Complex Projective Plane Research Articles

Geometry of symplectic flux and Lagrangian torus fibrations

AbstractSymplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux. We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand–Cetlin fibres of Fano varieties in terms of their polytopes. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications.

Plücker coordinates and the Rosenfeld planes

The exceptional compact hermitian symmetric space EIII is the quotient E6/Spin(10)×Z4U(1). We introduce the Plücker coordinates which give an embedding of EIII into CP26 as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.Our motivation is to understand EIII as the complex projective octonion plane (C⊗O)P2, whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety X∞ of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of X∞.We further decompose X=EIII into F4-orbits: X=Y0∪Y∞, where Y0∼(OP2)C is an open F4-orbit and is the complexification of OP2, whereas Y∞ has co-dimension 1, thus EIII could be more appropriately denoted as (OP2)C‾. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2].

On cubic-line arrangements with simple singularities

In the present note, we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and A5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{5}$$\\end{document} singular points. We deliver a Hirzebruch-type inequality for such arrangement and study the freeness of such arrangements providing an almost complete classification.

Examples of harmonic foliated currents and singular Levi-flats on the projective plane

We provide examples of foliations on the complex projective plane CP2 carrying positive foliated harmonic currents whose supports coincide with singular Levi-flats which, in turn, can be chosen real-analytic (but non-algebraic) or merely continuous with fractal transverse nature. Furthermore, non-trivial examples as above can already be found among foliations of degree 2 and 3. In addition, the space of positive foliated harmonic currents for these foliations is fully characterized and it contains a unique harmonic (non-closed) current supported on the Levi-flat in question. Finally, we also provide examples of foliations carrying diffuse positive foliated closed currents related to a theorem due to Brunella as well as a general criterion for the existence of singular real analytic Levi-flats for Riccati foliations.

Algebraic periods and minimal number of periodic points for smooth self-maps of 1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{1}$$\\end{document}-connected 4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{4}$$\\end{document}-manifolds with definite intersection forms

Let M be a closed 1-connected smooth 4-manifolds, and let r be a non-negative integer. We study the problem of finding minimal number of r-periodic points in the smooth homotopy class of a given map f:M→M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:M \\rightarrow M$$\\end{document}. This task is related to determining a topological invariant Dr4[f]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^4_r[f]$$\\end{document}, defined in Graff and Jezierski (Forum Math 21(3):491–509, 2009), expressed in terms of Lefschetz numbers of iterations and local fixed point indices of iterations. Previously, the invariant was computed for self-maps of some 3-manifolds. In this paper, we compute the invariants Dr4[f]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^4_r[f]$$\\end{document} for the self-maps of closed 1-connected smooth 4-manifolds with definite intersection forms (i.e., connected sums of complex projective planes). We also present some efficient algorithmic approach to investigate that problem

Geometry of certain foliations on 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.

Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the d-arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.

On maximizing line arrangements in the complex plane

Abstract In this note we provide a complete classification of weak combinatorics of the so-called maximizing line arrangements in the complex projective plane.

The Poincaré problem for reducible curves

We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are isolated, i.e., they are never contained in non-trivial analytic families of equisingular invariant curves. In this case, we show that the multiplicity of an invariant curve is at most twice the multiplicity of the foliation. Finally, we apply the local methods to foliations in the complex projective plane.

Almost complex manifolds with total Betti number three

In this paper, we show the minimal total Betti number of a closed almost complex manifold of dimension [Formula: see text] is four, thus confirming a conjecture of Sullivan except for dimension [Formula: see text]. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.

Pinwheels as Lagrangian barriers

The complex projective plane [Formula: see text] contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation, see for example [J. Evans and I. Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol. 22 (2018) 1143–1180]. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of [Formula: see text], meaning that pinwheels are Lagrangian barriers in the sense of [P. Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407–464]. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below [Formula: see text].

Klein’s arrangements of lines and conics

In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of 21 lines in the complex projective plane.

𝒬-conic arrangements in the complex projective plane

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.

The fundamental group of the complement of the Hesse configuration

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let [Formula: see text] be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane [Formula: see text]. Such [Formula: see text] has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of [Formula: see text] [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. [Formula: see text] has a 9-fold Galois covering space [Formula: see text] given by the complement of the Hesse configuration of 12 lines in [Formula: see text]. Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of [Formula: see text], which we carry out in this note.

Singularities of Holomorphic Codimension One Foliations of the Complex Projective Plane

We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singularity up to the action of an ad-hoc birational map. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.

Differential geometry of complex projective plane conics

Differential geometry of complex projective plane conics

Trinomials, torus knots and chains

Letn>mn>mbe fixed positive coprime integers. Forv>0v>0, we give a topological description of the setΛ(v)\Lambda (v), consisting of points[x:y:z][x:y:z]in the complex projective plane for which the equationxζn+yζm+z=0x\zeta ^n +y \zeta ^m+z=0has a root with normvv. It is shown that the setΩ(v)=PC2∖Λ(v)\Omega (v)= {\mathbb P_{\mathbb C}} ^2 \setminus \Lambda (v)hasn+1n+1components. Moreover, the topological type of each component is given. The same results hold forΛ\LambdaandΩ=PC2∖Λ\Omega ={\mathbb P_{\mathbb C}}^2 \setminus \Lambda, whereΛ\Lambdadenotes the set obtained as the union of all the complex tangent lines to the33-sphere at the points of the torus knot, that is, the knot obtained by intersecting{[x:y:1]∈PC2:|x|2+|y|2=1}\{[x:y:1] \in \mathbb {P}_{\mathbb C}^2 : |x|^2+|y|^2=1\}and the complex curve{[x:y:1]∈PC2:ym=xn}\{[x:y:1] \in {\mathbb P_{\mathbb C}} ^2 : y^m=x^n\}. Finally, we use the linking number of a distinguished family of circles and the torus knot to give a numerical invariant which determines the components ofΩ\Omegain a unique way.

Any oriented non-closed connected $4$-manifold can be spread holomorphically over the complex projective plane minus a point

Any oriented non-closed connected $4$-manifold can be spread holomorphically over the complex projective plane minus a point

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $$\mathbb {F}=SU(3)/T^2$$ , and their configuration relative to the twistor projection $$\pi $$ from $$\mathbb {F}$$ to the complex projective plane $$\mathbb {P}^{2}$$ , defined with the help of an anti-holomorphic involution $$j$$ . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $$\mathbb {P}^3$$ of the 4-dimensional sphere $$S^4$$ . Deformations of twistor fibers project to real surfaces in $$\mathbb {P}^{2}$$ , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in $$\mathbb {F}$$ of bidegree $$(1,1)$$ . These surfaces define orthogonal complex structures on specified dense open subsets of $$\mathbb {P}^{2}$$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $$(1,1)$$ are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in $$\mathbb {F}$$ .