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.

In 2004, Herzog, Hibi, and Zheng proved that a quadratic monomial ideal has a linear resolution if and only if all its powers have a linear resolution. We study a generalization of this result for square-free monomial ideals arising from facet ideals of a simplicial tree. We give a complete characterization of simplicial trees for which all powers of their facet ideal have a linear resolution. We compute the regularity of t-path ideals of rooted trees. In addition, we study the regularity of powers of t-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $\Delta $ is a d-dimensional simplicial tree connected in codimension one, then reg$(I(\Delta )^s) \leq (d+1)(s-1)~+$ reg$(I(\Delta ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees.

We introduce the notion of adjacency in three-manifolds. A three-manifold Y is n-adjacent to another three-manifold Z if there exists an n-component link in Y and surgery slopes for that link such that performing Dehn surgery along any nonempty sublink yields Z. We characterize adjacencies from three-manifolds to the three-sphere, providing an analogy to Askitas and Kalfagianni’s results on n-adjacency in knots.

We show that a divisor in a rational homogenous variety with split normal sequence is the preimage of a hyperplane section in either the projective space or a quadric.

In this article, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For a threefold, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one-dimensional Albanese fiber or one-dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a “bi-fibration” structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.

We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogs of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

Using metric techniques introduced by Berndtsson, we show a result on the constancy of families dominated by a constant variety and, on the opposite side, results on the strong non-isotriviality of certain families of surfaces with positive index and, in arbitrary dimension, in terms of the complex conjugate of a suitable representative of the Kodaira–Spencer class. We also give a metric interpretation of the liftability of relative volume forms.

The Askey–Wilson algebras illustrate the bispectral property of orthogonal polynomials in the Askey scheme. The universal Askey–Wilson algebra $\triangle _q$ is a central extension of the Askey–Wilson algebras associated with the most general orthogonal polynomials in the Askey scheme. The Verma $\triangle _q$-modules are a family of infinite-dimensional $\triangle _q$-modules with marginal weights. Under the condition that q is not a root of unity, it was shown that every finite-dimensional irreducible $\triangle _q$-module has a marginal weight and is isomorphic to a quotient of a Verma $\triangle _q$-module. Assume that q is a root of unity. We prove that every finite-dimensional irreducible $\triangle _q$-module with a marginal weight is isomorphic to a quotient of a Verma $\triangle _q$-module. More precisely, two natural families of finite-dimensional quotients of Verma $\triangle _q$-modules contain all finite-dimensional irreducible $\triangle _q$-modules with marginal weights up to isomorphism. Furthermore, we classify the finite-dimensional irreducible $\triangle _q$-modules with marginal weights up to isomorphism.

For the module category of an Artin algebra, we generalize the notion of torsion pairs to ideal torsion pairs. Instead of full subcategories of modules, ideals of morphisms of the ambient category are considered. We characterize the functorially finite ideal torsion pairs, which are those fulfilling some nice approximation conditions, first through corresponding functors and then through the notion of ideals determined by objects introduced in this work. As an application of this theory, we generalize preprojective modules, introduce a new homological dimension, the torsion dimension, and establish its connection with the Krull–Gabriel dimension. In particular, it is shown that both dimensions coincide for hereditary Artin algebras.

We present a unified construction of perfectoid towers from specific prisms which covers all the previous constructions of (p-torsion-free) perfectoid towers. By virtue of the construction, perfectoid towers can be systematically constructed for a large class of rings with Frobenius lift. Especially, any Frobenius lifting of a reduced $\mathbb {F}_p$-algebra has a perfectoid tower.

Let $S_k$ denote the space of cusp forms of weight k and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $\chi $ mod D, we introduce twisted periods $r_{t,\chi }$ on $S_k$. We show that for a fixed natural number n, if k is sufficiently large relative to n and D, then any n periods with the same twist but different indices are linearly independent. We also prove that if k is sufficiently large relative to $D,$ then any n periods with the same index but different twists mod D are linearly independent. These results are achieved by studying the trace of the products and Rankin–Cohen brackets of Eisenstein series of level D with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda’s conjecture implies a non-vanishing result on twisted central L-values of normalized Hecke eigenforms.

The notion of faithful flatness of a module over a commutative ring is studied for two R-modules M arising in functional analysis, where R is a Banach algebra and M is a Hilbert space. The following results are shown:

If X is a locally compact Hausdorff topological space, and $\mu $ is a positive Radon measure on X, then $L^2(X,\mu )$ is a flat $L^\infty (X,\mu )$-module. Moreover:

• • If $\mu $ is $\sigma $-finite, then for every finitely generated, nonzero, proper ideal $\mathfrak {n}$ of $L^\infty (X,\mu )$, there holds $\mathfrak {n}L^2(X,\mu )\subsetneq $ $L^2(X,\mu )$.

• • If X is the union of an increasing family of Borel sets $U_n$, $n\!\in \! {\mathbb {N}}$, such that for each $n\in {\mathbb {N}}$, $\overline {U_n}$ is compact and $\mu (U_{n+1}\setminus U_n)>0$, then $L^2(X,\mu )$ is not a faithfully flat $L^\infty (X,\mu )$-module.

$H^2$

$H^\infty $

We describe the behavior of a free reduced plane projective curve with respect to the addition, respectively, deletion, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.