Related papers: Non-commutative crepant resolutions for some toric…

We give an expository account of a conjecture, developed by Coates--Corti--Iritani--Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold X to the quantum cohomology of a crepant resolution Y of X. We explore some…

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with…

In this note we prove that the crepant transformation conjecture for a crepant birational transformation of Lawrence toric DM stacks studied in \cite{CIJ} implies the monodromy conjecture for the associated wall crossing of the symplectic…

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe…

We give a bound on the number of isolated, essential singularities of determinantal quartic surfaces in 3-space. We also provide examples of different configurations of real singularities on quartic surfaces with a definite Hermitian…

We establish the nonexistence of extremizers for a local Fourier restriction inequality on a certain class of planar convex curves whose curvature satisfies a natural assumption. We accomplish this by studying the local behavior of the…

In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the…

We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower…

We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space P(1,3,4,4). To compute the quantum corrected cohomology ring we combine the results of Coates-Corti-Iritani-Tseng on P(1,1,1,3) and our…

We construct modular resolutions of singularities for splitting loci, and use them to show that tame splitting loci have rational singularities. As a corollary of our results and Hurwitz-Brill-Noether theory, we prove that if $C$ is a…

The main purpose of this note is to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as, symmetric positive definite tensors and copositive tensors, we derive the existence theorems on a…

We show that every small resolution of a three-dimensional terminal hypersurface singularity can occur on a non-embeddable 1-convex manifold. We give an explicit example of a non-embeddable manifold containing an irreducible exceptional…

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant…

Let $R$ be a 3-dimensional complete local Gorenstein isolated singularity. For a basic maximal modifying $R$-module $M$, we construct a wall-and-chamber structure, denoted by ${\sf Cone}(M)$ and called the mutation cone of $M$, in the real…

In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.

In this article some noncommutative topological objects such as NC mapping cone and NC mapping cylinder are introduced. We will see that these objects are equipped with the NCCW complex structure of [PEDERSEN]. As a generalization we…

In this paper we prove criteria for a nonnormal toric variety to be flexible, to be rigid and to be almost rigid. For rigid and almost rigid toric varieties we describe the automorphism group explicitly.

Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian…

We give algebraic and geometric classifications of complex $4$-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative…

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without…