Let a finite group act faithfully by symplectic automorphisms on a manifold of $K3^{[n]}$ type for $n\geq2$. We prove Mongardi's conjecture that the strict inequality $\mathrm{rk}(S)+\ell(A_S)<24$ holds for its associated Leech coinvariant lattice $S$. The converse was already proved independently by Huybrechts and Mongardi: any Leech coinvariant lattice $S$ satisfying the strict inequality and of rank at most $20$ is realized by such an action for some $n\geq2$. The key idea is to consider the minimum norms of vectors representing discriminant classes of $S$. The absence of numerical walls gives lower bounds for certain classes, while the structure of Leech lattice implies upper bounds for all discriminant classes. Comparing these bounds reduces the proof to finitely many values of $n$. We exclude the remaining cases using further lattice arguments, together with H\"ohn--Mason's classification of fixed-point sublattices of Leech lattice, thereby proving Mongardi's conjecture.
Let $X=\mathbb{P}^n$, where $p=n+1$ is prime. Consider the orbit of the standard numerical exceptional basis in $K_0(X)$ under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors art congruent to $1$...
Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. We prove, uniformly over all primes $p$, that \[ \log_p f(p^m)=\frac{2}{27}m^3-\frac49m^2+O(m), \] with an absolute implied constant. This improves the classical upper bounds of Sims (\emph{Proc. Lond. Math. Soc.}, 1965) and Newman--Seeley and...
The classical Thompson's conjecture asserts that in any finite nonabelian simple group $G$, there exists a conjugacy class $S$ whose square is equal to the entire group. Shalev (Annals Math., 2009) conjectured that a much stronger quantitative version holds: there exists $\epsilon>0$ such that for any conjugacy class $...
For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$....
Let $Z$ be a compact K\"ahler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $\mu_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie alge...
Ball's weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{AG}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9/2}$. More precisely, a no...
Daniele Bartoli, A. Giannoni· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
