#### Volume 19, issue 3 (2015)

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Holomorphic Lagrangian branes correspond to perverse sheaves

### Xin Jin

Geometry & Topology 19 (2015) 1685–1735
##### 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.

##### Keywords
Fukaya category, holomorphic Lagrangian branes, perverse sheaves, constructible sheaves, Nadler–Zaslow correspondence
##### Mathematical Subject Classification 2010
Primary: 53D40, 32S60
##### Publication
Received: 29 March 2014
Revised: 18 July 2014
Accepted: 16 August 2014
Published: 21 May 2015
Proposed: Richard Thomas
Seconded: Ciprian Manolescu, Ronald Stern
##### Authors
 Xin Jin Department of Mathematics University of California, Berkeley Berkeley, CA 94720 USA http://math.berkeley.edu/~xjin/