On plus–one generated arrangements of plane conics

We study the geometry of Q \mathcal {Q} -conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such Q \mathcal {Q} -conic arrangements are never free. Moreover, we provide combinatorial constraints of the weak combinatorics of such arrangements.

Classification of Lagrangian surfaces of constant curvature in complex hyperbolic plane

Let C be a very generic smooth curve of degree d in the complex projective plane. In this paper, we show that for any curve D in the projective plane that does not contain C , the set C ∩ D contains at least d - 2 distinct points. Moreover, this bound is sharp when d ≥ 3. Our motivation for the study of this problem comes from the problem of hyperbolicity of the complement of a plane curve. As an application of our result, we give another proof of the following fact: if C is a very generic smooth curve of degree d ≥ 5 in the complex projective plane, then the intersection of the complement of C in the projective plane with any plane curve D which does not contain C is hyperbolic.

Three additional families of Lagrangian surfaces of constant curvature in complex projective plane

A fake projective plane is a smooth complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. The first example of such a surface was constructed by David Mumford in 1979 using p-adic uniformization. Two more examples were found by Ishida and Kato by related method. Keum has recently given an example which is possibly different from the three known earlier. It is an interesting problem in complex algebraic geometry to determine all fake projective planes. Using the arithmeticity of the fundamental group of fake projective planes, the formula for the covolume of principal arithmetic subgroups given by the first-named auhor, and some number theoretic estimates, we give a classification of fake projective planes in this paper. Twenty eight distinct classes are found. The construction given in the paper appears to be more direct and more natural. It does not use p-adic uniformization.

A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of AH . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP 2 as well as in complex hyperbolic plane CH 2. We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP 2 and 41 families in CH 2. All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP 2 or in CH 2 is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this thesis in real algebraic geometry, we obtain the classification of conjugacy classes of elements of prime order in the group of birational diffeomorphisms of the two-dimensional sphere. More precisely, the two-dimensional sphere S(\R) is the surface defined by the equation x^2+y^2+z^2=1 in \R^3. It corresponds to the real part of the smooth complex projective surface S_\C defined by the equation x^2+y^2+z^2=w^2 in the three-dimensional complex projective space \P^3, that is to say, to the part which is contained in the chart w=1. We can study the algebraic diffeomorphisms of S(\R): birational maps from the complex surface S_\C to itself such that the coefficients that define the map are real, and the restriction to the real part S(\R) is a diffeomorphism. Those maps are called birational diffeomorphisms of the sphere and they form a group denoted by Aut(S(\R)), the subject of this thesis. The interest in this group stems from many reasons. For instance, J. Kollar and F. Mangolte showed that Aut(X(\R)) is big - that is, dense in the group of all diffeomorphisms Diff(X(\R)) - if X is a smooth real compact rational surface, as in the case of the sphere. This is in stark contrast to the group of automorphisms of the complex surface, which is finite dimensional. One motivation to classify the conjugacy classes of the group Aut(S(\R)) is that, by definition, it is contained in the group of all birational maps of the complex surface S_\C, and this latter group is isomorphic to the Cremona group, i.e. the group of birational maps of the complex projective plane. The problem of classification of conjugacy classes of elements of finite order in the Cremona group has been of interest for a lot of mathematicians. L. Bayle and A. Beauville classified the elements of order two using the tools of the minimal model program developed in dimension two by Yu. Manin and V.A. Iskovskikh. T. de Fernex generalised the classification for elements of prime order (except for one case, done by A. Beauville and J. Blanc). Because of the relation of those two groups we can relate the conjugacy classes of birational diffeomorphisms of the sphere to the conjugacy classes of elements in the Cremona group. For example, we find that the sphere features an example of a large familiy of conjugacy classes of involutions that all correspond to one single class for the complex projective plane, to name just one of the differences between the two.

A new family of minimal Lagrangian tori in the complex projective plane is constructed. This family is characterized by its invariability by a one-parameter group of holomorphic isometries of the complex projective plane.

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.

We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is $\mathbf{CP}^2$.

An orientation preserving diffeomorphism over a surface embedded in a 4–manifold is called extendable, if this diffeomorphism is a restriction of an orientation preserving diffeomorphism on this 4–manifold. In this paper, we investigate conditions for extendability of diffeomorphisms over surfaces in the complex projective plane.

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension $7$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq3$ there are closed orbits in $\mathbf F(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{B}(\mathcal{F})$ of a foliation $\mathcal{F}\in\mathbf{F}(d)$ as the set of $\mathcal{G}\in\mathbf{F}(d)$ such that $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ We show that the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1)$, resp. $\dim \mathbf{F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, D\'eserti, Garba Belko and Meziani.

Under study is the energy functional on the set of Lagrangian tori in the complex projective plane. We prove that the value of the energy functional for a certain family of Hamiltonian minimal Lagrangian tori in the complex projective plane is strictly larger than for the Clifford torus.

Recently, we investigated real hypersurfaces in a n-dimentional complex projective space and complex hyperbolic space with respect to various structure Jacobi operator conditions. However these results necessitates dimension assumption n ≥ 3. The purpose of this paper is to study such real hypersurfaces in the complex projective plane and the complex hyperbolic plane.

Let F be a reduced foliation defined on ℙ2—the complex projective plane—given in homogeneous coordinates (X:Y:Z) by a 1-form of degree q, $$\Omega = AdX + BdY + CdZ,$$ i.e., such that A, B, C are homogeneous polynomials of degree q, with no common factors and satisfying Euler’s equation XA + YB + ZC ≡ 0. In this way we have a rational map $$\begin{gathered} \Omega :{\mathbb{P}_2} \to \mathbb{P}_2^{\text{v}} \hfill \\ Q \mapsto \Phi \left( Q \right) \hfill \\ \end{gathered} $$ defined on ℙ2\Sing(F) which associates to each Q the point in \(\mathbb{P}_2^{\text{v}}\) corresponding to the line defined by F at Q.