arXiv subjects
Publications and source records attributed to Tomasz Wawak.
We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-Kähler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension $4$ with $b_2 = 4$ and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.
We construct two examples of projective hyper-Kähler fourfolds of K3[2]-type with an action of the alternating group A7, making them some of the most symmetric hyper-Kähler fourfolds. They are realized as so called double EPW sextics and this allows us to construct an explicit family of irrational Gushel-Mukai threefolds.
We give a classification of finite groups of symplectic birational automorphisms on a manifold of K3^[3]-type with stable and stably saturated cohomological action. We describe the group of polarized automorphisms of a smooth double EPW-cube. Using this description, we exhibit examples of projective hyperkaehler manifolds of K3^[3]-type of maximal Picard rank with a symplectic action of a large group.
G. Höhn and G. Mason classified all finite groups acting faithfully and symplectically on a hyper-K{ä}hler fourfolds of type K3$^{[2]}$. There are 15 maximal among them, call them $\widetilde{G}_1,\ldots, \widetilde{G}_{15}$. Every manifold of type K3$^{[2]}$ admitting an action of $\widetilde{G}_i$ for some $i$ must necessarily have Picard rank 21 which is maximal. This fact allows us to use lattice-theoretic methods to classify all the finite groups $G$ acting faithfully on a hyper-K{ä}hler fourfold of type K3$^{[2]}$ $X$ such that $G$ contains $\widetilde{G}_i$ as a proper subgroup and $\widetilde{G}_i$ acts symplectically on $X$. We also describe examples of fourfolds of K3$^{[2]}$-type admitting an action of such groups.
This paper is the first of a series dealing with c-holomorphic functions defined on algebraic sets and having algebraic graphs. These functions may be seen as the complex counterpart of the recently introduced \textit{regulous} functions. Herein we study their growth exponent at infinity. A general result on injectivity on fibres of an analytic set together with a theorem of Tworzewski and Winiarski gives a bound for the growth exponent of a c-holomorphic function with algebraic graph in terms of the projective degrees of the sets involved. We prove also that algebricity of the graph is equivalent to the function being the restriction of a rational function (a Serre-type theorem). Then we turn to considering generically finite c-holomorphic mappings with algebraic graphs and we prove a Bézout-type theorem. We also study a particular case of the Łojasiewicz inequality at infinity in this setting.