International Journal of Mathematics

We give a characterization of modular curves by a single symmetry of Hecke type. In the proof, we use the theorem of André, which characterizes modular curves in terms of special points.

We construct a form of the swallowtail singularity in $R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtails.

There are two natural definitions of the Julia set for complex Hénon maps: the sets $J$ and $J^{\star }$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set $J^{\star }$, under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used.

We show that $J=J^{\star }$ also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of ${ℂ}^2$. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on $J^{\star }$. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set $J$ were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].

Let $X$ be a complex projective toric manifold. We associated to $X$, a positive and closed $(1,1)$-current called the canonical toric Kähler current of $X$ denoted by ${\omega }_{X,\mathrm{\text{can}}}$, and a new invariant called the canonical spectrum of $X$. This spectrum is obtained as the set of the eigenvalues of a singular Laplacian defined by ${\omega }_{X,\mathrm{\text{can}}}$ and which is described uniquely by the combinatorial structure of $X$. The construction of this Laplacian and the study of its spectral properties are the consequence of a generalized spectral theory of Laplacians on compact Kähler manifolds that we develop in this paper.

We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak $\mathbb{Q}$-Fano $3$-fold. We also treat a stacky proof of the unobstructedness of deformations of a $\mathbb{Q}$-Fano $3$-fold.