We first prove a certain kind of splitting theorem for an equifocal submanifold with non-flat section in a simply connected symmetric space of compact type, where an equifocal submanifold means a submanifold with parallel focal structure. By using the splitting theorem, we prove that there exists no equifocal submanifold with non-flat section in an irreducible simply connected symmetric space of compact type whose codimension is greater than the maximum of the multiplicities of roots of the symmetric space or the maximum added one. In particular, it follows that there exists no equifocal submanifold with non-flat section in some irreducible simply connected symmetric spaces of compact type and that there exists no equifocal submanifold with non-flat section in simply connected compact simple Lie group whose codimension is greater than two.

It is shown that the relative canonical linear system over a normal surface singularity has at most exceptional sets of rational singular points as its fixed part.

In this note, we use the combinatorial method of Goulden-Jackson-Vakil to give a simple proof of Witten conjecture-Kontsevich theorem.

Co-symplectic/co-Kähler manifolds are odd dimensional analog of symplectic/Kähler manifolds, defined early by Libermann in 1959/Blair in 1967 respectively. In this paper, we reveal their topology construction via symplectic/Kähler mapping tori. Namely,

Theorem. Co-symplectic manifold = Symplectic mapping torus; Co-Kähler manifold = Kähler mapping torus.