On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding \(\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} \) of the complex projective space ?P ⁿ considered as a 2n-dimensional, real-analytic manifold in the real projective space \(\mathbb{R}P^{n^2 + 2n} \). The image of the embedding σ is called the ?P ⁿ-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ?P-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ?P _? ⁿ . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ?P _? ⁿ is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.     

Parallelisms of projective spaces

New and old results on parallelisms of projective spaces are surveyed.     

Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization $\tilde{J}^d(Y)$ of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in $\tilde{J}^d(Y)$ . We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space M confirming the Vafa–Intriligator formulae in the nodal case.     

Monads on projective spaces

Let be a vector bundle on P ⁿ . There is a strong relationship between and its intermediate cohomology modules. In the case where has low rank, we exploit this relationship to provide various splitting criteria for . In particular, we give a splitting criterion for in terms of the vanishing of certain intermediate cohomology modules. We also show that the Horrocks-Mumford bundle is the only non-split rank two bundle on P ⁴ with a Buchsbaum second cohomology module.Partially supported by NSF Grants.     

Lines in projective spaces

Here we study finite unions, Y, of lines in a projective space PG(n, K). We prove that if K is an infinite field, Y spans PG(n, K) and a general hyperplane section of Y is not in linearly general position, then there exists at least one linear subspace M of PG(n, K) such that 2 dim(M) < n and M contains at least dim(M)+2 lines of Y.The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Cohomology of configuration spaces of complex projective spaces

In this paper we compute topological invariants for some configuration spaces of complex projective spaces. We shall describe Sullivan models for these configuration spaces.     

Lexicographic generation of projective spaces

Lexicographic or first choice constructions of geometric objects sometimes lead to amazingly good results. Usually it is difficult to determine the precise identity of the resulting geometries. Here we find infinitely many cases where the identification actually can be accomplished.     

Categories of exact sequences with projective middle terms

Let A be a finite-dimensional algebra over an algebraically closed field k,E the category of all exact sequences in A-mod,MP(respectively,MI)the full subcategory of E consisting of those objects with projective(respectively,injective)middle terms.It is proved that MP(respectively,MI)is contravariantly finite(respectively,covariantly finite)in E.As an application,it is shown that MP=MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective.     

Geodesics on weighted projective spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.