Extended hyperbolicity

Minimal 2-spheres in a complex projective space

We study curvature of Hopf hypersurfaces in a complex projective space or hyperbolic space. In particular, we prove that there are no real hypersurfaces in a non-flat complex space form whose Reeb-sectional curvature vanishes.

In the case where $\Sigma$ is a Riemann sphere, the above result was obtained by Siu [S-l, $Z$] (see also [B-R-S]). In the case where $M$ is a complex projective space and deg $f|\geqq m(p-1)/(m+1)$ where $p$ denotes the genus of $\Sigma$ , the above result was obtained by Eells and Wood [E-W]. They used algebraic geometric arguments (theorems of Riemann-Roch and Grothendieck). Recently, by using a twistor space over the domain manifold, Burns and de Bartolomeis have shown the above result in the case where $M$ is a complex projective space (cf. Remark 6 of [B-R-S]). We show the above theorem by a simple argument for the second variations used in [L-S].

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.

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved an analogue of Kronecker’s limit formula associated to any divisor D \mathcal D which is smooth in codimension one on any smooth Kähler manifold X X . In the present article, we apply the aforementioned Kronecker limit formula in the case when X X is complex projective space C P n \mathbb {C}\mathbb {P}^n for n ≥ 2 n \geq 2 and D \mathcal D is a hyperplane, meaning the divisor of a linear form P D ( z ) P_D({z}) for z = ( Z j ) ∈ C P n {z} = (\mathcal {Z}_{j}) \in \mathbb {C}\mathbb {P}^n . Our main result is an explicit evaluation of the Mahler measure of P D P_{D} as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L 2 L^{2} -norm of the vector of coefficients of P D P_{D} .

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

We study the high-dimensional asymptotics of the expected number of critical points of a given Morse index of Gaussian random holomorphic sections over complex projective space. We explicitly compute the exponential growth rate of the expected number of critical points of the largest index and of diverging indices at various rates as well as the exponential growth rate for the expected number of critical points (regardless of index). We also compute the distribution of the critical values for the expected number of critical points of smallest index.

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.

This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector $\xi$ is principal, the focal map has constant rank, and ${\nabla _\xi }S = 0$, where S is the Ricci tensor of the real hypersurface.

The study of homogeneous and evenly homogeneous complex supermanifolds of dimension whose reduction is the complex projective space is continued. All non-split supermanifolds of this type with retract admitting the full projective group as group of automorphisms are classified. The cohomology with values in the tangent sheaf is calculated for homogeneous supermanifolds satisfying the above-mentioned conditions.

Let M be 3 Semi-invariant submanifold of codimension 3 with lift-flat normal connection in a complex projective space. Further, if the mean curvature of M is constant, then we prove that M is a real hypersurface of a complex projective space of codimension 2 in the ambient space.

We give a more illuminating derivation of a well-known condition (obtained by M. Atiyah and J. A. Todd) for certain vector bundles over complex projective spaces to be stably fiber homotopic trivial, together with a generalization.

In the present paper we study Einstein Lagrangian submanifolds. We show that if M is a complete Einstein Lagrangian minimal submanifold of a complex projective space CPn(c), then M is parallel and M is one of the following conditions holds: a) M is totally geodesic, b) n = 2 and M is a finite Riemannian covering of a torus minimally embedded in CP2(c) with parallel second fundamental form, c) n > 2 and M is an embedded submanifold congruent to the standard embedding of: SU(3)/SO(3),n = 5;SU(3),n = 8;SU(6)/Sp(3),n = 14 or E6/F4,n = 26. We also study a compact Einstein Kaehler submanifold of a complex projective space.

This paper presents a method and a corresponding algorithm for constructing volume forms (and related forms that act as kernels of integral representations) on toric varieties from a convex integer polytope. The algorithm is implemented in the Maple computer algebra system. The constructed volume forms are similar to the volume forms of the Fubini---Study metric on a complex projective space and can be used for constructing integral representations of holomorphic functions in polycircular regions of a multidimensional complex space.

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomogeneity one. We characterize these submanifolds as the only minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ that are foliated by umbilical hypersurfaces of Lagrangian subspaces $\mathbb{R}\mathbb{H}^n$ of $\mathbb{C}\mathbb{H}^n$. By suitably generalizing this construction, we obtain new families of minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ from curves in $\mathbb{C}\mathbb{H}^1$ and $(n-1)$-dimensional minimal Lagrangian submanifolds of the complex space forms $\mathbb{C}\mathbb{P}^{n-1}$, $\mathbb{C}\mathbb{H}^{n-1}$ and $\mathbb{C}^{n-1}$. We give similar constructions in the complex projective space $\mathbb{C}\mathbb{P}^n$.