Positive curves in polarized manifolds

Seshadri positive submanifolds of polarized manifolds

“…(We thank John Abbott for providing us an explicit set of generators of degree 3 of I + (Y ′ ) using CoCoA-4.7.) Then by Corollary 1.6 in [5], ε(Y ′ , O P 4 (1)) ≥ 1 3 . Therefore for η ∈ (0, 1 3…”

“…In this section we give a series of relevant examples of submanifolds Y of dimension ≥ 2 and of codimension ≥ 2 in a polarized manifold (X, A) that are Seshadri A-big, giving a good motivation for the study of this notion in general. Examples of Seshadri A-big curves in a polarized k-fold can be found in [28], if k = 3, and subsequently in [5], if k ≥ 4. Proposition 3.16 at the end of this section will show that Seshadri A-bigness is stable under finite coverings.…”

“…Moreover, at the end of [5] it was left open the problem of finding the natural general setting for a theory of Seshadri positivity for a submanifold Y of any dimension y ≥ 1 in a complex polarized manifold of dimension k ≥ 3, which should generalize the case of Seshadri positive curves in a polarized manifold. And a possible such definition of Seshadri positivity was suggested in the last section of [5], but, unfortunately, it turned out to be much too strong to work with if y ≥ 2. polarized manifold (X, A) which are Seshadri positive. In Section 4 we show how the Seshadri positivity of a submanifold Y ⊂ X is related to the theory of formal-rational functions of X along Y , as well as to the cohomological dimension of the complement X \ Y .…”

“…Motivated by the study of the gonality of curves and by the behavior of restriction of the stable vector bundles (see [29]), the Seshadri constant ε(Y, A) was used by Paoletti in [28] to study the so-called Seshadri positive curves in a polarized threefold. Subsequently, this theory was generalized and completed to the case when Y is a smooth curve in a smooth polarized variety (X, A) of arbitrary dimension k ≥ 3 in [5]. Specifically, for every η ∈ (0, ε(Y, A)) define the numerical invariant δ η (Y, A)…”

“…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 the natural generalization of the theory initiated by Paoletti in [28] (which corresponds to the case (k, y) = (3, 1)) and subsequently generalized and completed in [5] (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.…”