Hermitian Surface Research Articles

A new combinatorial characterization of (quasi)-Hermitian surfaces

In this paper, we present a combinatorial characterization of a quasi-Hermitian surface as a set {mathcal {H}} of points of textrm{PG}(3,q), q=p^{2h}hge 1, p a prime number and qne 4, having the same size as the Hermitian surface and containing no plane, such that either a line is contained in {mathcal {H}} or intersects {mathcal {H}} in at most sqrt{q}+1 points and every plane intersects {mathcal {H}} in at least qsqrt{q}+1 points. Moreover, if there is no external line, the set {mathcal {H}} is a Hermitian surface.

On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces

The complete classification of the orbits on subspaces under the action of the projective stabilizer of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of PGL(2,q2) (and PSL(2,q2)) arising from the Hermitian Veronese curve in PG(3,q2), a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermitian surface.

New hemisystems of the Hermitian surface

Constructing hemisystems of the Hermitian surface is a well known, apparently difficult, problem in Finite geometry. So far, a few infinite families and some sporadic examples have been constructed. One of the different approaches relies on the Fuhrmann-Torres maximal curve and provides a hemisystem in \(PG(3,p^2)\) for every prime p of the form \(p=1+4a^2\), a even. Here we show that this approach also works in \(PG(3,p^2)\) for every prime \(p=1+4a^2\), a odd. The resulting hemisystem gives rise to two weight linear codes and strongly regular graphs whose properties are also investigated.

Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature

Motivated by a recent work of Chen–Zheng (J Geom Anal 32:141, 2022) on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection \(\nabla ^t \) is either Kähler, or an isosceles Hopf surface with an admissible metric and \(t=-1\) or \(t=3\). In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is Kähler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao–Zheng (On Gauduchon Kähler-like manifolds. ArXiv: 2108.08181), which extends a previous result by Apostolov–Davidov–Muškarov (Trans Am Math Soc 348:3051–3063, 1996).

Constructing Hermitian Hamiltonians for spin zero neutral and charged particles on a curved surface: physical approach

The Hamiltonians for a spin zero neutral particle and a charged particle in an electromagnetic field pinned to an arbitrary 2D surface embedded in 3D space are constructed. The new approach we follow starts with an expression for the 3D momentum operators whose components along the surface and the normal to the surface are separately Hermitian. The normal part of the kinetic energy operator is a Hermitian operator in this case. When this operator is dropped and the thickness of the layer is set to zero, one automatically gets the Hermitian surface Hamiltonian that contains the geometric potential term as expected. We show that Hermitian surface and normal momenta emerge automatically once one symmetrizes the usual normal and surface momentum operators; the geometrical potential originates from this symmetrization, too. Comparing our approach with the standard thin-layer quantization (TLQ), we show that Hermicity is also at the heart of TLQ.

A proof of Sørensen’s conjecture on Hermitian surfaces

In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of F q 2 \mathbb {F}_{q^2} -rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q . d \le q.

QCH Kähler surfaces II

In this paper we give new examples of QCH Kähler surfaces whose opposite almost Hermitian structure is Hermitian and not locally conformally Kähler. In this way we give also a large class of examples of Hermitian surfaces with J-invariant Ricci tensor which are not l.c.k.

Maximum Number of Points on Intersection of a Cubic Surface and a Non-Degenerate Hermitian Surface

In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case when $d=2$. In this paper, we prove that the conjecture is true for $d=3$ and $q \ge 8$. We further determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and second highest number of points in common with a non-degenerate Hermitian surface. This classifications disproves one of the conjectures proposed by Edoukou, Ling and Xing.

On the $$\mathrm{PSU}(4,2)$$-Invariant Vertex-Transitive Strongly Regular (216, 40, 4, 8) Graph

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $$\Gamma$$ with parameters (216, 40, 4, 8) and proved that it is the unique $$\mathrm{PSU}(4,2)$$-invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of $$\mathrm{PG}(3,4)$$, we provide a computer-free proof of the existence of the graph $$\Gamma$$. The maximal cliques of $$\Gamma$$ are also determined.

