Related papers: Shokurov's index conjecture for quotient singulari…

We prove the Milnor conjecture for Lie groups and the Friedlander conjecture for complex algebraic Lie groups.

We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…

A vector variational principle is proved.

In this paper we prove Shapiro's 1958 Conjecture on exponential polynomials, assuming Schanuel's Conjecture.

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang conjecture for curves. In particular we show that the relative…

Motivated by Lang-Vojta's conjectures on hyperbolic varieties, we prove a new version of the Shafarevich conjecture in which we establish the finiteness of pointed families of polarized varieties. We then give an arithmetic application to…

We prove that good quotients of algebraic varieties with 1-rational singularities also have 1-rational singularities. This refines a result of Boutot on rational singularities of good quotients.

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a simple combinatorial proof of his formula. In addition, we show that it is equivalent with the alternating convolution formula of…

We present a relative form of the Toponogov comparison theorem.

We give an example for the Kuznetsov-Shinder conjecture, with infinitely many non-isomorphic D-equivalent and L-equivalent varieties.

We prove formulas for the core of ideals that apply in arbitrary characteristic.

We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.

We prove the Invariant Subspace Conjecture for separable Hilbert spaces.

We prove the main conjecture of arXiv:2102.13459 for integral coefficients in the case of tori and prove that it is t-exact. Along the way we prove that the spectral action as constructed in that manuscript is compatible with the action of…

We prove the positivity conjecture for all skew-symmetric cluster algebras.

In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers and the Tjurina numbers for isolated hypersurface…

We complete the proof of Oka's conjecture on the Alexander polynomial of an irreducible plane sextic. We also calculate the fundamental groups of irreducible sextics with a singular point adjacent to $J_{10}$.

The article presents the procedure of the index calculation for the elements of the algebra generated by one dimensional singular integral operators with discontinuous oscillating coefficients.

We resolve Manin's conjecture for all Ch\^atelet surfaces over $\mathbb{Q}$.

In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components…