相关论文: E8 lattice and the Kodaira dimension of orthogonal…

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there…

This paper aims to derive new anomaly cancellation formulas by combining modular forms with E8 and E8*E8 bundles. To this end, we systematically twist and generalize known SL(2,Z) modular forms to define new modular forms associated with…

We survey results concerning special elements of nine types (modular, lower-modular, upper-modular, cancellable, distributive, codistributive, standard, costandard and neutral elements) in the lattice of all semigroup varieties and certain…

Let $(X,B)$ be a log canonical pair over a normal variety $Z$ with maximal Albanese dimension. If $K_X+B$ is relatively abundant over $Z$ (for example, $K_X+B$ is relatively big over $Z$), then we prove that $K_X+B$ is abundant. In…

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly…

We exhibit a simple uniruledness criterion for general orthogonal modular varieties in terms of invariants of the corresponding lattice. As an application, we obtain the uniruledness of almost all Nikulin--Vinberg moduli spaces…

Let $L$ be a finite $n$-element lattice. We prove that if $L$ has at least $83\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar lattice with exactly $83\cdot 2^{n-8}-1$ sublattices.

Let $N\subset \RR^{r}$ be a lattice, and let $\deg\colon N \to \CC$ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on $\deg$, the data $(N,\deg)$ determines a…

The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete…

The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H+E_8+E_7. Key ingredients for this classification are: a normal form for these lattice polarized K3 surfaces, a…

We prove that if $\Gamma$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $\Gamma$-equivariantly homotopy equivalent to a proper cocompact $\Gamma$-CW complex of dimension the virtual cohomological…

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular $(k,n)$ frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the…

Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this paper we proved that there exists a residual $\mathcal{G}$ of $H^{1}\left( M;\mathbb{R}\right)$ such that the property: \begin{equation*}…

We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.

A general result by Jackson (Flat algebras and the translation of universal Horn logic to equational logic, J. Symb. Log. 73(1) (2008) 90--128) implies that the lattice of all quasivarieties of groups of exponent dividing $n$ embeds into…

Let $R$ be a commutative ring. Roughly speaking, we prove that an $R$-module $M$ is flat iff it is a direct limit of $R$-module affine algebraic varieties, and $M$ is a flat Mittag-Leffler module iff it is the union of its $R$-submodule…

We prove that the Kodaira dimension of the moduli space M_{23} of curves of genus 23 is at least 2. We also present some evidence for the hypothesis that the Kodaira dimension of the moduli space is actually equal to 2. Note that for g > 23…

We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that…

In this paper, we study a class of generalized intersection matrix Lie algebras $\gim(M_n)$, and prove that its every finite-dimensional semi-simple quotient is of type $M(n,{\bf a}, {\bf c},{\bf d})$. Particularly, any finite dimensional…

We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre…