Related papers: On semi-continuity problems for minimal log discre…

Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, \Delta)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an…

We construct a moduli space of stable projective pairs with a nontrivial action of a connected reductive group. These stable reductive pairs are higher-dimensional analogs of stable n-pointed curves and generalize to the non-commutative…

A conjecture by Corti, Filip and Petracci, inspired by mirror symmetry, states that smoothing types of affine Gorenstein toric 3-folds correspond to zero mutable Laurent polynomials. We propose a method to prove this conjecture via log…

We prove the existence of positive solutions for a class of semipositone problem with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.

We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.

We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.

We give conditions for a n-connective quasicoherent obstruction theory on a Deligne-Mumford stack to come from the structure of a connective spectral Deligne-Mumford stack on the underlying topos.

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne-Mumford stability. For every pair of integers 0<m<n, we…

Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant…

In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental…

After establishing suitable notions of stability and Chern classes for singular pairs, we use K\"ahler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal…

In this paper, we prove the termination of 4-fold semi-stable log flips under the assumption that there always exist 4-fold (semi-stable) log flips.

We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation…

L. Szpiro and T. Tucker recently proved that under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between…

We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror…

In the famous paper of Deligne and Mumford, they proved that a proper hyperbolic curve over a discrete valuation field has stable reduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue…

We present an extension of several results on pairs and varieties to foliated surface pairs. We prove the boundedness of local complements, the local index theorem, and the uniform boundedness of minimal log discrepancies (mlds), as well as…

We translate the equivariant decomposition theorem (in the case of a proper morphism of toric varieties) in to the language of combinatorially defined ``shifted minimal complexes''.

In this article we treat two closely related problems: 1) the upper semi continuity property for Almgren minimal sets in regions with regular boundary, which guanrantees that the uniqueness property is well defined; and 2) the Almgren…

The Deligne-Mumford stable reduction theorem asserts that for a family of stable curves over the punctured disk, after a finite base change, the family can be completed in a unique way to a family of stable curves over the disk. In this…