Holomorphic symmetric differentials and parallelizable compact complex manifolds

We provide a characterization of complex tori using holomorphic symmetric differentials. With the same method we show that compact complex manifolds of Kodaira dimension 0 having some symmetric power of the cotangent bundle globally generated are quotients of parallelizable manifolds, therefore have an infinite fundamental group

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.Comment: UPDATED: more details added on main proo

Linear series on curves: stability and Clifford index

We study concepts of stabilities associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the associated Dual Span Bundle and linear stability. Our result implies a stability condition related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Eventually using our results we obtain stable vector bundles of integral slope 3, and prove that they admit theta-divisors.Comment: 24 page

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections

On Stability of Tautological Bundles and their Total Transforms

Through the use of linearized bundles, we prove the stability of tautological bundles over the symmetric product of a curve and of the kernel of the evaluation map on their global sections

On linear stability and syzygy stability

In previous works, the authors investigated the relationships between linear stability of a generated linear series |V| on a curve C, and slope stability of the syzygy vector bundle MV OC → L) . In particular, the second named author and L. Stoppino conjecture that, for a complete linear system |L|, linear (semi)stability is equivalent to slope (semi)stability of ML . The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7

Iitaka Fibrations for Vector Bundles

A vector bundle on a smooth projective variety, if it is generically generated by global sections, yields a rational map to a Grassmannian, called Kodaira map. We investigate the asymptotic behaviour of the Kodaira maps for the symmetric powers of a vector bundle, and we show that these maps stabilize to a map dominating all of them, as it happens for a line bundle via the Iitka fibration. Through this Iitaka-type construction, applied to the cotangent bundle, we give a new characterization of Abelian varieties.Comment: Edit for second version: the final remark was removed, as it was not correct. Some questions and a theorem added in the last sectio

Linear stability for line bundles over curves

Let C be a smooth irreducible projective curve and let (L, H^0(L)) be a complete and globally generated linear series on C. Denote by M_L the syzygy bundle, kernel of the evaluation map H^0(L) 97 O_C \u2192 L. In this work we restrict our attention to the case of globally generated line bundles L over a curve with h0(L) = 3. The purpose of this short note is to connect Mistretta-Stoppino Conjecture on the equivalence between linear (semi)stability of L and slope (semi)stability of M_L with the existence of extensions of line bundles of L by certain quotients Q of M_L. Also, we give numerical conditions to produce examples of line bundles L which are linearly semistables but with syzygy bundle M_L unstable, that is, we find numerical conditions to look for counter-examples to Mistretta-Stoppino Conjecture of rank 2