arXiv research

Character variety of Whitehead link described in detail.

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 subgroup found in Lie groups with unusual properties.

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…

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…

Study shows convergence of cscK surfaces in Hilbert scheme.

Proves properties of complex algebraic varieties and local systems.

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

Odd-dimensional SL(n,Q) contains dense surface subgroups.

New domains of discontinuity found for Anosov representations.

In this paper, we study the $SL_2(\mathbb{C})$ character variety of a hyperbolic link in $S^3$. We analyze a special smooth projective variety $Y^h$ arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in $K_2({…

We provide a simple algebraic construction of the twistor spaces of arbitrary Joyce's self-dual metrics on the 4-manifold H^2 x T^2 that extend smoothly to nCP^2, the connected sum of complex projective planes. Indeed, we explicitly realize projective models of the twistor spaces of arbitrary Joyce metrics on nCP^2 in …

Paper solves Hermitian-Einstein equations on noncompact manifolds.

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

Survey on minimal rational curves and their geometric structures.

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

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…

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…

Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.

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 show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $SO^* (2n)$, $n$ odd) and the surface group is maximal in some $S(U(p,p)\times U(q-p))\subset SU(p,q)$ (resp. $SO^* (2n-2)\times SO(2)\subset SO^* (2n)$)…

We define a subset of an almost complex manifold (M,J) to be a holomorphic shadow if it is the image of a J-holomorphic map from a compact complex manifold. Notice that a J-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold. We show that under some conditions on an a…

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

We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit a cohomology class $[α] \in H^{1,1}(Y,\mathbb R)$ which is numerically effective (nef) and has positive self-intersection (meaning $\int_Y α^n \,>\, 0$, where $n\,=\,\di…

Study critical exponents in normal subgroups of higher rank 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…

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…

Classifies Zariski closures of positive representations in Lie groups.

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

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

Combination theorems for convex projective geometry subgroups.

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 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…

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.

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

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…

Study Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.

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…

Paper finds surface groups can deform in reductive symmetric spaces.

An open subset U of a complex surface can be topologically perturbed to yield an open subset whose inherited complex structure is Stein, if and only if U is homeomorphic to the interior of a handlebody whose handles all have index equal or less than 2.

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