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/