## Kyoto Journal of Mathematics

### Brane involutions on irreducible holomorphic symplectic manifolds

#### Abstract

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three Kähler structures of the associated hyper-Kähler structure. Starting from a brane involution on a $\mathrm{K3}$ or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a $\mathrm{K3}$ surface and $\mathrm{K3}^{[2]}$-type manifolds.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 1 (2019), 195-235.

Dates
Received: 5 July 2016
Accepted: 10 February 2017
First available in Project Euclid: 8 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1546916422

Digital Object Identifier
doi:10.1215/21562261-2018-0009

Mathematical Reviews number (MathSciNet)
MR3934628

Zentralblatt MATH identifier
07081627

#### Citation

Franco, Emilio; Jardim, Marcos; Menet, Grégoire. Brane involutions on irreducible holomorphic symplectic manifolds. Kyoto J. Math. 59 (2019), no. 1, 195--235. doi:10.1215/21562261-2018-0009. https://projecteuclid.org/euclid.kjm/1546916422

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