##
**Lagrangian submanifolds in hyper-Kähler manifolds, Legendre transformation.**
*(English)*
Zbl 1057.53034

This is a beautiful, excellently written paper throwing new and rather unexpected insight on the geometry of hyper-Kähler manifolds.

Let \(M\) be a hyper-Kähler manifold of real dimension \(4n\). Denote with \(\Omega\) the holomorphic 4-form and with \(\omega\) the Kähler form for a fixed complex structure (with respect to which the complex submanifolds are intended) of the hyper-Hermitian structure.

The first striking result is that if \(M\) is compact, then the notions of isotropic and co-isotropic (in particular Lagrangian) submanifolds, are cohomological: the Poincaré dual of the submanifold is annihilated by the cup product with \([\Omega]\) (theorem 3). Therefore, such properties are invariant under deformations.

Another beautiful result is that complete intersection co-isotropic submanifolds can be characterized by the vanishing of a Chern number. In fact, for complete intersections \(C\) of dimension \(n+k\), \(k\leq n-2\), the author proves that \(\int_C c_2(C)(\Omega\overline{\Omega})^k \omega^{n-k-2}\geq0,\) with equality if and only if \(C\) is coisotropic (theorem 8).

A whole section of the paper is devoted to Lagrangian submanifolds. The author first shows that any \(c\in H^{2n}(M,\mathbb{Z})\) determines the rational cobordism class of any possible Lagrangian submanifold representing \(c\) (theorem 10). He then studies the moduli space of Lagrangian submanifolds (theorem 12). Here the tool is the second fundamental form of a Lagrangian submanifold viewed as a cubic form on \(H^0(C, T^*C)\), thus inducing special Kähler structures on the moduli space [cf. N. J. Hitchin, Asian J. Math. 3, No. 1, 77–91 (1999; Zbl 0958.53057)]. A Lagrangian category, holomorphic analogue of the Fukaya category on real symplectic manifolds, is then defined and studied.

The last section of the paper introduces a Legendre transformation of Lagrangian subvarieties along a co-isotropic exceptional subvariety in \(M\). This leads to a new Plücker type formula relating intersection numbers of Lagrangian subvarieties under the Legendre transformation.

Let \(M\) be a hyper-Kähler manifold of real dimension \(4n\). Denote with \(\Omega\) the holomorphic 4-form and with \(\omega\) the Kähler form for a fixed complex structure (with respect to which the complex submanifolds are intended) of the hyper-Hermitian structure.

The first striking result is that if \(M\) is compact, then the notions of isotropic and co-isotropic (in particular Lagrangian) submanifolds, are cohomological: the Poincaré dual of the submanifold is annihilated by the cup product with \([\Omega]\) (theorem 3). Therefore, such properties are invariant under deformations.

Another beautiful result is that complete intersection co-isotropic submanifolds can be characterized by the vanishing of a Chern number. In fact, for complete intersections \(C\) of dimension \(n+k\), \(k\leq n-2\), the author proves that \(\int_C c_2(C)(\Omega\overline{\Omega})^k \omega^{n-k-2}\geq0,\) with equality if and only if \(C\) is coisotropic (theorem 8).

A whole section of the paper is devoted to Lagrangian submanifolds. The author first shows that any \(c\in H^{2n}(M,\mathbb{Z})\) determines the rational cobordism class of any possible Lagrangian submanifold representing \(c\) (theorem 10). He then studies the moduli space of Lagrangian submanifolds (theorem 12). Here the tool is the second fundamental form of a Lagrangian submanifold viewed as a cubic form on \(H^0(C, T^*C)\), thus inducing special Kähler structures on the moduli space [cf. N. J. Hitchin, Asian J. Math. 3, No. 1, 77–91 (1999; Zbl 0958.53057)]. A Lagrangian category, holomorphic analogue of the Fukaya category on real symplectic manifolds, is then defined and studied.

The last section of the paper introduces a Legendre transformation of Lagrangian subvarieties along a co-isotropic exceptional subvariety in \(M\). This leads to a new Plücker type formula relating intersection numbers of Lagrangian subvarieties under the Legendre transformation.

Reviewer: Liviu Ornea (Bucureşti)

### MSC:

53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |

53D12 | Lagrangian submanifolds; Maslov index |

53D40 | Symplectic aspects of Floer homology and cohomology |