DEGENERATIONS OF AUTOMORPHISMS ON IRREDUCIBLE HOLOMORPHIC SYMPLECTIC VARIETIES

The aim of this thesis is to study the degeneration of special automorphisms (termed non-symplectic) on certain families of irreducible holomorphic symplectic varieties (IHS varieties).\newline\newline In fact, the moduli space of smooth cubic threefolds is isomorphic to a moduli space $\mathcal{N}^{\rho, \zeta}_{\langle 6\rangle}$ of IHS varieties equipped with a special order 3 non-symplectic automorphism. This is a result by Boissière--Camere--Sarti, \cite{BCS}. In the same paper, the authors extended the result to the general case with a so-called nodal singularity. Extending this map primarily means studying the limit of a one-parameter degeneration whose central point is the period of a nodal cubic threefold. What happens is that if we consider a one-parameter family in $\mathcal{N}^{\rho, \zeta}_{\langle 6\rangle}$ that has a nodal period as its limit, it turns out that at the limit the family degenerates into a family of IHS varieties with an order 3 non-symplectic automorphism with a larger invariant. In this sense, we say the automorphism \emph{degenerates}. Secondly, it means providing a map (which in this case is birational) between the nodal locus and a suitable moduli space of IHS varieties possessing a non-symplectic order 3 automorphism, with precisely a larger invariant.\newline\newline The first part of the thesis is dedicated to finding a similar result for non-generic nodal cubics. In particular, by going to increasingly higher codimension, a birational map is found between the subspaces of the nodal locus whose generic element is a cubic with a single singularity $A_i$ for $i=2,3,4$, and a moduli space of IHS varieties of type $K3^{[2]}$ with a non-symplectic $\rho_i$ automorphism of order 3. To achieve this, we employ techniques developed by Boissière, Camere, and Sarti to arrive at a bijective restriction of the period map.\newline\newline The second part, in collaboration with Boissière and Comparin, is dedicated to a detailed study of the geometry of the limit which can in fact be expressed as the symplectic resolution of the Fano variety of lines lying on the cyclic cubic fourfold, that is, a cyclic 3-to-1 cover of $\mathbb{P}^4$ branching over a cubic threefold which in this case has isolated singularities of type $A_i$ for $i=2,3,4$.