Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain generalization...

We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: For a smooth projective curve $C$ of genus $g$ in characteristic 0, the condition $\text{Cliff}C>l$ is equivalent to the fact that $K_{g-l^{\text{'}}-2,1}(C,K_C)=0,\forall l^{\text{'}}\le l$. We propose a new approach, which allows up to prove this result for generic curves $C$ of genus $g\left(C\right)$ and gonality $\text{gon(C)}$ in the range $$\frac{g\left(C\right)}{3}+1\le \text{gon(C)}\le \frac{g\left(C\right)}{2}+1.$$

We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

Let $n$ be a positive integer and $A^{\prime }$ a complete characteristic zero discrete valuation ring with maximal ideal $𝔪$, absolute ramification index $e\<p-1$ and perfect residue field $k$ of characteristic $p\>2$. In this paper we classify smooth finite dimensional formal $p$-faithful groups over $A_n^{\prime }=A^{\prime }/{𝔪}^n{A}^{\prime }$, i.e. groups on which the “multiplication by $p$” morphism is faithfully flat, in particular $p$-divisible groups. As applications, we prove that $p$-divisible groups over $k$, and the morphisms between them, lift canonically to $A^{\prime }/p{A}^{\prime }$, and...

Soient $k$ un corps et $X$ une $k$-variété projective et lisse. Si $X$ est géométriquement rationnelle, on dispose d’une application injective du quotient de groupes de Brauer $Br\left(X\right)/Br\left(k\right)$ dans le premier groupe de cohomologie galoisienne du réseau défini par le groupe de Picard géométrique de $X$. Dans cette note on donne des cas où cette application est toujours surjective. Pour les espaces homogènes de certains tores algébriques, on donne des générateurs explicites dans $Br\left(X\right)$. On applique cela à l’étude du principe de...

Le sujet de cet article est le groupe de Picard de la variété de modules $M(r,c_1,c_2)$ des faisceaux algébriques semi-stables de rang $r$ et de classes de Chern $c_1,c_2$ sur ${\mathbf{P}}_2\left(\mathbf{C}\right)$. Le premier résultat est que $M(r,c_1,c_2)$ est localement factorielle, ce qui permet d’identifier Pic$\left(M\right(r,c_1,c_2\left)\right)$ et le groupe des classes d’équivalence linéaire des diviseurs de Weil de $M(r,c_1,c_2))$. Il existe une unique application $\delta :\mathbf{Q}\to \mathbf{Q}$ telle que dim$\left(M\right(r,c_1,c_2\left)\right)\>0$ si et seulement si $(c_2-(r-1)c_1^2/2r)/r\>\delta (c_1/r)$. Si on a égalité, Pic$\left(M\right(r,c_1,c_2\left)\right)$ est isomorphe à $\mathbf{Z}$, et si l’inégalité est stricte, Pic$\left(M\right(r,c_1,c_2\left)\right)$ est isomorphe à ${\mathbf{Z}}^2$. On donne ensuite...