: =Corollary \ref{['cor:existence_of_bundle']}
• Theorem 1.10: =Theorem \ref{['thm:LRP_higher_rank']}
• Theorem 1.11: =Theorem \ref{['thm:general_case']}
• Remark 1.12