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We show that any continuous $\mathbf{C}$-linear Lie algebra splitting of the symbol map from the Atiyah algebra of a vector bundle on a complex manifold is given by a differential operator of order at most the rank of the bundle plus one.…

Let $B$ be a smooth projective curve of genus $g$, and $S \subset B$ be a finite subset of cardinality $s$. We give an effective upper bound on the number of deformation types of admissible families of canonically polarized manifolds of…

Let $C$ be a curve with two smooth components and a single node. Let $\mathcal{U}_C(r,w,\chi)$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $\chi$. In…

We reformulate base point free theorems. Our formulation is flexible and has some important applications. One of the main purposes of this paper is to prove a generalization of the base point free theorem in Fukuda's paper: On numerically…

The factorial hull of the projective variety X (or its cone) is a graded algebra R(X) that can be used in some situations to consider simultaneously all divisor classes on X. Associated with X is a certain cone H in the divisor class group…

Let $(X,H)$ be a smooth, projective, polarized surface over $\mathbb{C}$, and let $v \in K_{\mathrm{num}}(X)$ be a class of positive rank. We prove that for certain Bridgeland stability conditions $\sigma = (\mathcal{A}, Z)$ "on the…

We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact K\"ahler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that…

Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of…

We establish criteria for when Noether--Lefschetz divisors generate an extremal ray in the cone of pseudoeffective divisors of an orthogonal modular variety. In particular, we exhibit many extremal rays of the cone of pseudoeffective…

Let $\pi:Z\rightarrow\mathbb{P}^{n-1}$ be a general minimal $n$-fold conic bundle with a hypersurface $B_Z\subset\mathbb{P}^{n-1}$ of degree $d$ as discriminant. We prove that if $d\geq 4n+1$ then $-K_Z$ is not pseudo-effective, and that if…

We prove an analogue of the Lefschetz (1,1) Theorem characterizing cohomology classes of Cartier divisors (or equivalently first Chern classes of line bundles) in the second integral cohomology. Let $X$ be a normal complex projective…

We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the…

We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions…

The main goal of this article is to compute the class of the divisor of $\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of $\Omega\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the…

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$…

Let $X$ be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety $Y$ that is birational to $X$ and such that some multiple of its anticanonical divisor…

Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to $k$ times the canonical divisor for $k \in \mathbb{Z}_{\geq 1}$. In this article we give a purely combinatorial criterion…

We show that the total cohomology of the canonical bundle of a smooth projective variety, seen as a module over an exterior algebra, splits into a natural direct sum of submodules which are generated in degree zero and have a linear free…

Let $E$ be the Whitney sum of complex line bundles over a topological space $X$. Then, the projectivization $P(E)$ of $E$ is called a \emph{projective bundle} over $X$. If $X$ is a non-singular complete toric variety, so is $P(E)$. In this…

We prove an equivalence between two approaches to characterizing complex-projective varieties $X$ with klt singularities and ample canonical divisor that are uniformized by bounded symmetric domains. In order to do so, we show how to…