Holomorphic Lagrangian branes correspond to perverse sheaves

Abstract

Let $X$ be a compact complex manifold, $D_c^b\left(X\right)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk\left(T^{\ast }X\right)$ be the Fukaya category of $T^{\ast }X$. A Lagrangian brane in $Fuk\left(T^{\ast }X\right)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^{\ast }{X}_{ℂ}$, the holomorphic cotangent bundle of $X$. We prove that under the quasiequivalence between $D_c^b\left(X\right)$ and $DFuk\left(T^{\ast }X\right)$ established by Nadler and Zaslow, holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.