Related papers: Fixed divisors on hyperk\"ahler manifolds

In this paper, we prove that an algebraic fiber space $f:X\to Y$ over a perfect field $k$ of characteristic $p>0$ with nef relative anti-canonical divisor $-K_{X/Y}$ splits into the product after taking the base change along a finite cover…

We consider projective Hyper-K\"ahler manifolds of dimension four that are deformation equivalent to Hilbert squares of K3 surfaces. In case such a manifold admits a divisorial contraction, the exceptional divisor is a conic bundle over a…

We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: 1. We describe the nef cone, the movable cone, and the effective…

This note presents basic restrictions on the topology "general" Lagrangian surfaces of hyper-K\"ahler $4$-folds and a remark on the interaction of a Lagrangian subvariety with a Lagrangian fibration of the associated hyper-K\"ahler variety.

We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\geq 1$ on a homogeneous compact K\"ahler manifold $X$ of dimension $n>k$. Assuming that the singular set $Sing(\mathcal{F})$ of $\mathcal{F}$ is contained in an absolutely…

By means of a certain exact non-abelian duality transformation, we show that there is a natural embedding, dense in the sense of the distributions in the large-N limit, of parabolic Higgs bundles of rank N on a fiber two-dimensional torus…

We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The basis of the fibration is a general hypersurface of low degree.

We show that if a stable variety (in the sense of Koll\'ar and Shepherd-Barron) admits a fibration with stable fibers and base, then this fibration structure deforms (uniquely) for all small deformations. During our proof we obtain also a…

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The…

The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that…

We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb{A}_k^1$, the derived critical locus has a natural Lagrangian fibration…

Let $\pi : X \to B$ be a Lagrangian fibration of a compact hyper-K\"ahler manifold. We prove that every codimension $1$ fiber of $\pi$ has multiplicity $1$.

We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing…

We prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. The main tools are (some known cases of) the Kawamata's effective…

Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$…

We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we ``classify'' in dimension 2 nef divisors…

Let $f:X\to S$ be a projective morphism of normal varieties. Assume $U$ is an open subset of $S$ and $L_U$ is a $\mathbb{Q}$-divisor on $X_U:=X\times_S U$ such that $L_U\equiv_U 0$. We explore when it is possible to extend $L_U$ to a global…

The Abels-Margulis-Soifer lemma states that if a semigroup $\Gamma$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $\Gamma$ can be multiplied by an element of some fixed…

In this work lipid ordering phase changes arising in planar membrane bilayers is investigated both accounting for elas- ticity alone and for effective viscoelastic response of such assemblies. The mechanical response of such membranes is…

We investigate effectiveness and ampleness of adjoint divisors of the form $aL+bK_X$, where $L$ is a suitably positive line bundle on a smooth projective variety $X$ and $a,b$ are positive integers.