We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup WExc, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kähler cone as a fundamental domain for the WExc action on the positive cone. A proof of a weak version of Morrison's movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV.
More generally, a Poisson structure on a manifold M is a Lie algebra bracket { , } on C ∞ (M) which acts as a derivation in each variable:Since the value at a point m of a given derivation acting on a function g is a linear function of d m g, we see that a Poisson structure on M determines a global 2-vectoror equivalently a skew-symmetric homomorphismConversely, any 2-vector ψ on M determines an alternating bilinear bracket on C ∞ (M), by {f, g} := (df ∧ dg, ψ), and this acts as a derivation in each variable. An equivalent way of specifying a Poisson structure is thus to give a global 2-vector ψ satisfying an integrability condition (saying that the above bracket satisfies the Jacobi identity, hence gives a Lie algebra).We saw that a symplectic structure σ determines a Poisson bracket { , }. The corresponding homomorphism Ψ is just (⌋σ) −1 ; the closedness of σ is equivalent to integrability of ψ. Thus, a Poisson structure which is (i.e. whose 2-vector is) everywhere non-degenerate, comes from a symplectic structure.A general Poisson structure can be degenerate in two ways: first, there may exist non-constant functions f ∈ C ∞ (M), called Casimirs, satisfying{f, g} = 0 for all g ∈ C ∞ (M).This implies that the rank of Ψ is less than maximal everywhere. In addition, or instead, rank Ψ could drop along some strata in M. For even r, letThen a basic result [We] asserts that the M r are submanifolds, and they are canonically foliated into symplectic leaves, i.e. r-dimensional submanifolds Z ⊂ M r which inherit a symplectic structure. (This means that the restriction ψ | Z is the image, under the inclusion Z ֒→ M r , of a two-vector ψ Z on Z which is everywhere nondegenerate, hence comes from a symplectic structure on Z.) These leaves can be described in several ways:• The image Ψ(T * M r ) is an involutive subbundle of rank r in T M r ; the Z are its integral leaves.• The leaf Z through m ∈ M r is Z = {z ∈ M r |f (m) = f (z) for all Casimirs f on M r }.
Let S be a K3 surface and AutD(S) the group of auto-equivalences of the derived category of S. We construct a natural representation of AutD(S) on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on S. The main result of this paper is the precise relation of this action with the monodromy of the Hilbert schemes S [n] of points on the surface. A formula is provided for the monodromy representation, in terms of the Chern character of the universal sheaf. Isometries of the second cohomology of S [n] are lifted, via this formula, to monodromy operators of the whole cohomology ring of S [n] .
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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