scholarly journals Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry

2020 ◽  
Author(s):  
Roger Bielawski
Keyword(s):  
Hamiltonian Systems ◽  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Poisson Structures ◽  
Hilbert Schemes ◽  
Hyperkähler Manifolds ◽  
Symplectic Surface ◽  
Monopole Moduli Spaces ◽  
Hyperkähler Geometry

Abstract Dedicated to the memory of Sir Michael Francis Atiyah (1929-2019) We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.

scholarly journals Transverse Hilbert schemes and completely integrable systems

2017 ◽  
Vol 4 (1) ◽  
pp. 263-272 ◽  
Author(s):  
Niccolò Lora Lamia Donin
Keyword(s):  
Integrable Systems ◽  
Hilbert Scheme ◽  
Symplectic Form ◽  
Initial Surface ◽  
Hilbert Schemes ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
Symplectic Surface ◽  
Complex Symplectic ◽  
Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

scholarly journals Generalized punctual Hilbert schemes and 𝔤-complex structures

2021 ◽  
Author(s):  
Alexander Thomas
Keyword(s):  
Lie Algebras ◽  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Complex Structure ◽  
Complex Structures ◽  
Hilbert Schemes ◽  
Simple Complex ◽  
Geometric Structures ◽  
Punctual Hilbert Scheme ◽  
Alternative Description

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin–Kostant–Rallis section.

scholarly journals Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures

2016 ◽  
Vol 27 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Poisson Structure ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Smooth Surfaces ◽  
Poisson Structures ◽  
Hilbert Schemes ◽  
Hilbert Scheme Of Points

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

scholarly journals NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

2020 ◽  
pp. 1-25
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ANDREA CATTANEO
Keyword(s):  
Moduli Spaces ◽  
Quadratic Forms ◽  
Symplectic Manifolds ◽  
Hilbert Schemes ◽  
Small Rank ◽  
Twisted Sheaves ◽  
Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

2014 ◽  
Vol 6 (01) ◽  
pp. 87-106
Author(s):  
Xueyang Li ◽  
Aiguo Xiao ◽  
Dongling Wang
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Generating Function ◽  
Poisson Structure ◽  
Poisson Structures ◽  
Volterra Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

Moduli spaces of bundles and Hilbert schemes of scrolls over $$\nu $$ ν -gonal curves

2018 ◽  
Vol 70 (2) ◽  
pp. 295-321
Author(s):  
Youngook Choi ◽  
Flaminio Flamini ◽  
Seonja Kim
Keyword(s):  
Moduli Spaces ◽  
Hilbert Schemes

The Effective Cone of the Kontsevich Moduli Space

2008 ◽  
Vol 51 (4) ◽  
pp. 519-534 ◽  
Author(s):  
Izzet Coskun ◽  
Joe Harris ◽  
Jason Starr
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete Characterization ◽  
Stable Maps ◽  
Linear Combinations ◽  
Effective Divisors ◽  
Effective Cone

AbstractIn this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, , stabilize when r ≥ d. We give a complete characterization of the effective divisors on . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

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