國立清華大學 數學系

Abstract:
Mirror symmetry is a duality between complex and symplectic geometry. Kontsevich proposed a mathematical definition for mirror symmetry which is now known as homological mirror symmetry (HMS). HMS predicts that the Fukaya category of a symplectic manifold is quasi-equivalent to the derived category of its mirror complex manifold. Despite HMS has been proved in many interesting cases, it's usually hard to give an exact geometric correspondence between objects due to its homological nature. Tropical geometry serves as a tool to understand how geometric objects in the two categories can correspond. In this talk, I will introduce the notion of tropical Lagrangian multi-sections over rational fans and the two realization problems in toric equivariant mirror symmetry. Given a tropical Lagrangian multi-section $L^{trop}$, the A-realization problem asks if we can find an embedded exact Lagrangian multi-section whose asymptotics are prescribed by $L^{trop}$, and the B-realization problem asks if one can find a toric equivariant bundle whose Chern data is prescribed by $L^{trop}$. I will provide the affirmative answer to both realization problems for toric surfaces and the degree of $L^{trop}$ is $r=2$. If time permits, I will discuss the case $r>2$.