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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]