Counter-example to global Torelli problem for irreducible symplectic manifolds

By Y. Namikawa. 2001

69 citations recorded by the source corpus.

A simply connected compact Kaehler manifoldX is an irreducible symplectic manifold if there is an everywhere non-degenerate holomorphic 2-form Ω on X with H0(X,Ω2X) = C[Ω]. By definition, X has even complex dimension. There is a canonical symmetric form qX on H (X,Z), which is called the BeauvilleBogomolov form (cf. [Be]). On the other hand, since X is Kaehler, H(X,Z) has a natural Hodge structure of weight 2. If two irreducible symplectic manifolds X and Y are bimeromorphically equivalent, then there is a natural Hodge isometry between (H(X,Z), qX) and (H (Y,Z), qY ) (cf. [O, Proposition (1.6.2)], [Huy, Lemma 2.6]). Debarre [De] has constructed bimeromorphically equivalent irreducible symplectic manifolds X and Y such that X and Y are not isomorphic. This is a counter-example to the following problem.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9fb96625dcd4ad4098d10ce39cb1e0a9d61f18ff

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Citations in the corpus

A global Torelli theorem for hyperkaehler manifolds (after Verbitsky) (2011, background): “This hope was shattered by Namikawa’s example in [22]: • There exist compact hyperkähler manifolds X and X ′ (projective and of dimension four) with isometric Hodge structures H(2)(X,Z) ∼= H(2)(X ,Z) but without X and X ′ being bimeromorphic (birational).”
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, background): “This shows that Namikawa’s example gives a counterexample even to the birational global Torelli theorem.”
A note on automorphisms and birational transformations of holomorphic symplectic manifolds (2009, background): “In higher dimensions, Debarre [14] and Namikawa [24] provided counterexamples to various analogues of the global Torelli theorem.”
Monodromy and birational geometry of O'Grady's sixfolds (2019, methodology): “For every v = (v0, v2, v4) = (2w0, 2w2, 2w4) = 2w as in Setting 4.1, let v⊥ = w⊥ be the perpendicular lattice to v in the Mukai lattice of A and let H2(Kv(A),Z) be endowed with the lattice structure given by the Beauville-Bogomolov-Namikawa form.”
Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces (2004, unknown): “njecture 24. The generalized Kummer varieties constituteanothersequence ofcompacthyperk¨ahler varieties [B1]. Namikawa found a counter example to the Torelli question for generalized Kummer fourfolds [Na]. Our second application relates to the monodromy group of hyperk¨ahler varieties deformation eq”
A survey of Torelli and monodromy results for holomorphic-symplectic varieties (2011, background): “ by the author in an unpublished work. When n= 2, the equality N(X) = Mon2(X) follows from the Global Torelli Theorem 2.2 and Namikawa’s counter example to the naive Hodge theoretic Torelli statement [Nam2]. If H1,1(X,Z) is cyclic, generated by a class uwith (u,u) = 2, then ρu is induced by a regula”
Holomorphic symplectic geometry: a problem list (2010, background): “This is not the only obstruction: Namikawa observed [28] that if T is a 2-dimensional complex torus, and T∗ its dual torus, the Kummer manifolds K2(T) andK2(T) (1.”
K3 surfaces via almost-primes (2001, background): “ In addition, there are several lattices S, S′ and M such that S ≃ S′ but S ⊕ M ≃ S′ ⊕ M. So, Hilbert schemes of K3 surfaces found in (1.7) may not satisfy the condition of Propsition A. (2) Namikawa [Nm] ﬁnds a counter example of the birational injectivity of the period mapping for hyperka¨hler man”
A global Torelli theorem for hyperkahler manifolds (2009, unknown): “theorem holds. Remark 7.23: For other examples of hyperk¨ahler manifolds, the Hodge-theoretic global Torelli theorem is known to be false. For generalizedKummer varieties this was proven by Namikawa ([Na]), and for M= K3[n] this observation is due to Markman ([M2]). For O’Grady’s examples of hyperk¨”