On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

Sur l'anneau de cohomologie du schéma de Hilbert de C2

We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.     

An example of an SL2-Hilbert scheme with multiplicities

In [AB05], Alexeev and Brion have introduced the notion of invariant Hilbert schemes. We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of SL₂ on ( \mathbbC² ) ^?6 {\left( {{\mathbb{C}^2}} \right)^{ \oplus 6}} as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for realizing these calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.     

Equivariant deformations of the affine multicone over a flag variety

We prove that the invariant Hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. More specifically, we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well-determined wonderful variety.     

Twisted rings and moduli stacks of “fat” point modules in non-commutative projective geometry

The Hilbert scheme of point modules was introduced by Artin–Tate–Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general “fat” point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin?s conjecture on the birational classification of non-commutative surfaces.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic‐symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation‐equivalent to the Hilbert scheme of length‐3 subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two‐torsion. © 2011 Wiley Periodicals, Inc.     

Tropical quadrics through three points

We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these objects inside the tropical Grassmannian of planes in projective 5-space, and we explore a small tropical Hilbert scheme.     

Poisson structures on Hilbert schemes of points¶of a surface and integrable systems

In this paper we prove that if S is a Poisson surface, i.e., a smooth algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville [B1] and S. Mukai [M2] in the symplectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the simple case S=ℙ² we obtain by this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke in [V1] and, more generally, in [V2]. Received: 9 March 1998 / Revised version: 19 June 1998     

Harmonic analysis and systems of covariance for phase space representation of the poincaré group

Continuing some earlier work on the Galilei group, the spectral resolution of phase space representations of the Poincaré group is achieved by deriving all possible decompositions into irreducible representations corresponding to reproducing, kernel Hilbert spaces. Systems of covariance related to quantum measurements performed with extended test particles are analyzed, and questions of global unitarity discussed.Supported in part by NSERC Research Grants.     

On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.The last three authors are partially sponsored by CNPq.     

Hilbert space representations of decoherence functionals and quantum measures

We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.     

Smooth and irreducible multigraded Hilbert schemes

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x,y], which establishes a conjecture of Haiman and Sturmfels.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Higher asymptotics of unitarity in ��quantization commutes with reduction��

Let M be a compact K?hler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515?C538, 1982, no. 3), showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M //G. This map, though, is not in general unitary, even to leading order in ${\hslash}$ . Hall and Kirwin (Commun Math Phys 275:401?C422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed ${\hslash}$ , becomes unitary in the semiclassical limit ${\hslash\rightarrow0}$ (cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297?C302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). The unitarity of the classical Guillemin?CSternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M //G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as ${\hslash\rightarrow0}$ .     

On the McKay correspondences for the Hilbert scheme of points on the affine plane

The quotient of a finite-dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety. Under appropriate assumptions, the McKay correspondence relates the geometry of nice resolutions of singularities and the representations of the group. For the Hilbert scheme of points on the affine plane, we study how different correspondences (McKay, dual McKay and multiplicative McKay) are related to each other.     

On the Cartan Map for Crossed Products and Hopf–Galois Extensions

We study certain aspects of the algebraic K-theory of Hopf–Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules.     

On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

In this paper we study the multigraded Hilbert and Poincaré-Betti series of A=S/a, where S is the ring of polynomials in n indeterminates divided by the monomial ideal a. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x₁,…,x_n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A. We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case.The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A.     

自伴矩阵代数上约当半三可乘映射

In this paper,we show that every injective Jordan semi-triple multiplicative map on the Hermitian matrices must be surjective,and hence is a Jordan ring isomorphism.