We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a generalized version of Hochschild homology. Furthermore, using techniques coming from birational geometry, we establish the derived invariance of the Albanese dimension for varieties having nonnegative Kodaira dimension. We apply our machinery to study the derived invariance of the holomorphic Euler characteristic and of certain Hodge numbers for special classes of varieties. Further applications concern the behavior of particular types of fibrations under derived equivalence.

We determine the limiting distribution of the normalized Euler factors of an abelian surface $A$ defined over a number field $k$ when $A$ is $\overline{\mathbb{Q}}$-isogenous to the square of an elliptic curve defined over $k$ with complex multiplication. As an application, we prove the Sato–Tate conjecture for Jacobians of $\mathbb{Q}$-twists of the curves $y^2=x^5-x$ and $y^2=x^6+1$, which give rise to 18 of the 34 possibilities for the Sato–Tate group of an abelian surface defined over $\mathbb{Q}$. With twists of these two curves, one encounters, in fact, all of the $18$ possibilities for the Sato–Tate group of an abelian surface that is $\overline{\mathbb{Q}}$-isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato–Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato–Tate group of its Jacobian.

We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no nontrivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski-dense and any infinite subset of a single orbit is also Zariski-dense, thereby verifying the dynamical “Manin–Mumford” conjecture of Zhang and the dynamical “Mordell–Lang” conjecture of Denis and Ghioca and Tucker in this case.

Consider tuples $\left(K_1,\dots ,K_r\right)$ of separable algebras over a common local or global number field $F$, with the $K_i$ related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of $K_i∕F$. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.

We apply inequalities from the theory of linear forms in logarithms to deduce effective results on $S$-integral points on certain higher-dimensional varieties when the cardinality of $S$ is sufficiently small. These results may be viewed as a higher-dimensional version of an effective result of Bilu on integral points on curves. In particular, we prove a completely explicit result for integral points on certain affine subsets of the projective plane. As an application, we generalize an effective result of Vojta on the three-variable unit equation by giving an effective solution of the polynomial unit equation $f\left(u,v\right)=w$, where $u$, $v$, and $w$ are $S$-units, $\left|S\right|\le 3$, and $f$ is a polynomial satisfying certain conditions (which are generically satisfied). Finally, we compare our results to a higher-dimensional version of Runge’s method, which has some characteristics in common with the results here.

We prove a Lefschetz hypersurface theorem for abelian fundamental groups allowing wild ramification along some divisor. In fact, we show that isomorphism holds if the degree of the hypersurface is large relative to the ramification along the divisor.

We prove some fundamental structural results for spherical varieties in arbitrary characteristic. In particular, we study Luna’s two types of localization and use them to analyze spherical roots, colors, and their interrelation. At the end, we propose a preliminary definition of a $p$-spherical system.

Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with residue field ${\mathbb{F}}_q$, and let $\ell$ be a prime such that $q\equiv 1\left(mod\ell \right)$. We investigate the cohomology of the Lubin–Tate towers of $K$ with coefficients in ${\overline{\mathbb{F}}}_{\ell }$, and we show how it encodes Vignéras’ Langlands correspondence for unipotent ${\overline{\mathbb{F}}}_{\ell }$-representations.

After constructing a splitting tower for separable commutative ring objects in tensor-triangulated categories, we define and study their degree.