The relative Bruce–Roberts number of a function on a hypersurface

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$ . We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$ . As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$ . We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$ , obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

Let (V,0)={(z1,…,zn)∈Cn:f(z1,…,zn)=0} be an isolated hypersurface singularity with mult(f)=m. Let Jk(f) be the ideal generated by all k-th order partial derivatives of f. For 1≤k≤m−1, the new object Lk(V) is defined to be the Lie algebra of derivations of the new k-th local algebra Mk(V), where Mk(V):=On/((f)+J1(f)+…+Jk(f)). Its dimension is denoted as δk(V). This number δk(V) is a new numerical analytic invariant. In this article we compute L4(V) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ4(V). We also verify a sharp upper estimate conjecture for the δ4(V) for large class of singularities. Furthermore, we verify another inequality conjecture: δ(k+1)(V)<δk(V),k=3 for low-dimensional fewnomial singularities.

Schur sector of Argyres-Douglas theory and $W$-algebra

We study the Schur index, the Zhu’s C_2C2 algebra, and the Macdonald index of a four dimensional \mathcal{N}=2𝒩=2 Argyres-Douglas (AD) theories from the structure of the associated two dimensional WW-algebra. The Schur index is derived from the vacuum character of the corresponding WW-algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu’s C_2C2 algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu’s C_2C2 algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the WW-algebra.

NOTES ON VANISHING CYCLES AND APPLICATIONS

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.

The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

Lipschitz Normal Embedding Among Superisolated Singularities

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if its projectivized tangent cone has only ordinary singularities. This provides an infinite family of LNE singularities, which is radically different from the class of minimal singularities.

JET SCHEMES OF QUASI-ORDINARY SURFACE SINGULARITIES

We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on $\mathbb{A}^{3}$ ), that computes the log-canonical threshold of the singularity embedded in $\mathbb{A}^{3}$ . This provides us with pairs $X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair $C\subset \mathbb{A}^{2}$ , where $C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of $\mathbb{A}^{2}$ ).