In this article, we investigate the $L^2$-Dolbeault cohomology of the symmetric power of cotangent bundles of ball quotients with finite volume, as well as their toroidal compactification. Moreover, by proving the finite dimensionality of these cohomologies, through the application of Hodge theory for complete Hermitian manifolds, we establish the existence of Hodge decomposition and Green’s operator.

Consider a pair of elements f and g in a commutative ring Q. Given a matrix factorization of f and another of g, the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum $f+g$. We will study the tensor product of d-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen–Macaulay and Ulrich modules over hypersurface domains of a certain form.

This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called “level” in a triangulated category can be used to measure the failure of a variety to have a prescribed singularity type. We provide explicit computations of this invariant for reduced Nagata schemes of Krull dimension one and for affine cones over smooth projective hypersurfaces. Furthermore, these computations are utilized to produce upper bounds for Rouquier dimension on the respective bounded derived categories.

Boundary points on the moduli space of pointed curves corresponding to collisions of marked points have modular interpretations as degenerate curves. In this paper, we study degenerations of orbifold projective curves corresponding to collisions of stacky points from the point of view of noncommutative algebraic geometry.

Given a presilting object in a triangulated category, we find necessary and sufficient conditions for the existence of a complement. This is done both for classic (pre)silting objects and for large (pre)silting objects. The key technique is the study of associated co-t-structures. As a consequence of our techniques we recover some known cases of the existence of complements, including for derived categories of some hereditary abelian categories and for silting-discrete algebras. Moreover, we also show that a finite-dimensional algebra is silting discrete if and only if every bounded large silting complex is equivalent to a compact one.

We study the singularities of varieties obtained as infinitesimal quotients by $1$-foliations in positive characteristic. (1) We show that quotients by (log) canonical $1$-foliations preserve the (log) singularities of the MMP. (2) We prove that quotients by multiplicative derivations preserve many properties, amongst which most F-singularities. (3) We formulate a notion of families of $1$-foliations, and investigate the corresponding families of quotients.

We propose and present evidence for a conjectural global-local phenomenon concerning the p-rationality of height-zero characters. Specifically, if $\chi $ is a height-zero character of a finite group G and D is a defect group of the p-block of G containing $\chi $, then the p-rationality of $\chi $ can be captured inside the normalizer ${\mathbf {N}}_G(D)$.

We construct a monoidal version of Lurie’s un/straightening equivalence. In more detail, for any symmetric monoidal $\infty $-category $\mathbf {C}$, we endow the $\infty $-category of coCartesian fibrations over $\mathbf {C}$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\infty $-category of functors from $\mathbf {C}$ to $\mathbf {Cat}_\infty $. In fact, we do this over any $\infty $-operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\infty $-category $\mathbf {C}$ its category of coCartesian fibrations on the one hand, and its category of functors to $\mathbf {Cat}_\infty $ on the other hand.