Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{\mathrm {reg}}$, and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.

The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$-liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$.

We prove a relative Lefschetz–Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$ -category of cohomological correspondences. We show that local acyclicity is equivalent to dualisability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalising a theorem of Gabber.

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

We consider the possible disentanglements of holomorphic map germs f: (ℂn, 0) → (ℂN, 0), 0 < n < N, with nonisolated locus of instability Inst (f). The aim is to achieve lower bounds for their (homological) connectivity in terms of dim Inst (f). Our methods apply in the case of corank 1.

It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

We consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb ), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x 2 + y 5) has two topological orbits.

An affine symplectic singularity $X$ with a good $\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers $N$ and $d$, there are only a finite number of conical symplectic varieties of dimension $2d$ with maximal weights $N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$-algebra.

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites $f\,{\circ}\,F$ with isolated singularities is studied, where $f\,{:}\,Y\,{\longrightarrow}\,\mathbb{C}$ is a function with (possibly non-isolated) singularity and $F:X\,{\longrightarrow}\, Y$ is a map into the domain of $f$, and $F$ only is deformed. The corresponding $T^1(F)$ is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that \[\tau=\mu(f\circ F)-\beta_0+\beta_1\], where $\tau$ is the length of $T^1(F)$ and $\beta_i$ is the length of $\mbox{Tor}_i^{\cal{O}_Y}({\cal O}_Y{/}J_f,\cal{O}_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When $f$ has Cohen–Macaulay singular locus (for example when $f$ is the determinant function), relations between $\tau$ and the rank of the vanishing homology of the zero locus of $f\circ F$ are obtained.

We study adjacency of equisingularity types of planar complex curve singularities in terms of their Enriques diagrams. The goal is, given two equisingularity types, to determine whether one of themis adjacent to the other. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also obtain new examples of exceptional deformations, i.e, singular curves of type ${\mathcal{D}}'$ that can be deformed to a curve of type $\mathcal{D}$ without ${\mathcal{D}}'$ being adjacent to $\mathcal{D}$.

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

We give dimension formulas for the cotangent cohomology groups Ti for all rational surface singularities. The computations take place on the first blow-up of the singularity.

It is shown that a function whose critical locus is an isolated complete intersection singularity of arbitrary dimension, and that has finite codimension (in the sense of R. Pellikaan, Proc. London Math. Soc. (3) 57 (1998) 357–382) with respect to the ideal defining the isolated complete intersection singularity, can be approximated by a function whose critical locus is a finite number of Morse points together with the Milnor fibre of the isolated complete intersection singularity, having there well-known types of singularities.