INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE

By E. Markman. 2006

105 citations recorded by the source corpus.

Let S[n] be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL[H*(S[n],ℤ)] generated by monodromy operators, and let Mon2 be its image in OH2(S[n],ℤ). We prove that Mon2 is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2 does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power. As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKahler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(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 2ρ(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 H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29590d9cbe4e0ae486cf3268d05793cce08b456c

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In Geometry and Topology

1. D. Huybrechts
2. E. Markman
3. Misha Verbitsky
4. Ekaterina Amerik
5. K. Hulek
6. G. Sankaran
7. V. Gritsenko
8. Giovanni Mongardi

In Mathematical Physics

1. D. Huybrechts
2. E. Markman
3. K. Hulek
4. G. Sankaran
5. V. Gritsenko
6. Misha Verbitsky
7. Ekaterina Amerik
8. Nicolas Addington

Citations in the corpus

• Morrison-Kawamata cone conjecture for hyperkahler manifolds (2014, methodology): “d by monodromy transform maps for Gauss-Manin local systems obtained from all deformations of (M,I) over a complex base ([V2, Definition 7.1]). This is how this group was originally defined by Markman ([M2], [M3]). Definition 2.28: Let z ∈ H2(M,Z) be an integral cohomology class. The space Teichz is th”
• Rational curves on hyperkahler manifolds (2014, unknown): “eneratedbymonodromytransformmaps for Gauss-Manin local systems obtained from all deformations of (M,I) over a complex base([V4, Definition 7.1]). This is how this group wasoriginallydefined by Markman ([M2], [M3]). The fact that it is of finite index in O(H2(M,Z),q) is crucial for the Morrison-Kawamata”
• Ergodic complex structures on hyperkähler manifolds (2013, background): “e monodromy transform mapsfor the Gauss-Manin local systems obtained from all deformations of (M,I) over a complex base ([V2, Definition 7.1]). This is how this group was originally defined by Markman ([Ma1], [Ma2]). Remark 2.21: A caution: usually “the global Torelli theorem” is understood as a theor”
• Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type (2013, background): “ces the isometry g. When X= X1 = X2, we call ga monodromy operator. The monodromy group Mon2(X) of X is the subgroup, of the isometry group of H2(X,Z), generated by monodromy operators. Theorem 2.4. ([Ma3], Theorem 1.2 and Lemma 4.2) The subgroup Mon2(X) of O(Λ) consists of orientation preserving is”
• A global Torelli theorem for hyperkaehler manifolds (after Verbitsky) (2011, methodology): “The monodromy group Mon(X) has been computed by Markman for X = Hilb(S) and arbitrary n (see [19, 20]).”
• A survey of Torelli and monodromy results for holomorphic-symplectic varieties (2011, methodology): “Indeed, these moduli spaces are not birational in general ([ Ma5 ], Proposition 4.10).”
• A global Torelli theorem for hyperkahler manifolds (2009, unknown): “ogomolov-BeauvilleFujiki form and the natural spin orientation on H2(M,I1) and H2(M,I2). (Remark 7.15). Then (M,I1) is bimeromorphic to (M,I2). Proof: This is Theorem 7.19. Remark 1.24: E. Markman in [M2] constructed counterexamples to the Hodgetheoretic global Torelli problem for K3[n], where n−1 i”
• Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, background): “Markman first noticed the importance of these operators for the study of irreducible symplectic manifolds, developing his ideas in a series of papers [84], [83], [85].”