The automorphism groups of compact complex manifolds are an interesting and classical topic. In this paper, the authors study such groups for two isolated classes of hyperkähler manifolds, the so-called O’ Grady’s manifolds of dimensions 6 and 10. Hyperkähler manifolds are generalisations to higher dimensions of \(K3\) surfaces, and have very special properties. So it is expected that their automorphism groups have also special properties. The current paper contributes to vindicate this speculation, showing that to some extend the automorphisms of O’Grady’s manifolds can be understood from the pullbacks on the second cohomology group. More precisely, let \(X\) be a compact Kähler manifold and denote by \(\mathrm{Aut}(X)\) the group of automorphisms of \(X\). By pulling back we obtain a representation \(\nu :\mathrm{Aut}(X)\rightarrow \mathrm{GL}(H^2(X,\mathbb{Z}))\). The main result of the paper under review is the following: If \(X\) is a deformation of O’Grady’s manifold of dimension \(10\) then \(\nu\) is injective. On the other hand, if \(X\) is a deformation of O’Grady’s manifold of dimension \(6\) then the kernel of \(\nu\) is isomorphic to \((\mathbb{Z}/2)^8\).
This is a new addition to this subject, besides previous results by A. Beauville [Prog. Math. 39, 1–26 (1983; Zbl 0537.53057)] (who showed that if \(X\) is a deformation of Hilbert scheme of points on a \(K3\) surfaces then \(\nu\) is injective) and S. Boissière et al. [J. Math. Pures Appl. (9) 95, No. 5, 553–563 (2011; Zbl 1215.14046)] (who showed that if \(X\) is a deformation of generalised Kummer varieties of dimension \(n\) then \(\nu\) is generated by translations by points of order \(n\) and the map \(-\mathrm{id}\), both coming from the underlying abelian surface). Note that the above mentioned manifolds are an exhausting list of known hyperkähler manifolds. Note also that in the case of deformations of generalised Kummer surfaces, Oguiso showed that if we consider instead the representation on the whole cohomology ring: \(\tau :\mathrm{Aut}(X)\rightarrow \mathrm{GL}(\bigoplus _iH^i(X,\mathbb{Z}))\), then \(\tau\) is injective.
The key tools in the proofs of the main results in this paper are the following. The first one is a result due to B. Hassett and Y. Tschinkel [Mosc. Math. J. 13, No. 1, 33–56 (2013; Zbl 1296.14008)], which asserts that the kernel of the above representation \(\nu\) is a deformation invariant, hence it suffices to prove the results for special manifolds in the deformation. Then the special manifolds chosen are those constructed via the help of moduli spaces of stable sheaves associated with a Mukai vector on a \(K3\) surface or abelian variety.
In the case of dimension 10, the special manifold \(X\) mentioned above is the resolution of singularity such a moduli space on a special \(K3\) surface \(S\) which is a double cover of \(\mathbb{P}^2\) branched along a sextic curve \(\Gamma\) with a unit tritangent. In this case, the moduli space in question is the relative Jacobian \(J^4(|2H|)\) of degree \(4\) over the linear system \(|2H|\), where \(H\) is the pullback to \(S\) of a line in \(\mathbb{P}^2\). That is, we have a map \(\pi :J^4(|2H|)\rightarrow |2H|\), where for a general curve \(C\) in \(|2H|\), the fibre \(\pi ^{-1}(C)\) is isomorphic to the Jacobian. [Note that such a curve \(C\) is a covering map of degree \(2\) over a smooth quadric curve in \(\mathbb{P}^2\), and is branched over \(12\) points (\(=\) the intersection between the quadric curve and the sextic \(\Sigma\)), hence is of genus \(5\) because of the Riemann-Hurwitz theorem.]
Let \(\psi\) be an automorphism of \(X\) which acts trivially on the cohomology group \(H^2(X,\mathbb{Z})\). Then \(\psi \) fixes the classes of exceptional divisors of the blowup \(X\rightarrow J^4(|2H|)\), hence it descends to an automorphism \(\psi '\) of \(J^4(|2H|)\), and still acts trivially over \(H^2(J^4(|2H|),\mathbb{Z})\). In particular, \(\psi '\) fixes the class of \(\pi ^*(\mathcal{O} (1))\). Therefore, \(\psi '\) maps fibres to fibres.
