Let (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

In this paper, I construct noncompact analogs of the Chern classes for equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the (topological) Euler characteristic of complete intersections in reductive groups. In the case where a complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve. I also prove that the higher Chern classes vanish. The first...

Lorsqu’un tore $\mathbf{T}$ agit sur une variété algébrique complexe $\mathbf{X}$ munie de la topologie transcendante, nous définissons la classe d’Euler $\mathbf{T}$-équivariante d’un point fixe isolé $x\in {\mathbf{X}}^{\mathbf{T}}$, qu’il soit lisse ou non. Cette classe est une fraction rationnelle à un nombre fini de variables et lorsque $x$ est rationnellement lisse dans $\mathbf{X}$, c’est un polynôme qui s’identifie canoniquement à la classe d’Euler équivariante usuelle, mais, réciproquement, lorsque la classe d’Euler équivariante est polynomiale, il n’est pas toujours...

We explain the philosophy behind the computations in [BDP] and place them in a wider conceptual setting. We also outline, for toric varieties, the resulting equivalent approach to some key results in that theory.

In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to solve the problem when $n=2$ and for any $n$ when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier...

We describe how the graded minimal resolution of certain semigroup algebras is related to the combinatorics of some simplicial complexes. We obtain characterizations of the Cohen-Macaulay and Gorenstein conditions. The Cohen-Macaulay type is computed from combinatorics. As an application, we compute explicitly the graded minimal resolution of monomial both affine and simplicial projective surfaces.

Les variétés abéliennes principalement polarisées admettent un espace des modules grossier qu’on sait compactifier de plusieurs façons (compactification de Satake, compactifications toroïdales). Cependant, le problème s’est posé de construire une compactification “modulaire”en termes d’objets géométriques qui permettent de décrire les points du bord. On souhaite aussi compactifier l’application de Torelli qui à chaque courbe algébrique, projective et lisse, associe sa jacobienne. L’exposé présente...

A compact complex space $X$ is called complex-symmetric with respect to a subgroup $G$ of the group ${\mathrm{Aut}}_0\left(X\right)$, if each point of $X$ is isolated fixed point of an involutive automorphism of $G$. It follows that $G$ is almost $G^0$-homogeneous. After some examples we classify normal complex-symmetric varieties with $G^0$ reductive. It turns out that $X$ is a product of a Hermitian symmetric space and a compact torus embedding satisfying some additional conditions. In the smooth case these torus embeddings are classified using...

Consider a rational representation of an algebraic torus $T$ on a vector space $V$. Suppose that $\{f_1,\cdots ,f_p\}$ is a homogeneous minimal generating set for the ring of invariants, $\mathbf{k}[V{]}^T$. New upper bounds are derived for the number $N_{V,T}:=\max \{\mathrm{deg}{f}_i\}$. These bounds are expressed in terms of the volume of the convex hull of the weights of $V$ and other geometric data. Also an algorithm is described for constructing an (essentially unique) partial set of generators $\{f_1,\cdots ,f_s\}$ consisting of monomials and such that $\mathbf{k}[V{]}^T$ is integral over $k[f_1,\cdots ,f_s]$.