Related papers: Siegel Disks on Rational Surfaces

We solve the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces.

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…

Rationally convex topological embeddings of compact surfaces (closed or with boundary) into $\mathbb{C}^2$ are constructed.

We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every…

Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

In this short note, we prove the existence of infinitely many pairwise non-isomorphic non-hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also found all those compact Riemann surfaces…

We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact…

We will show that there is a smooth complex projective surface, birational to some Enriques surface, such that the automorphism group is discrete but not finitely generated.

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle…

This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good…

This thesis deals with automorphisms of real algebraic surfaces, which are polynomial transformations with a polynomial inverse. The main concern is whether their restriction to the real locus reflects all the richness of the complex…

We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that…

We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing…

This paper is a survey about $K3$ surfaces with an automorphism and log rational surfaces, in particular, log del Pezzo surfaces and log Enriques surfaces. It is also a reproduction on my talk at "Mathematical structures of integrable…

We survey our recent papers (some being joint ones) about the relation between the geometry of a compact K\"ahler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program…

We construct a symplectic flow on a surface of genus g greater than one with exactly 2g-2 hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly non-degenerate) symplectomorphism of a surface (with genus g…

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an…

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…

We prove that the morphism that maps a rational ruled surface to its singular locus is genericaly injective modulo isomophism and duality. We also calculate the dimension and the degre of its image.

We prove gap theorems for entropy norms on automorphism groups of K3 surfaces, Enriques surfaces, and irreducible holomorphic symplectic manifolds. We also study the achirality of automorphisms of K3 surfaces and Enriques surfaces in terms…