Rational curves on a smooth Hermitian surface

We study the set $R$ of nonplanar rational curves of degree $d<q+2$ on a smooth Hermitian surface $X$ of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$. We prove that $R$ is the empty set when $d<q+1$. In the case where $d=q+1$, we count the number of elements of $R$ by showing that the group of projective automorphisms of $X$ acts transitively on $R$ and by determining the stabilizer subgroup. In the special case where $X$ is the Fermat surface, we present an element of $R$ explicitly.

Hemisystems of the Hermitian surface

We present a new method for the study of hemisystems of the Hermitian surface U3 of PG(3,q2). The basic idea is to represent generator-sets of U3 by means of a maximal curve naturally embedded in U3 so that a sufficient condition for the existence of hemisystems may follow from results about maximal curves and their automorphism groups. In this paper we obtain a hemisystem in PG(3,p2) for each p prime of the form p=1+16n2 with an integer n. Since the famous Landau's conjecture dating back to 1904 is still to be proved (or disproved), it is unknown whether there exists an infinite sequence of such primes. What is known so far is that just 18 primes up to 51000 with this property exist, namely 17,257,401,577,1297,1601,3137,7057,13457,14401,15377,24337,25601,30977,32401,33857,41617,50177. The scarcity of such primes seems to confirm that hemisystems of U3 are rare objects.

Symmetric Surface Momentum and Centripetal Force for a Particle on a Curved Surface

The Hermitian surface momentum operator for a particle confined to a 2D curved surface spanned by orthogonal coordinates and embedded in 3D space is expressed as a symmetric expression in derivatives with respect to the surface coordinates and so is manifestly along the surface. This is an alternative form to the one reported in the literature and usually named geometric momentum, which has a term proportional to the mean curvature along the direction normal to the surface, and so “apparently” not along the surface. The symmetric form of the momentum is the sum of two symmetric Hermitian operators along the two orthogonal directions defined by the surface coordinates. The centripetal force operator for a particle on the surface of a cylinder and a sphere is calculated by taking the time derivative of the momentum and is seen to be a symmetrization of the well-known classical expressions.

Ovoids of [formula omitted], [formula omitted] odd, admitting a group of order [formula omitted

A new construction of an ovoid O of the Hermitian surface H(3,q2), q>3 odd, is provided. In particular, O is left invariant by a group of order (q+1)32 and it cannot be obtained from a Hermitian curve by means of multiple derivation and it is not locally Hermitian.

On the curvature of Einstein–Hermitian surfaces

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

Intriguing sets of quadrics in PG(5, q)

Abstract A hemisystem on the Hermitian surface ℋ(3, q 2), q ≥ 7 odd, admitting a subgroup of PΩ-(4, q) of order q 2(q +1) is constructed. Also, a new family of Cameron-Liebler line classes of PG(3, q), q ≥ 5 odd, with parameter (q 2 + 1)/2 is provided.

On maximal cliques of polar graphs

The maximal cliques of the graph NU(4,q2) related to the Hermitian surface of PG(3,q2) and of the graph NO±(2n+2,q), q even, n≥1, are classified.

A relative m-cover of a Hermitian surface is a relative hemisystem

An m-cover of the Hermitian surface $$\mathrm {H}(3,q^2)$$H(3,q2) of $$\mathrm {PG}(3,q^2)$$PG(3,q2) is a set $$\mathcal {S}$$S of lines of $$\mathrm {H}(3,q^2)$$H(3,q2) such that every point of $$\mathrm {H}(3,q^2)$$H(3,q2) lies on exactly m lines of $$\mathcal {S}$$S, and $$0

A New Infinite Family of Hemisystems of the Hermitian Surface

In this paper, we construct an infinite family of hemisystems of the Hermitian surface $\mathsf{H}(3,q^2)$. In particular, we show that for every odd prime power $q$ congruent to $3$ modulo $4$, there exists a hemisystem of $\mathsf{H}(3,q^2)$ admitting $C_{(q^3+1)/4} : C_3$.

Intersections of the Hermitian Surface with Irreducible Quadrics in Even Characteristic

We determine the possible intersection sizes of a Hermitian surface ${\mathcal H}$ with an irreducible quadric of $PG(3,q^2)$ sharing at least a tangent plane at a common non-singular point when $q$ is even.

Rigidity results for Hermitian-Einstein manifolds

A differential operator introduced by A. Gray on the unit sphere bundle of a K\ahler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a K\ahler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms amongst compact Hermitian surfaces and the product of two projective lines amongst all K\ahler-Einstein surfaces are then derived.