The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space

Let M be a real hypersurface with almost contact metric structure ${(\phi, \xi, \eta, g)}$ in a complex projective space ${P_{n}\mathbb{C}}$ . A Real hypersurface M is said to be a Hopf hypersurface if ξ is principal. In this paper we investigate real hypersurfaces of ${P_{n}\mathbb{C}}$ whose Ricci tensors S satisfy ${\nabla_{\phi\nabla_{\xi}\xi}S = 0}$ . Under some further conditions we characterize Hopf hypersurfaces of ${P_{n}\mathbb{C}}$ .     

On the Riemannian curvature tensor of a real hypersurface in a complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface.     

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

We classify real hypersurfaces of complex projective space , m3, with -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.     

On characterizations of real hypersurface in complex space form with Codazzi type structure Lie operator

In this paper, we prove that if $(\nabla _{X} L_{\xi })Y= (\nabla _{Y} L_{\xi })X$ holds on $M$ , then $M$ is a Hopf hypersurface, where $L_\xi $ denote the induced operator from the Lie derivative with respect to the structure vector field $\xi $ . We characterize such Hopf hypersurfaces of $M_n(c)$ .     

The structure vector field and structure Jacobi operator of real hypersurfaces in nonflat complex space forms

In this paper we determine the real hypersurfaces for which the structure Jacobi operator commutes over both the Ricci tensors and structure tensors (for a definition of the operator see Sect. 1). We prove that such hypersurfaces are homogneous real hypersurfaces of type (A) and are a special class of Hopf hypersurfaces.     

Real hypersurfaces in a complex projective space with pseudo-$$ \mathbb{D} $$-parallel structure Jacobi operator

We introduce the new notion of pseudo-$ \mathbb{D} $ \mathbb{D} -parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.     

Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in the direction of the structure vector field.     

Commuting structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.     

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface \(X \subset \mathbf {CP}^{n}\). Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface \(X\): we construct a configuration of monotone Lagrangian spheres in \(X\), and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of \(X\) (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for \(X\). We also explain how to extract non-trivial information about Gromov–Witten invariants of \(X\) from its Fukaya category.     

On the structure vector field of a real hypersurface in complex two-plane Grassmannians

Considering the notion of Jacobi type vector fields for a real hypersurface in a complex two-plane Grassmannian, we prove that if a structure vector field is of Jacobi type it is Killing. As a consequence we classify real hypersurfaces whose structure vector field is of Jacobi type.     

Real hypersurfaces in a complex space form with a condition on the structure Jacobi operator

In this paper we classify the real hypersurfaces in a non-flat complex space form with its structure Jacobi operator R _ξ satisfying (? _X R _ξ )ξ = 0, for all vector fields X in the maximal holomorphic distribution D. With this result, we prove the non-existence of real hypersurfaces with D-parallel as well as D-recurrent structure Jacobi operator in complex projective and hyperbolic spaces. We can also prove the non-existence of real hypersurfaces with recurrent structure Jacobi operator in a non-flat complex space form as a corollary.     

On special quadratic birational transformations of a projective space into a hypersurface

We study transformations as in the title with emphasis on those having smooth connected base locus, called ??special??. In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.     

On the ricci tensor of a real hypersurface of quaternionic hyperbolic space

Summary We prove the non-existence of Einstein real hypersurfaces of quaternionic hyperbolic space. This article was processed by the author using the IAT_EX style filecljour1 from Springer-Verlag.