Enumerative geometry of dormant opers

The purpose of the present paper is to develop the enumerative geometry of dormant $G$-opers for a semisimple algebraic group $G$. In the present paper, we construct a compact moduli stack admitting a perfect obstruction theory by introducing the notion of a dormant faithful twisted $G$-oper (or a "$G$-do'per" for short. Moreover, a semisimple $2$d TQFT (= $2$-dimensional topological quantum field theory) counting the number of $G$-do'pers is obtained by means of the resulting virtual fundamental class. This $2$d TQFT gives an analogue of the Witten-Kontsevich theorem describing the intersection numbers of psi classes on the moduli stack of $G$-do'pers.Comment: 64 pages, the title is changed, some mistakes are correcte

On the cuspidalization problem for hyperbolic curves over finite fields

In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobenius-preserving isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-$l$ fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.Comment: 44 pages, to appear in Kyoto Journal of Mathematic

Duality for dormant opers

In the present paper, we prove that on a fixed pointed stable curve of characteristic $p>0$, there exists a duality between dormant $\mathfrak{sl}_n$-opers ($1 < n <p-1$) and dormant $\mathfrak{sl}_{(p-n)}$-opers. Also, we prove that there exists a unique (up to isomorphism) dormant $\mathfrak{sl}_{(p-1)}$-oper on a fixed pointed stable curve of characteristic $p>0$.Comment: 41 page