Miguel Ángel Barja

On the topological index of irregular surfaces

We study the topological index of some irregular surfaces that we call generalized Lagrangian. We show that under certain hypotheses on the base locus of the Lagrangian system the topological index is non-negative. For the minimal surfaces of general type with q = 4 and pg = 5 we prove the same statement without any hypothesis.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluri- canonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this pa- per we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional gen- eralization to this context of curves of genus 2: varieties birationally equivalent to the theta-divisor of an indecomposable principally polar- ized abelian variety. The proof is based on the (generalized) Fourier- Mukai transform.

On the slope of bielliptic fibrations

Let

LOWER BOUNDS OF THE SLOPE OF FIBRED THREEFOLDS

\pi :S\longrightarrow B

Xiao’s conjecture for general fibred surfaces

be a bielliptic fibration. We prove

Stability Conditions and Positivity of Invariants of Fibrations

S

Stability and singularities of relative hypersurfaces

is, up to base change, a rational double cover of an elliptic fibration and that

Linear stability of projected canonical curves with applications to the slope of fibred surfaces

\pi

Surfaces on the Severi line

is isotrivial provided it is smooth. Finally, we prove that the slope of

On the slope of fibred surfaces

\pi