Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMBM loci in families of hyperkahler manifolds and centers of birational contractions
85
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation space has minimal possible dimension 2n-2, where 2n is the complex dimension of M. We study the MBM loci, defined as the subvarieties covered by deformations of an MBM curve within M. When M is projective, MBM loci are centers of birational contractions. For each MBM class z, we consider the Teichmuller space $Teich^{min}_z$ of all deformations of M such that $z^{bot}$ contains a face of the Kahler cone. We prove that for all $I,Jin Teich^{min}_z$, the MBM loci of (M, I) and (M,J) are homeomorphic under a homeomorphism preserving the MBM curves, unless possibly the Picard number of I or J is maximal.
rate research
Read More
An MBM locus on a hyperkahler manifold is the union of all deformations of a minimal rational curve with negative self-intersection. MBM loci can be equivalently defined as centers of bimeromorphic contractions. It was shown that the MBM loci on deformation equivalent hyperkahler manifolds are diffeomorphic. We determine the MBM loci on a hyperkahler manifold of K3-type of low dimension using a deformation to a Hilbert scheme of a non-algebraic K3 surface.
The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperkahler manifolds. These manifolds are interesting from several points of view: dynamical (some have interesting automorphism groups), arithmetical (although we will not say anything on this aspect of the theory), and geometric. It is also one of those rare cases where the Torelli theorem allows for a powerful link between the geometry of these manifolds and lattice theory. We do not prove all the results that we state. Our aim is more to provide, for specific families of hyperkahler manifolds (which are projective deformations of punctual Hilbert schemes of K3 surfaces), a panorama of results about projective embeddings, automorphisms, moduli spaces, period maps and domains, rather than a complete reference guide. These results are mostly not new, except perhaps those of Appendix B (written with E. Macr`i), where we give an explicit description of the image of the period map for these polarized manifolds.
Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families, so that this contraction may induce maps between moduli spaces of curves. In order to do this, we introduce the notion of mesa curve, a nodal curve $C$ with a logarithmic structure and a piecewise linear function $overline{lambda}$ on the tropicalization of $C$. This piecewise linear function determines a subcurve $E$. We then construct a contraction of $E$ inside of $C$ for families of mesa curves. Resulting singularities include the elliptic Gorenstein singularities.
This paper is a survey of finiteness results in hyperkahler geometry. We review some classical theorems by Sullivan, Kollar-Matsusaka, Huybrechts, as well as theorems in the recent literature by Charles, Sawon, and joint results of the author with Verbitsky. We also strengthen a finiteness theorem of the author. These are extended notes of the authors talk during the closing conference of the Simons Semester in the Banach Center in Bc{e}dlewo, Poland.
In this paper, we discuss the cycle theory on moduli spaces $cF_h$ of $h$-polarized hyperkahler manifolds. Firstly, we construct the tautological ring on $cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperkahler manifolds with $nleq 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperkahler manifolds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
