Related papers: Blow-$ADE$ singularities and $\mu^*$-constant defo…
In the context of holomorphic families of ${\mathbb P}^k$ endomorphisms, we show that various notions of stability are equivalent. This allows us to both extend and simplify the architecture of the proof of certain results of [BBD]
In 2001, de Oliveira, Katzarkov, and Ramachandran conjectured that the property of smooth projective varieties having big fundamental groups is stable under small deformations. This conjecture was proven by Beno\^it Claudon in 2010 for…
In this paper, Brou\'e's conjecture is reduced to simple groups, with an additional stability condition.
We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and…
We are concerned with the focusing $L^2$-critical nonlinear Schr\"odinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$…
Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic…
A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C^3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The…
When identified with sequences of irreducible Hermitian-Einstein connections, sequences of stable holomorphic bundles of fixed topological type and bounded degree on a compact complex surface equipped with a Gauduchon metric are shown to…
Let $f : X\to \Delta$ be a $1$-parameter family of $2$-dimensional isolated hypersurface singularities. In this paper, we show that if the Milnor number is constant, then any semistable model, obtained from $f$ after a sufficiently large…
Studying crepant blow-ups of (compound) du Val singularities, we classify complexes of coherent sheaves which admit no negative self-extensions -- such a complex, up to flops and mutation equivalences, must either be (1) a module over a…
We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of…
We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as…
We classify normal stable surfaces with $K_X^2 = 1$, $p_g = 2$ and $q=0$ with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli…
We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the…
Main topic of the paper is the determination, for a compact complex manifold $M$, of the class of manifolds $X$ which are deformation equivalent to it. If $M$ is a complex torus, then also $X$ is so. After describing the structure of…
We study the spaces of locally-finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of $A_n$-singularities supported at the exceptional sets. Our main theorem is that they are connected and…
In this paper, we investigate the relations among various results concerning the minimal resolution of cyclic quotient singularities of the form $\mathbb{C}^2/G$. We refer to these as "bamboo-type" singularities, since the dual graphs of…
We study the geometry of Gorenstein curve singularities of genus two, and of their stable limits. These singularities come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1\leq m…
We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable…
We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric $\alpha$-stable process under H\"older regularity conditions for the drift term. We partially recover the…