Hypersurfaces In Complex Projective Space Research Articles

New characterizations of ruled real hypersurfaces in complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections. For any nonnull constant k and any symmetric tensor field of type (1, 1) L on M, we can define two tensor fields of type (1, 2) on M, L_F^{(k)} and L_T^{(k)}, related to both connections. We study the behaviour of the structure operator phi with respect to such tensor fields in the particular case of L=A, the shape operator of M, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.

Some Conditions Concerning the Shape Operator of a Real Hypersurface in Complex Projective Space

Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, B_F^{(k)} and B_T^{(k)}. We will classify real hypersurfaces in complex projective space for which B_F^{(k)} and B_T^{(k)} either take values in the maximal holomorphic distribution mathbb {D} or are parallel to the structure vector field xi , in the particular case of B=A, where A denotes the shape operator of M. We also introduce the concept of A_F^{(k)} and A_T^{(k)} being mathbb {D}-recurrent and classify real hypersurfaces such that either A_F^{(k)} or A_T^{(k)} are mathbb {D}-recurrent.

On two tensors associated to the structure Jacobi operator of a real hypersurface in complex projective space

A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian structure of the ambient space. This almost contact metric structure allows us to define, for any nonzero real number k, the so-called k-th generalized Tanaka–Webster connection. With this connection and the Levi-Civita one we can associate two tensors of type (1,2) to the structure Jacobi operator R_{xi } of M. We classify real hypersurfaces in complex projective space for which such tensors satisfy a cyclic property.

Derivatives of the Lie structure operator on a real hypersurface in complex projective space

Derivatives of the Lie structure operator on a real hypersurface in complex projective space

Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces II

Let M be a real hypersurface in complex projective space. The almost contact metric structure on M allows us to consider, for any nonnull real number k, the corresponding k-th generalized Tanaka-Webster connection on M and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi operator Rξ on M a tensor field of type (1,2), RξT(k). We obtain some classifications of real hypersurfaces for which RξT(k) is either symmetric or skew symmetric.

Normal Bundles of Lines on Hypersurfaces

If X is a smooth hypersurface in complex projective space, the Fano variety of lines on X is stratified by the splitting type of the normal bundle of the line. We show that for general hypersurfaces, these strata have the expected dimension and, in this case, compute the class of the closure of the strata in the Chow ring of the Grassmannian of lines in projective space. For certain splitting types, we also provide upper bounds on the dimension of the strata that hold for all smooth X.

Totally geodesic real hypersurfaces in complex quadric

The Complex quadric is a complex hypersurface in complex projective space. It also can be regarded as a kind of real Grassmann manifold of compact type with rank 2. On the other hand Jacobi fields along geodesics of a given Riemannian manifold satisfy a well known differential equation, called Jacobi equation. This equation naturally inspires the so-called Jacobi operator. The purpose of this article is to give a non-existence result for any totally geodesic real hypersurfaces with recurrent normal Jacobi operator RN and also with recurrent structure Jacobi operator Rξ in Qm. In the continuation, we obtain classifications of totally geodesic real hypersurfaces in Qm with recurrent Ricci tensor Ric.

Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex 2-plane Grassmannians II

We show that Hopf hypersurfaces in complex projective space are constructed from half-dimensional totally complex submanifolds in complex 2-plane Grassmannian and Legendrian submanifolds in the twistor space.

Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces

On a real hypersurface M in a complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the k-th g-Tanaka–Webster connection. Associated to g-Tanaka–Webster connection we can define a differential operator of first order. We classify real hypersurfaces such that both the Lie derivative and this differential operator, either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the structure Jacobi operator of M.

Comparing Lie Derivatives on Real Hypersurfaces in Complex Projective Spaces

On a real hypersurface M in a complex projective space, we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. Therefore, we can also consider their associated Lie derivatives. We classify real hypersurfaces such that both the Lie derivatives associated with the Levi-Civita connection and the kth g-Tanaka–Webster connection either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the shape operator of M.

Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

Holomorphic projection and duality for domains in complex projective space

We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C \mathbb {C} -convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform.

Hyperbolicity of generic high-degree hypersurfaces in complex projective space

We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.

Lie and generalized Tanaka–Webster derivatives on real hypersurfaces in complex projective spaces

On a real hypersurface M in complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. We classify real hypersurfaces such that both the Lie derivative associated to the Levi-Civita connection and the kth g-Tanaka–Webster derivative in the direction of the structure vector field ξ coincide when we apply them to either the shape operator or the structure Jacobi operator of M.

Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex 2-plane Grassmannian I

We define Gauss map from a real hypersurface in complex projective space to complex 2-plane Grassmannian. We show that if a real hypersurface is Hopf, then the image of the Gauss map is a half-dimensional totally complex submanifold with respect to quaternionic Kähler structure of complex 2-plane Grassmannian.

A characterization of certain geodesic hyperspheres in complex projective space

We characterize geodesic hyperspheres of radius r such that cot^2(r)=\frac{1}{2} as the unique real hypersurfaces in complex projective space whose structure Jacobi operator satisfies a pair of conditions.

A characterization of type $A$ real hypersurfaces in complex projective space

A characterization of type $A$ real hypersurfaces in complex projective space

Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie 𝔻-parallel

AbstractWe prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie 𝔻-parallel and satisfies a further condition.

Two Conditions on the Structure Jacobi Operator for Real Hypersurfaces in Complex Projective Space

AbstractWe classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.