Related papers: Factorization for stacks and boundary complexes

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks $D^n$, when the later are given their natural…

To any affine scheme with a $\mathbb{G}_m$-action, we provide a Bousfield colocalization on the equivariant derived category of modules by constructing, via homotopical methods, an idempotent integral kernel. This endows the equivariant…

This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then…

Let $\cMx$ be the moduli space of stable vector bundles of rank $n\geq 3$ and determinant $\xi$ over a connected Riemann surface $X$, with $n$ and $d(\xi)$ coprime. Let $D$ be a Calabi-Yau hypersurface of $\cMx$. Denote by $U_D$ the…

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a…

The Gromov-Witten theory of Deligne-Mumford stacks is a recent development, and hardly any computations have been done beyond 3-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed…

Discrete transforms such as the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) are important tools in numerical analysis. The successful application of transform techniques relies on the existence of efficient…

Isolated hypersurface singularities come equipped with a Milnor lattice, a ${\mathbb Z}$-lattice of finite rank, and a set of $distinguished$ ${\mathbb Z}$-bases of this lattice. Usually these bases are constructed from $one$ morsification…

Gradient descent for matrix factorization exhibits an implicit bias toward approximately low-rank solutions. While existing theories often assume the boundedness of iterates, empirically the bias persists even with unbounded sequences. This…

We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective…

Given a compact complex manifold $X$, we prove a Baum-Bott type formula for one-dimensional holomorphic foliations on $X$ that are logarithmic along a hypersurface with isolated singularities. We show that the residues of these foliations…

We consider a Deligne-Mumford stack $X$ which is the quotient of an affine scheme $\operatorname{Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$…

Families of stable curves of genus $\gamma$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_\gamma$. It is natural to compactify the corresponding moduli problem using stable maps…

We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We…

A number of compactifications familiar in complex-analytic geometry, in particular, the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose nonempty…

We show that the moduli stacks of smooth complete intersections in P^N polarized by O(1) are separated (except for quadrics) and Deligne-Mumford (apart from a few exceptions). ----- On montre que les champs de modules d'intersections…

We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we…

Studying crepant blow-ups of (compound) du Val singularities, we classify complexes of coherent sheaves which admit no negative self-extensions -- such a complex, up to flops and mutation equivalences, must either be (1) a module over a…

We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new…

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…