The authors proceed to show that the relative theta divisor \(\Theta _{|2H|}\) of \(J^4(|2H|)\) (whose fibre over a general curve \(C\) is the theta divisor of \(C\)) is rigid. This means that \(\Theta _{|2H|}\) is the unique effective divisor in its cohomology class. Hence, the relative theta divisor is fixed by \(\psi '\). In particular, if the fibre \(J^4(C_1)\) is mapped to the fibre \(J^4(C_2)\), where \(C_1\) and \(C_2\) are general elements in \(|2H|\), then the class of \(\Theta _{C_1}\) is mapped to the class of \(\Theta _{C_2}\). Since theta divisors give rise to polarisation on Jacobian of smooth curves, by Torelli theorem we have that \(C_1\) and \(C_2\) are isomorphic. The authors then show that with the special choice of \(\Gamma\) as above, then this implies that \(C_1=C_2\). That is, the map \(\psi '\) maps each fibre into itself. Since \(\psi '\) fixes the relative theta divisor, the map \(\psi ':J^4(C)\rightarrow J^4(C)\) cannot be given by translations. Therefore, by Torelli’s theorem again and the assumption that \(\psi '\) acts trivially on cohomology, it follows that \(\psi '\) is the identity.
The case of dimension 6 is more complicated. The author works similarly as above, but now the \(K3\) surface \(S\) is replaced by an abelian surface \(A\). Here \(A\) is the Jacobian of a curve of genus \(2\), and choose \(H\) the principal polarisation on \(A\) given by a symmetric theta divisor. By adjunction formula, it follows that \(H.H=2\) and hence for a curve \(C\) in \(|2H|\), the arithmetic genus of \(C\) is \(5\) (again by the adjunction formula). By Hirzerbruch-Riemman-Roch theorem and Kodaira vanishing theorem, it follows that \(h^0(2L)=4\). Hence the similar constructed \(J^4(|2H|)\) has dimension \(8\). There is an associated Albanese map \(J^4(|2H|)\rightarrow A^*\) (\(A^*\) is the dual of \(A\)), and the O’Grady’s manifold \(X_6\) of dimension \(6\) is the resolution of a generic fibre \(K^4(|2H|)\) of this map. The author then proceeds, using a result by K. Yoshioka [“Albanese map of moduli of stable sheaves on abelian surfaces”, arXiv:math/9901013], that if \(G_0\) is the group generated by points of order \(2\) in \(A\times A^*\), then \(G_0\) induces an action on \(X_6\) which acts trivially on \(H^2(X_6,\mathbb{Z})\). To finish the proof, it suffices to show that \(G_0\) is exactly the kernel of \(\nu\). If \(\psi\) is in the kernel of \(\nu\), then \(\psi\) descends to an automorphism of \(K^4(|2H|)\), and the fibration \(\pi :K^4(|2H|)\rightarrow |2H|\) is \(\psi\)-equi-invariant. In particular, \(\psi\) induces an automorphism \(\psi _0\) of \(|2H|=\mathbb{P}^3\). In \(|2H|\), there is a natural stratification, depending on the type of the curve \(C\in |2H|\), and it is observed that \(\psi _0\) must preserve that stratification. In this stratification, there are types \(R(1)\) and \(D\): the closure of \(R(1)\) is isomorphic to the Kummer surface \(\mathrm{Kum}_s(A)\) associated to \(A\), and \(D\) corresponds to the \(16\) notes of this Kummer surface. Then the automorphism \(\psi _0\) preserves \(\mathrm{Kum}_s(A)\) and also nodes, and hence it can be lifted to an automorphism of the abelian surface \(A\) given by translations of points of order \(2\) in \(A\). Hence, we may assume that the action on \(2H\) is trivial, by composing \(\psi\) with an appropriate element in \(A[2]\). This means that for smooth generic \(C\) in \(|2H|\), the automorphism \(\psi\) preserves the fibre \(K^4(C)\). Then the authors showed similarly that the relative theta divisor on \(K^4(|2H|)\) is an effective rigid divisor. Let \(\psi _C\) be the restriction of \(\psi\) on \(K^4(C)\). Then \(\psi _C\) acts trivially on \(H^2\), and cannot be given as \(-id\), and hence since \(K^4(C)\) is a simple abelian variety, it follows that \(\psi _C\) must be the translation \(t_x\) of some point \(x\) of finite order (since kernel of \(\nu\) is finite). This, together with the fact that the relative theta divisor is rigid, finishes the proof.
In the last section (Section 6) of the paper, the authors consider the fixed locus of the kernel of \(\nu\) on \(X_6\).