Related papers: Twisted cubics on cubic fourfolds and stability co…
We characterize the birational geometry of some hyperk\"ahler fourfolds of Picard rank $3$ obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we identify all of…
We use a recently proposed formulation of stable holomorphic vector bundles $V$ on elliptically fibered Calabi--Yau n-fold $Z_n$ in terms of toric geometry to describe stability conditions on $V$. Using the toric map $f: W_{n+1} \to…
We show that the Charney--Davis strict hyperbolization procedure can preserve stable tangent bundles, answering a question of Charney and Davis. The key input is the construction of many hyperbolizing pieces, obtained using separability…
We prove that every Hassett's Noether-Lefschetz divisor of special cubic fourfolds contains a union of three codimension-two subvarieties, parametrizing rational cubic fourfolds, in the moduli space of smooth cubic fourfolds.
The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism type. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. In Voisin's original proof of the global…
The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated…
The twisted Lie-algebraically deformed relativistic and nonrelativistic phase spaces are constructed with the use of Heisenberg double procedure. The corresponding Heisenberg uncertainty principles are discussed as well.
This work deals with the study of embeddings of toric Calabi-Yau fourfolds which are complex cones over the smooth Fano threefolds. In particular, we focus on finding various embeddings of Fano threefolds inside other Fano threefolds and…
This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for…
Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasi periodic motions in phase space is…
We exhibit examples of slope-stable and modular vector bundles on a hyperk\"ahler manifold of K3$^{[2]}$-type which move in a 20-dimensional family and study their algebraic properties. These are obtained by performing standard linear…
By using the classification of Fano 3-folds we prove: Let $X$ be Fano 3-fold. Assume that the tangent bundle $T_X$ of $X$ is not stable (i.e. semi-stable or unstable). Then $b_2\geq 2$ and a relative tangent sheaf $T_{X/Y}$ of a contraction…
A very good cubic fourfold is a smooth cubic fourfold that does not contain a plane, a cubic scroll, or a hyperplane section with a corank 3 singularity. We prove that the normalization of the relative compactified Prym variety associated…
Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for…
We investigate the dynamics of travelling oscillating solitons of the cubic NLS equation under an external spatiotemporal forcing of the form $f(x,t) = a \exp[iK(t)x]$. For the case of time-independent forcing a stability criterion for…
A pair of disjoint lines on a smooth cubic threefold determines an irreducible component of the Hilbert scheme. We prove that this component is smooth and isomorphic to the blow-up of the symmetric product of Fano varieties of lines on the…
We study Fano fourfolds of K3 type with a conic bundle structure. We construct direct geometrical links between these fourfolds and hyperK\"ahler varieties. As a result we describe families of nodal surfaces that can be seen as…
We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in locally constant systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a…
We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is…
We shall obtain unobstructed deformations of four geometric structures: Calabi-Yau, HyperK\"ahler, $\G$ and Spin(7) structures in terms of closed differential forms (calibrations). We develop a direct and unified construction of smooth…