Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

Markman, Eyal

doi:10.1016/j.aim.2006.03.006

Cited by 54 publications

(67 citation statements)

References 29 publications

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“…Sections 4 and 5 are logically independent. This is the final paper in a sequence of four, and it relies heavily on the preceeding three paper [Ma1,Ma2,Ma3]. In fact, the results of the current paper, with the exception of Theorem 1.5 and section 5, were posted originally as the second part of the eprint arXiv:math.AG/0406016 v1 containing the motivation to and main application of the determination of the generators for the integral cohomology ring H * (S [n] , Z) in the first part.…”

Section: Theorem 114 A) Let I Be An Even Integer In the Rangementioning

confidence: 99%

“…Otherwise, there is a weaker notion of a twisted universal sheaf, denoted also by E, where the twisting is encoded by a class α inČech cohomology H 2 (M, O * ), in the classical topology. For a triple S, v, H, as above, the class α is always topologically trivial; it maps to 0 in H 2 (M, F ), where F is the sheaf of continuous complex valued invertible functions [Ma3]. Consequently, E defines a class…”

Section: Notationmentioning

confidence: 99%

“…The odd integral cohomology groups of X vanish ([Ma3] or Theorem 2.4 below). We set Q 2 (X, Z) := H 2 (X, Z).…”

Section: The Mukai Lattice As An Invariant Of a Hyper-kähler Varietymentioning

confidence: 99%

“…In fact, the results of the current paper, with the exception of Theorem 1.5 and section 5, were posted originally as the second part of the eprint arXiv:math.AG/0406016 v1 containing the motivation to and main application of the determination of the generators for the integral cohomology ring H * (S [n] , Z) in the first part. The original version was split, following the recommendation of a referee, and the first part was published as [Ma3]. We are forced to include a somewhat lengthy summary of background material and previous results in sections 2 and 3.2.…”

Section: Theorem 114 A) Let I Be An Even Integer In the Rangementioning

confidence: 99%

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Integral Constraints on the Monodromy Group of the Hyperkähler Resolution of a Symmetric Product of a K3 Surface

Markman

2010

Int. J. Math.

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Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface 1 Eyal Markman 2 Abstract: Let S [n] be the Hilbert scheme of length n subschemes of a K3 surface S. H 2 (S [n] , Z) is endowed with the Beauville-Bogomolov bilinear form. Denote by M on the subgroup of GL[H * (S [n] , Z)] generated by monodromy operators, and let M on 2 be its image in OH 2 (S [n] , Z). We prove that M on 2 is the subgroup generated by reflections with respect to +2 and −2 classes (Theorem 1.2). Thus M on 2 does not surject onto OH 2 (S [n] , Z)/(±1), when n − 1 is not a prime power.As a consequence, we get counter examples to a version of the weight 2 Torelli question for hyperkähler varieties X deformation equivalent to S [n] . The weight 2 Hodge structure on H 2 (X, Z) does not determine the bimeromorphic class of X, whenever n − 1 is not a prime power (the first case being n = 7). There are at least 2 ρ(n−1)−1 distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n − 1) is the Euler number of n − 1.The second main result states, that if a monodromy operator acts as the identity on H 2 (S [n] , Z), then it acts as the identity on H k (S [n] , Z), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism M on → M on 2 , if n ≡ 0, or n ≡ 1 modulo 4 (Corollary 1.6).

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Section: Theorem 114 A) Let I Be An Even Integer In the Rangementioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

“…The odd integral cohomology groups of X vanish ([Ma3] or Theorem 2.4 below). We set Q 2 (X, Z) := H 2 (X, Z).…”

Section: The Mukai Lattice As An Invariant Of a Hyper-kähler Varietymentioning

confidence: 99%

Section: Theorem 114 A) Let I Be An Even Integer In the Rangementioning

confidence: 99%

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Integral Constraints on the Monodromy Group of the Hyperkähler Resolution of a Symmetric Product of a K3 Surface

Markman

2010

Int. J. Math.

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“…The integral cohomology of the Hilbert scheme is more subtle than the rational cohomology. Markman computed the integral cohomology of the Hilbert schemes X [a] for X of dimension 2 with effective anticanonical divisor [10]. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology.…”

Section: Introductionmentioning

confidence: 99%

The Integral Cohomology of the Hilbert Scheme of Two Points

Totaro

2016

Forum of Mathematics, Sigma

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The Hilbert scheme X [a] of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X . The integral cohomology of X [a] is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology.2010 Mathematics Subject Classification: 14C05 (primary); 55R80 (secondary)For a complex manifold X and a natural number a, the Hilbert scheme X [a] (also called the Douady space) is the space of 0-dimensional subschemes of degree a in X . It is a compactification of the configuration space B(X, a) of a-element subsets of X . The Hilbert scheme is smooth if and only if X has dimension at most 2 or a 3 [3, equation (0.2.1)]. The integral cohomology of the Hilbert scheme is more subtle than the rational cohomology. Markman computed the integral cohomology of the Hilbert schemes X [a] for X of dimension 2 with effective anticanonical divisor [10]. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology. In one way, things are unexpectedly good: the Hilbert scheme X [2] has torsionfree cohomology if X does (Theorem 2.2). On the other hand, the details are intricate, and it was not clear that complete answers would be possible. The behavior of the inclusion of the exceptional divisor E X into X [2] is related to the Steenrod operations on the mod 2 cohomology of X (Theorem 2.1). To explain one difficulty: some cohomology classes on X [2] can be defined as the classes

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Supersingular Irreducible Symplectic Varieties

2021

Rationality of Varieties

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

Cited by 54 publications

References 29 publications

Integral Constraints on the Monodromy Group of the Hyperkähler Resolution of a Symmetric Product of a K3 Surface

Integral Constraints on the Monodromy Group of the Hyperkähler Resolution of a Symmetric Product of a K3 Surface

The Integral Cohomology of the Hilbert Scheme of Two Points

Supersingular Irreducible Symplectic Varieties

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