On vector bundles on $3$-folds with sectional genus $1$

Notes on very ample vector bundles on 3 - folds

Let E be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of E is provided when KX+detE is nef but not big, and when a suitable positive multiple of KX+det E defines a morphism X → B with connected fibers onto a smooth projective curve B, where KX is the canonical bundle of X. As an application, the case where the genus...

Vector Bundles on Fano 3-folds without Intermediate Cohomology

A well-known result of Horrocks (see [Ho]) states that a vector bundle on a projective space has not intermediate cohomology if and only if it decomposes as a direct sum of line bundles. There are two possible generalizations of this result to arbitrary varieties. The first one consists of giving a cohomological characterization of direct sums of line bundles. This has been done for quadrics an...

On Genus-1 Gromov-Witten Invariants of Complete Intersection 3-Folds

Notes on Vector Bundles

−→ V, (v1, v2) −→ v1+v2, are smooth. Note that we can add v1, v2∈V only if they lie in the same fiber over M , i.e. π(v1)=π(v2) ⇐⇒ (v1, v2) ∈ V ×M V. The space V ×M V is a smooth submanifold of V ×V , as follows immediately from the Implicit Function Theorem or can be seen directly. The local triviality condition means that for every point m∈M there exist a neighborhood U of m in M and a diffeo...

Vector bundles on Orbispaces

Orbispaces are spaces with extra structure. The main examples come from compact Lie group actionsX G and are denoted [X/G], their underlying space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. To be more precise, an orbispace is a topological stack which is locally equivalent to [X/G] for G a Lie group and X a G-CW-complex [2] [4] [5]. Howev...