Hyperkaehler.info

Monodromy group

The monodromy group is generated by parallel transport operators in smooth families of hyperkähler manifolds containing $X$ as a fiber. One distinguishes the full cohomological monodromy \[ \operatorname{Mon}(X)\subseteq\operatorname{GL}(\operatorname{H}^\ast(X,\mathbb{Z})) \] and the degree-two monodromy \[ \operatorname{Mon}^2(X)\subseteq\operatorname{O}(\operatorname{H}^2(X,\mathbb{Z}),\mathrm{q}_X), \] where the Beauville–Bogomolov form $\mathrm{q}_X$ depends only on the deformation type. By definition, monodromy is a deformation invariant.

The discriminant group of an integral lattice $L$ is \[ \operatorname{A}_L := L^\vee\!/L, \] where $L^\vee=\operatorname{Hom}(L,\mathbb{Z})$ is the dual lattice. It is a finite abelian group whose order equals $|\det\operatorname{Gram}(L)|$. For $L=\operatorname{H}^2(X,\mathbb{Z})$ equipped with $\mathrm{q}_X$, the discriminant group $\operatorname{A}_{\operatorname{H}^2(X,\mathbb{Z})}$ is the group listed in the BBF table (e.g. $\mathbb{Z}/(2n-2)\mathbb{Z}$ for $\mathrm{K3}^{[n]}$-type).

Let \[ \mathrm{W}_X:=\{f\in \operatorname{O}^{+}(\operatorname{H}^2(X,\mathbb{Z}),\mathrm{q}_X)\mid f\text{ acts on }\operatorname{A}_{\operatorname{H}^2(X,\mathbb{Z})}\text{ as }\pm\operatorname{id}\}, \] be an index-$2^a$ subgroup of $\operatorname{O}^{+}$.

K3 surfaces

\[ \operatorname{Mon}^2(S)=\operatorname{Mon}(S)=\operatorname{O}^{+}(\operatorname{H}^2(S,\mathbb{Z}),\mathrm{q}_S). \] Here $\operatorname{O}^{+}$ denotes orientation-preserving isometries of the positive 3-plane in $\operatorname{H}^2(S,\mathbb{R})$, see [Chapter 7, Proposition 5.5, MR3586372].

K3^[n]-type

The restriction map $\operatorname{Mon}(X)\to\operatorname{Mon}^2(X)$ is an isomorphism, see [Lemma 2.1, MR4260432]. Moreover, \[ \operatorname{Mon}^2(X)=\mathrm{W}_X, \] by [Theorem 1.2 and Lemma 4.2, MR2650367]. In particular, $\operatorname{Mon}^2(X)=\operatorname{O}^{+}(\operatorname{H}^2(X,\mathbb{Z}),\mathrm{q}_X)$ when $n-1$ is a prime power.

Kumⁿ-type

Write $\chi$ for the character induced by the discriminant action. One has \[ \operatorname{Mon}^2(X)=\ker(\det\chi\colon\mathrm{W}_X\to\{\pm1\}), \] where $\mathrm{W}_X$ is as above (here $a+1$ is the number of distinct prime factors of $n+1$). See [Theorem 4.3, MR3504537].

OG₆-type

The monodromy is maximal: \[ \operatorname{Mon}^2(X)=\operatorname{O}^{+}(\operatorname{H}^2(X,\mathbb{Z}),\mathrm{q}_X), \] see [MR4197280].

OG₁₀-type

The monodromy is maximal: \[ \operatorname{Mon}^2(X)=\operatorname{O}^{+}(\operatorname{H}^2(X,\mathbb{Z}),\mathrm{q}_X), \] see [MR4484546]. The statement of [Theorem 5.3, MR3504537] is corrected in [MR4563000].

References

MR3586372

Huybrechts, D. (2016). Lectures on K3 surfaces (Vol. 158, p. xi+485). Cambridge University Press, Cambridge. doi:10.1017/CBO9781316594193 MR3586372

MR4260432

Markman, E. (2021). On the existence of universal families of marked irreducible holomorphic symplectic manifolds. Kyoto J. Math., 61(1), 207–223. doi:10.1215/21562261-2019-0075 MR4260432

MR2650367

Markman, E. (2010). Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of a K3 surface. Internat. J. Math., 21(2), 169–223. doi:10.1142/S0129167X10005957 MR2650367

MR3504537

Mongardi, G. (2016). On the monodromy of irreducible symplectic manifolds. Algebr. Geom., 3(3), 385–391. doi:10.14231/AG-2016-017 MR3504537

MR4197280

Mongardi, G., & Rapagnetta, A. (2021). Monodromy and birational geometry of O’Grady’s sixfolds. J. Math. Pures Appl. (9), 146, 31–68. doi:10.1016/j.matpur.2020.12.006 MR4197280

MR4484546

Onorati, C. (2022). On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces. J. Algebraic Geom., 31(3), 425–465. doi:10.1090/jag/802 MR4484546

MR4563000

Mongardi, G. (2023). Erratum: On the monodromy of irreducible symplectic manifolds (Algebraic Geometry 3 (2016), no. 3, 385–391). Algebr. Geom., 10(2), 259–261. doi:10.14231/AG-2023-008 MR4563000