Related papers: A Refined Derived Torelli Theorem for Enriques sur…

A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface. We deal with results obtained in this last…

Let X and X' be compact Riemann surfaces of genus at least three. Let G and G' be nontrivial connected semisimple linear algebraic groups over C. If some components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated Deligne--Hitchin…

Enriques manifolds are complex spaces whose universal coverings are hyperkaehler manifolds. We introduce period domains for Enriques manifolds, establish a local Torelli theorem, and apply period maps in various situations, involving…

We give a short and "classical" proof of Borcherds' theorem that the moduli space of Enriques surfaces is quasi-affine. The use of the Borcherds' product is replaced in our proof by an application of the Grothendieck-Riemann-Roch theorem.

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential…

Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a…

We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over…

We consider the derived category of an Artin-Mumford quartic double solid blown-up at ten ordinary double points. We show that it has a semi-orthogonal decomposition containing the derived category of the Enriques surface of a Reye…

We present a short proof of the fact that two irreducible germs of plane analytic curves are isotopic if they are equisingular, without recourse to the structure of the associated knots.

Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for…

We give a proof of the Neilsen-Thurston classification theorem of a homeomorphism f of a standard surface of finite type as either periodic, pseudo-Anosov, or reducible. In the periodic case, we show that there exists an integer n>0 such…

We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on…

We study the first homology group of the mapping class group and Torelli group with coefficients in the first rational homology group of the universal abelian cover of the surface. We prove two contrasting results: for surfaces with one…

This is the first paper of two ones. Here we prove that two compact Alexandrov surfaces of bounded integral curvature having no peak points are bi-Lipschitz equivalent if they are homeomorphic one to the other. Also conditions under that…

We study the universal cover of the complex one-dimensional torus as a model-theoretic structure in a natural language. We consider also abstract covers of one-dimensional tori over algebraically closed fields of characteristic zero. The…

We classify smooth surfaces whose higher cohomologies of i-forms for all i vanish. We show that if such a surface is not affine, then it has essentially two possibilities.

A set of locally finite perimeter $E \subset \mathbb{R}^{n}$ is called an anisotropic minimal surface in an open set $A$ if $\Phi(E;A) \le \Phi(F;A)$ for some surface energy $\Phi(E;A) = \int_{\partial^{*}E \cap A} \| \nu_{E}\| d…

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric…

Let $Y$ be a smooth Enriques surface. A $K3$ carpet on $Y$ is a locally Cohen-Macaulay double structure on $Y$ with the same invariants as a smooth $K3$ surface (i.e., regular and with trivial canonical sheaf). The surface $Y$ possesses an…

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such…