arXiv subjects
Publications and source records attributed to Morgan Brown.
We study the higher dimensional geometry of Berkovich spaces using virtual open disks, which are given by fibration of relative dimension $1$. Inspired by birational geometry, we conjecture that the Berkovich skeleton is the complement of the union of all virtual open disks, and prove this conjecture for $\mathcal{X}$ admitting a strictly semistable model with semiample canonical class.
We study the problem of how the dual complex of the special fiber of an snc degeneration $\cX_R$ changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to $X_K$. Using the Kato fan, we define a skeleton $\Sk(\cX_R)$ when the model $\cX_R$ is log-regular. We show that if $\cX_R$ and $\cY_R$ are log-regular, and at least one is semistable, then $\Sk(\cX_R\times_R \cY_R) \simeq \Sk(\cX_R)\times \Sk(\cY_R)$. The essential skeleton $\Sk(X_K)$, defined by Musta\c{t}\u{a} and Nicaise, is a birational invariant of $X_K$ and is independent of the choice of $R$-model. We extend their definition to pairs, and show that if both $X_K$ and $Y_K$ admit semistable models, $\Sk(X_K\times_K Y_K) \simeq \Sk(X_K)\times \Sk(Y_K)$. As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-K{\"a}hler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the $2n$-dimensional degeneration is homeomorphic to either a point, $n$-simplex, or $\mathbb{C}\mathbb{P}^n$, depending on the type of the degeneration.
We prove a conjecture of Shokurov which characterises toric varieties using log pairs.
Let k be an algebraically closed field of characteristic 0, and let f be a morphism of smooth projective varieties from X to Y over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the Berkovich analytifications of X and Y are homotopy equivalent. Two important consequences of this result are that the homotopy type of the Berkovich analytification of any smooth projective variety X over k((t)) is a birational invariant of X, and that the Berkovich analytification of a rationally connected smooth projective variety over k((t)) is contractible.
We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.