Cyclic Lagrangian submanifolds and Lagrangian fibrations (Q2566542)

Recall that a symplectic manifold \((M,\omega)\) is called prequantizable if there exists a nonnegative number \(\gamma_\omega\) such that the period group \(\Gamma_\omega= \{[\omega](A)= \int_A \omega: A\in H_2(M,\mathbb{Z})\}\subset \mathbb{R}\) is of the form \(\Gamma_\omega= \gamma_\omega\cdot\mathbb{Z}\). It is known that if \((M,\omega)\) is prequantizable, then there is a complex line bundle \(E\to M\) with a connection \(\nabla\) such that its curvature \(R_\nabla\) satisfies \[ R_\nabla= {2\pi i\over \gamma_\omega} \omega.\tag{1} \] \((E,\nabla)\) is called a prequantization bundle. If \(H_1(M,\mathbb{Z})= 0\) then there exists an unique equivalence class of connections on \(E\) which satisfies (1) and the holonomy of \(\nabla\) along a loop \(\ell\in M\) is equal to \[ \exp\Biggl({2\pi i\over\gamma_\omega} \int_S \omega\Biggr), \] where \(S\subset M\) is a surface with \(\partial S= \ell\). Let \(i: L\to M\) be a Lagrangian submanifold. Then \((i^* E,i^*\nabla)\) is flat and we have a holonomy homomorphism \(\text{Hol}_L: \pi_1(L)\to S^1\). One calls \(L\) cyclic if the image of \(\text{Hol}_L\) is cyclic in \(S^1\). The main results of the paper are the following ones: 1) If \((M,\omega)\) is Kähler-Einstein with nonzero scalar curvature then \(L\subset M\) is cyclic iff the mean curvature cohomology class of \(L\) is rational. 2) Let \((M,I,\omega)\) be a prequantizable Ricci-flat \(K3\) surface. Then one of the following holds: a) There are not embedded Lagrangian tori \(L\) with \([L]\neq 0\) in \(H_2(M,\mathbb{Z})\). b) There are noncyclic minimal Lagrangian tori. 3) Let \((M,I,\omega)\) be a compact Kähler manifold of complex dimension \(n\) with an effective isometric Hamiltonian action of the real torus \(T^n\) on \(M\). Then, any connected regular fiber of the moment map is Hamiltonian trivial.