arXiv research

Study shows convergence of cscK surfaces in Hilbert scheme.

Character variety of Whitehead link described in detail.

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…

We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of…

The Kähler-Ricci flow smooths positive currents on Kähler manifolds.

New domains of discontinuity found for Anosov representations.

Classifies Zariski closures of positive representations in Lie groups.

We build examples of properly convex projective manifold $Ω/ Γ$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$ which are not lattice, nor Schottky groups. Moreover, the open properly convex set $Ω$ is…

The study identifies two minimal orbits of foliations on complex projective plane and explores their properties.

We study meromorphic actions of unipotent complex Lie groups on compact Kähler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable Kähler structures obtained by symplectic reduct…

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on $π_1$ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?

We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is $\mathbb{P}^{m}$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers…

We study the Kähler-Ricci flow on compact Kähler manifolds whose canonical bundle is big. We show that the normalized Kähler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular Kähler-Einstein metric in the canonical class. The key ingredi…

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Generic Hitchin representations generate dense subgroups.

The paper explores mapping class group quotients by Dehn twists and their representations.

New subgroup found in Lie groups with unusual properties.

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

Finite-gap solutions approximate jets of initial data for certain BKM systems.

Complex projective manifolds without rational curves are quotients of Abelian varieties.

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the delta-invariant) is a recently introduced invariant that is known to detect the K-s…

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

Study trisections on rational elliptic surfaces to find new Zariski pairs.

The study explores deformations of discrete subgroups in non-compact homogeneous spaces.

A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically fi…

The study finds conditions for certain groups to be dense in a specific mathematical space.

Research examines coamenable subgroups in higher rank groups.

New lattices in higher dimensions have dense surface subgroups.

We give a new proof of the fact that the condition of a Fano manifold admitting a Kähler-Einstein metric is Zariski-open (provided that the automorphism group is discrete). This proof does not use the characterisation involving stability. The arguments involve estimates of Futaki invariants obtained from a differential…

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.

Deforms surface groups to be Zariski dense in SL(n,R)

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

Two unique conic-line arrangements with degree 9 are found.

The paper finds dense subgroups in certain Lie groups.

The study explores deformations of standard locally homogeneous spaces.

Study conic line arrangements of degree 7, finding their topology and connected components.

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

Paper finds surface groups can deform in reductive symmetric spaces.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding in…

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Pro…

Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy takin…

Maximal representations in symplectic lattices proven for most cases.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techni…