# On subvarieties with ample normal bundle

@article{Ottem2013OnSW, title={On subvarieties with ample normal bundle}, author={John Christian Ottem}, journal={arXiv: Algebraic Geometry}, year={2013} }

We show that a pseudoeffective R-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, giving an affirmative answer to a question of Peternell. We also give other positivity properties such subvarieties.

## 16 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

### Compact subvarieties with ample normal bundles, algebraicity and cones of cycles

- Mathematics
- 2011

In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact K\"ahler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic…

### Ample subvarieties of algebraic varieties

- Mathematics
- 1970

Ample divisors.- Affine open subsets.- Generalization to higher codimensions.- The grothendieck-lefschetz theorems.- Formal-rational functions along a subvariety.- Algebraic geometry and analytic…

### Submanifolds with ample normal bundles and a conjecture of Hartshorne

- Mathematics
- 2008

The Hartshorne conjecture predicts that two submanifolds X and Y in a projective manifold Z with ample normal bundles meets as soon as dim X + dim Y is at least dim Z. We mostly assume slightly…

### Ample Vector Bundles on Curves

- MathematicsNagoya Mathematical Journal
- 1971

In our earlier paper [4] we developed the basic sheaftheoretic and cohomological properties of ample vector bundles. These generalize the corresponding well-known results for ample line bundles. The…

### The ² ∂-method, weak Lefschetz theorems, and the topology of Kähler manifolds

- Mathematics
- 1998

In [No], Nori studied the fundamental group of complements of nodal curves with ample normal bundle in smooth projective surfaces. The main tool was the following weak Lefschetz theorem: Theorem…

### The pseudo-effective cone of a compact K\

- Mathematics
- 2004

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a…

### Line bundles with partially vanishing cohomology

- Mathematics
- 2010

Define a line bundle L on a projective variety to be q-ample, for a natural number q, if tensoring with high powers of L kills coherent sheaf cohomology above dimension q. Thus 0-ampleness is the…

### Coniveau 2 Complete Intersections and Effective Cones

- Mathematics
- 2008

Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface X of degree d in $${\mathbb{P}^n}$$ shows that it has coniveau ≥ c once n ≥ dc. The generalized Hodge…

### The topology of smooth divisors and the arithmetic of abelian varieties.

- Mathematics
- 2000

We have three main results. First, we show that a smooth complex projective variety which contains three disjoint codimension-one subvarieties in the same homology class must be the union of a whole…