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We prove a conjecture proposed by Berman–Boucksom–Eyssidieux–Guedj–Zeriahi, affirming that the Demailly–Lelong number can be determined through a combination of intersection numbers given by the divisorial part of the potential and the SNC divisors over a log resolution of the maximal ideal of a given point. Moreover, this result establishes a pointwise comparison of two different notions of Lelong numbers of plurisubharmonic functions defined on singular complex spaces. We also provide an estimate for quotient singularities and sharp estimates for two-dimensional ADE singularities. [ABSTRACT FROM AUTHOR]

In this paper, we construct a crepant resolution for the quotient singularity A4/A4 in characteristic 2, where A4 is the alternating group of degree 4 with permutation action on A4. By computing the Euler number of the crepant resolution, we obtain a new counter-example to an analogous statement of McKay correspondence in positive characteristic. [ABSTRACT FROM AUTHOR]

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group W acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups. [ABSTRACT FROM AUTHOR]

The wild McKay correspondence, a variant of the McKay correspondence in positive characteristics, shows that stringy motives of quotient varieties equal some motivic integrals on the moduli space of the Galois covers of a formal disk. In this paper, we determine when the integral converges for the case of cyclic groups of prime power order. As an application, we give a criterion for the quotient variety being canonical or log canonical. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds Y Γ that are supposed to be the crepant resolution of quotient singularities C 3 / Γ with Γ a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms ω 2 , 1 . Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on Y Γ . We study the issue of Ricci-flat Kähler metrics on such resolutions Y Γ , with particular attention to the case Γ = Z 4 . We advance the conjecture that on the exceptional divisor of Y Γ the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on tot K W P [ 112 ] that we construct, i.e., the total space of the canonical bundle of the weighted projective space W P [ 112 ] , which is a partial resolution of C 3 / Z 4 . For the full resolution, we have Y Z 4 = tot K F 2 , where F 2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on F 2 produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown. [ABSTRACT FROM AUTHOR]

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities C n / G , where G is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity A n leads to a remarkable formula. [ABSTRACT FROM AUTHOR]

We use the methods introduced by Cheltsov–Rubinstein–Zhang (Sel Math (N.S.) 25(2):25–34, 2019) to estimate δ -invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov–Park–Shramov (J Geom Anal 20(4):787–816, 2010) that have α -invariants less than 2 3 . As a result, we verify that each of these surfaces admits an orbifold Kähler–Einstein metric. [ABSTRACT FROM AUTHOR]

We classify all n-dimensional reduced Cohen-Macaulay modular quotient varieties A F n / C 2 p and study their singularities, where p is a prime number and C 2 p denotes the cyclic group of order 2p. In particular, we present an example that demonstrates that the problem proposed by Yasuda has a negative answer if the condition that "G is a small subgroup" was dropped. [ABSTRACT FROM AUTHOR]

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings k[x1, . . .,xn]G, where G is a finite small subgroup of GL(n, k), and for hypersurface rings k[x1, . . ., xn]/(f) of dimension ≥ 3 with an isolated singularity. In the first case, we obtain the value 1/|G|, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value 0, providing an example of a ring where differential symmetric signature and F-signature are different. [ABSTRACT FROM AUTHOR]

On a normal Stein variety , we study the thickening problem, i.e. the problem whether the assumption that a compact set is contained in the interior of another compact set, implies that the same inclusion holds for their holomorphic hulls. An affirmative answer is given for with isolated quotient singularities. On any Stein space with isolated singularities, we prove thickening for those hulls which have analytic structure at the singular points, obtaining a limitation for possible counter-examples. In dimension , we finally relate the holomorphic hulls to analytic extension from parts of strictly pseudoconvex boundaries. [ABSTRACT FROM AUTHOR]

A singularity is said to be weakly exceptional if it has a unique purely log terminal blow-up. This is a natural generalization of the surface singularities of types Dn, E6, E7 and E8. Since this idea was introduced, quotient singularities of this type have been classified in dimensions up to at most 4. This note extends that classification to dimension 5. [ABSTRACT FROM PUBLISHER]

Abstract: The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal Cohen–Macaulay; second, to see that in that situation, a different, but Morita equivalent, noncommutative crepant resolution can be constructed by using Frobenius morphisms; finally, to study the relation between Frobenius morphisms of noncommutative rings and the finiteness of global dimension. As a byproduct, we will obtain a result on wild quotient singularities: If the smooth cover of a wild quotient singularity is unramified in codimension one, then the singularity is not strongly F-regular. [Copyright &y& Elsevier]

A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D, E, E, E. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4. [ABSTRACT FROM AUTHOR]

Abstract: We prove two conjectures from Cautis and Logvinenko (2009) which describe the geometrical McKay correspondence for a finite, abelian subgroup of . We do it by studying the relation between the derived category mechanics of computing a certain Fourier–Mukai transform and a piece of toric combinatorics known as ‘Reid''s recipe’, effectively providing a categorification of the latter. [Copyright &y& Elsevier]

Abstract: Let A be a F-algebra where F is a field, and let W be an A-module of finite presentation. We use the linear Lie-Rinehart algebra V W of W to define the first Chern-class c 1(W) in , where U in Spec(A) is the open subset where W is locally free. We compute explicitly algebraic V W -connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities B mn2 = x m + y n + z 2, and show that these connections are integrable, hence the first Chern-class c 1(W) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanish. [Copyright &y& Elsevier]

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued. [ABSTRACT FROM AUTHOR]