We prove that the functor of noncommutative deformations of every flipping or flopping
irreducible rational curve in a 3-fold is representable, and hence, we associate to every such …

We show that the moduli stacks of semistable sheaves on smooth projective varieties are
analytic locally on their coarse moduli spaces described in terms of representations of the …

We study derived categories of Gorenstein varieties X and X^+ connected by a flop. We
assume that the flopping contractions f: X\to Y, f^+: X^+\to Y have fibers of dimension …

We define and study refined Gopakumar-Vafa invariants of contractible curves in complex
algebraic 3-folds, alongside the cohomological Donaldson--Thomas theory of finite …

Noncommutative Deformation Theory is aimed at mathematicians and physicists studying
the local structure of moduli spaces in algebraic geometry. This book introduces a general …

Suppose that $ f $ is a projective birational morphism with at most one-dimensional fibres
between $ d $-dimensional varieties $ X $ and $ Y $, satisfying ${\bf R} f_*\mathcal {O} …

The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-
1) $-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d …

This is an expository survey article on compound Du Val (cDV) singularities, with emphasis
on recent homological approaches, including: noncommutative resolutions, tilting theory …

Suppose that f is a projective birational morphism with at most one-dimensional fibres
between d-dimensional varieties X and Y, satisfying ${\bf R} f_*\mathcal {O} _X=\mathcal {O} …

We define the functor $\textrm {ncDef} _ {(Z_1,\ldots, Z_n)} $ of non-commutative
deformations of an $ n $-tuple of objects in an arbitrary $ k $-linear abelian category …