Lucian Bădescu

Seshadri positive submanifolds of polarized manifolds

Let Y be a submanifold of dimension y of a polarized complex manifold (X, A) of dimension k ≥ 2, with 1 ≤ y ≤ k−1. We define and study two positivity conditions on Y in (X, A), called Seshadri A-bigness and (a stronger one) Seshadri A-ampleness. In this way we get a natural generalization of the theory initiated by Paoletti in [Paoletti R., Seshadri positive curves in a smooth projective 3-fold, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1996, 6(4), 259–274] (which corresponds to the case (k, y) = (3, 1)) and subsequently generalized and completed in [Bădescu L., Beltrametti M.C., Francia P., Positive curves in polarized manifolds, Manuscripta Math, 1997, 92(3), 369–388] (regarding curves in a polarized manifold of arbitrary dimension). The theory presented here, which is new even if y = k − 1, is motivated by a reasonably large area of examples.

A Connectedness Theorem for Products of Weighted Projective Spaces

We prove a connectedness result for products of weighted projective spaces.

Connectivity and a Problem of Formal Geometry

Let \(P = \mathbb{P}^{m}(e) \times \mathbb{P}^{n}(h)\) be a product of weighted projective spaces, and let Δ P be the diagonal of P × P. We prove an algebraization result for formal-rational functions on certain closed subvarieties X of P × P along the intersection X ∩Δ P .

A Barth-Lefschetz theorem for submanifolds of a product of projective spaces ∗

Let X be a complex submanifold of dimension d of ℙm × ℙn (m ≥ n ≥ 2) and denote by α: Pic(ℙm × ℙn) → Pic(X) the restriction map of Picard groups, by NX|ℙm × ℙn the normal bundle of X in ℙm × ℙn. Set t := max{dim π1(X), dim π2(X)}, where π1 and π2 are the two projections of ℙm × ℙn. We prove a Barth–Lefschetz type result as follows: Theorem. If

Proof of Theorem 1.3

d \geq \frac{m+n+t+1}{2}

Extensions of Projective Varieties

then X is algebraically simply connected, the map α is injective and Coker(α) is torsion-free. Moreover α is an isomorphism if

The Zak Map of a Curve. Gaussian Maps

d \geq \frac{m+n+t+2}{2}

Formal Functions on Homogeneous Spaces

, or if

Quasi-lines on Projective Manifolds

d=\frac{m+n+t+1}{2}

Basic Definitions and Results

and NX|ℙm×ℙn is decomposable. These bounds are optimal. The main technical ingredients in the proof are: the Kodaira–Le Potier vanishing theorem in the generalized form of Sommese ([18, 19]), the join construction and an algebraization result of Faltings concerning small codimensional subvarieties in ℙN (see [9]).