Related papers: Curves on K3 surfaces in divisibility two

Let $X$ be a $K3$ surface over a $p$-adic field $k$ such that for some abelian surface $A$ isogenous to a product of two elliptic curves, there is an isomorphism over the algebraic closure of $k$ between $X$ and the Kummer surface…

In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus over…

We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this…

We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we…

We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every…

We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many…

Given two hyperbolic curves over p-adic local fields, the absolute anabelian conjecture claims that any isomorphism between their \'etale fundamental group comes from an isomorphism of schemes. This conjecture was proven by S. Mochizuki for…

We prove a conjecture of Odaka--Oshima, which says that there is an algebraic description of the Gromov--Hausdorff compactification of all unit-diameter hyperk\"ahler metrics on K3 surfaces. As a corollary, we obtain a classification of the…

A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.

A 2015 conjecture of Codesido-Grassi-Mari\~no in topological string theory relates the enumerative invariants of toric CY 3-folds to the spectra of operators attached to their mirror curves. We deduce two consequences of this conjecture for…

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0), without saddle-nodes, has invariant surface. We extend the argument of Cano-Cerveau, done for the…

Using the isomorphism between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. Precisely, we show…

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and…

In this paper, we prove quasi-modularity property for the twisted Gromov-Witten theory of $\mathcal{O}(3)$ over $\mathbb{P}^2$. Meanwhile, we derive its holomorphic anomaly equation.

We prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is a Kervaire-Milnor invariant for surfaces and we give a…

If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the…

We test the refined distance conjecture in the vector multiplet moduli space of 4D $\mathcal{N}=2$ compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the…

The notion of constant cycle curves on K3 surfaces is introduced. These are curves that do not contribute to the Chow group of the ambient K3 surface. Rational curves are the most prominent examples. We show that constant cycle curves…