In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $\mathit{\phi }$ with an isolated singularity at 0 in an open subset of ${\mathbb{C}}^n$. This threshold is defined as the supremum of constants c > 0 such that $e^{-2c\mathit{\phi }}$ is integrable on a neighborhood of 0. We relate $c(\mathit{\phi })$ to the intermediate multiplicity numbers $e_j(\mathit{\phi })$, defined as the Lelong numbers of ${(d{d}^c\mathit{\phi })}^j$ at 0 (so that in particular $e_0(\mathit{\phi })=1$). Our main result is that $c(\mathit{\phi })⩾{\sum }_{j=0}^{n-1}{e}_j(\mathit{\phi })/e_{j+1}(\mathit{\phi })$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(\mathit{\phi })⩾1/e_1(\mathit{\phi })$ due to Skoda, as well as the lower estimate $c(\mathit{\phi })⩾n/e_n{(\mathit{\phi })}^{1/n}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t+{(u_{xx}+u^5)}_x=0,$$which is a canonical mass critical problem, for initial data in H¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\Vert u_x{(t)\Vert }_{L^2}\sim \frac{C(u_0)}{T-t}\mathrm{as}t\to T.$$Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].

We prove that for any free ergodic probability measure-preserving action ${\mathbb{F}}_n\curvearrowright (X,\mathit{\mu })$ of a free group on n generators ${\mathbb{F}}_n,2\le n\le \infty$, the associated group measure space II₁ factor $L^{\infty }(X)⋊{\mathbb{F}}_n$ has L^∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II₁ factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II₁ factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.