Frédéric Campana (Univ. Lorraine)

Title: Numerical dimension of the cotangent bundle of projective manifolds

Abstract: We define k1(X) and v1(X), the Kodaira and numerical dimensions of a the cotangent bundle of a complex projective manifold X as the ones of the tautological bundle on the projectivisation, and conjecture that they coincide (a version of abundance for the canonical bundle). This Kodaira dimension was already definied and studied long time ago by Fumio Sakai. Horing-Peternell independently conjectured and studied the particular case when k1=-∞. We can show this equality in a certain number of quite peculiar cases: Sub-manifolds with ample normal bundles of Abelian varieties, and complete inter-sections in projective spaces of dimension at least 2n if X is n-dimensional. But already the case of surfaces of general type is completely open in general. We show that k1=v1 for any uniruled X if it holds for the base of its MRC. For smooth models of klt varieties with c1=0, we show that κ1=v1=q’, the maximum irregularity of an étale cover. In particular, it is then a topological and deformation invariant. We conjecture that κ1=v1=q’ remains true for “special” manifolds. F. Sakai noticed that κ1 is not a deformation invariant (already for elliptic surfaces over curves of genus at least 2).

Yi Gu (Soochow Univ.)

Title: On the reduction of hyperelliptic curves

Abstract: Let f: X→C be a surface fibration of fibre genus g over the complex numbers, the semistable reduction theorem asserts there exists a morphism of curves h: C'→C such that the base change of f over h admits a birational semistable model over C'. We denote by N(f) the smallest possible degree of such h. An interesting question is to find uniform bounds of N(f) depending only on g. In the 1980s, Xiao Gang proved such an effective uniform upper Ng and asked if his bound is also optimal. In this talk, we will give a similar bound N'g under an extra assumption that f is hyperelliptic. Our bound N'g is optimal in the hyperelliptic sense and hence answers Xiao's question for genus g when Ng=N'g . Finally, we will show those g such that Ng=N'g has a positive density in natural numbers in a certain sense. This is a joint work with C. Gong, J. Lu, and P. Pollack.

Chunyi Li (Warwick Univ.)

Title: A conjectural Bogomolov-Type inequality for higher Chern characters

Abstract: For a stable vector bundle on a smooth polarized variety, the classical Bogomolov inequality provides a bound on its second Chern character. Building on the framework established by Bertram, Bayer, Macri, Stellari, and Toda, this inequality has been extended to a conjectural Bogomolov-type bound for the third Chern character. In this talk, I will present a proposed generalization of this conjectural inequality to higher Chern characters and discuss its potential applications.

Qifeng Li (Shandong Univ.)

Title: Local equivalence of VMRT-structures

Abstract: In this talk, we will firstly introduce the local equivalence problem of VMRT strctures, and then introduce the theory of cone structures, applying it to study VMRT-structures. This is based on joint works with Jun-Muk Hwang.

Claire Voisin (CNRS-Jussieu)

Title: Universally defined cycles

Abstract: I introduce the notion of universally defined cycles (for smooth varieties of dimension d) and prove that any universally defined cycle is given on generic fibers by a polynomial in the Chern classes, which can be seen as a higher dimensional version of the Franchetta conjecture. I will also give the motivation for the main definition and explain a conjectural extension of that result to universally defined cycles on powers of varieties of given dimension.