We answer an open problem posed by Iarrobino, Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert', in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276] $\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$ with dimension less than $(n-1)(d-1)$ ? For each $n\geq 4$ , we construct an infinite class of elementary components in $\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$ producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev, Power sums, Gorenstein algebras, and determinantal loci (Springer-Verlag, Berlin, 1999), 221–226] of the form with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension $2$ .

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$-Kähler structures with the $(p,p+1)$th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$-lemma under small differentiable deformations.

We analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.

We construct an algebraic almost closed 1-form with zero scheme not expressible (even locally) as the critical locus of a holomorphic function on a non-singular variety. The result answers a question of Behrend–Fantechi. We correct here an error in our paper (D. Maulik, R Pandharipande and R. P. Thomas, Curves on K3 surfaces and modular forms, J. Topol.3 (2010) 937–996. arXiv:1001.2719v3), where an incorrect construction with the same claimed properties was proposed.

We study infinitesimal deformations of holomorphic maps of compact, complex, Kähler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain.

In this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

We show that the deformation functor of a maximal Cohen–Macaulay module M = coker($\phi$) over the hypersurface singularity det($\phi$) is given by deformations of the presenting matrix which keep the determinant constant. A simplified expression for an edge map in the canonical five-term exact sequence to a change of rings spectral sequence is obtained, including the tangent and obstruction spaces (H1 and H2). We relate the edge map to the Scandinavian complex$\mathcal{S}$ of $\phi$ which yields relations between the homology of $\mathcal{S}$ and Hi for i = 1, 2. This gives (infinitesimal) rigidity and non-rigidity results and a dimension estimate for the formally (mini-)versal formal hull H of the deformation functor.