# May 31-June 1, 2018

## Titles and abstracts ▾

• Francesco Bastianelli Effective cycles on symmetric products of curves

In this talk, I will be concerned with the convex-geometric properties of the cone of $n$-dimensional (pseudo)effective cycles in the symmetric product $C_d$ of a smooth curve $C$. I will firstly show that the $n$-dimensional diagonal cycles span a perfect face of the pseudoeffective cone. Then, I will describe a series of subcones related to the contractibility properties of the Abel-Jacobi morphism, and I will show that they form a maximal chain of perfect faces of the pseudoeffective cone, provided that $C$ is a very general curve. This is a joint work with A. Kouvidakis, A. F. Lopez and F. Viviani.

• Vicente Cortés Compact homogeneous Einstein manifolds in dimension 6

I will present some recent progress concerning the classification of compact homogeneous six-dimensional Einstein manifolds. The talk is based on joint work with Florin Belgun, Alexander Haupt and David Lindemann, see J. Geom. Phys. 128 (2018), 128-139, arXiv:1703.10512.

• Alessandro Ghigi Compactifying automorphism groups of Kaehler manifolds

Around 1978 Akira Fujiki and David Lieberman independently introduced a natural compactification of the connected component of the identity in the automorphism group of a compact Kaehler manifold. In the talk I will recall the construction of this compactification using Barlet cycle space. Then I will describe some recent results obtained jointly with Leonardo Biliotti. The main issue is the interpretation of boundary points in the compactification in terms of non-dominant meromorphic maps of the manifold in itself.

• Victor Lozovanu Convex geometry, local positivity, and singular divisors of an ample line bundle

Seshadri constants measure the local positivity of an ample line bundle at a fixed point. Due to this, they show up in many areas of mathematics ranging from Kähler to algebraic or arithmetic geometry. In this talk I will try to explain how we can translate algebro-geometric data into convex geometric conditions, allowing us a better (visual) understanding of Seshadri constants. As a consequence, together with ideas pioneered in birational geometry, we are able to study syzygetic global properties of an ample line bundle on abelian varieties in terms of numerical data on abelian submanifolds.

• Diego Matessi Lagrangian submanifolds lifting tropical hypersurfaces

I will describe a construction of Lagrangian submanifolds in the cotangent bundle of a real torus which lift tropical hypersurfaces in the affine space. This leads to new constructions of Lagrangian submanifolds in toric varieties and in symplectic Calabi-Yau manifolds. It is expected that these manifolds are "mirror" to the complex varieties arising from tropical to complex correspondences.

• Angela Ortega Generic injectivity of the Prym map for double ramified coverings

Given a finite morphism of smooth curves one can canonically associate it a polarized abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarized abelian varieties, known as the Prym map. In this talk we will consider the Prym map between the moduli space of double coverings over a genus g curve ramified at r points, and the moduli space of polarized abelian varieties of dimension $(g-1+r)/2$ with polarization of type D. We will show the generic injectivity of the Prym map in the cases (a) $g=2$, $r=6$ and (b) $g=5$, $r=2$. In the first case the proof is constructive and can be extended to the range $r > \max{6, 2(g+2)/3}$. This a joint work with Juan Carlos Naranjo.

• Tommaso Pacini Variations on the theme of volume

The standard volume functional can be defined for any submanifold: the "flip side" of this generality is that it generally does not capture any special geometric features of the submanifold. Lagrangian submanifolds are an important exception: in this case the volume functional has several important properties related to ambient curvature, maslov indices, etc. Joint work with J Lotay (UCL) has showed that, by adding an appropriate weight function, we can extend this theory from the symplectic to the complex category, also obtaining several new features. The seminar will provide a broad survey of these results.

• Nicola Pagani Pull-backs of Brill-Noether cycles from universal Jacobians

Let Mgn be the moduli space of smooth $n$-pointed curves of genus $g$. For a given vector of integers $(d_1,...,d_n)$ one can define a natural locus of pointed curves $(C, p_i)$ such that $O_C(\sum d_i p_i)$ admits a nonzero global section. We discuss how this can be extended to the moduli space of stable curves by interpreting it as the pullback of a cohomology class on (compactified) universal Jacobians. Because there are multiple ways to compactify the Jacobian, this leads to multiple classes related by wall-crossing. We explain why this gives an effective approach for the computation of the cohomology class of these cycles (in terms of tautological classes). The case when $d_1 + ... + d_n = 0$ is known in the literature as the "double ramification cycle" and it has attracted the attention of several mathematicians. A joint work with Jesse Kass.

• Luis Ugarte Complex geometry of nilmanifolds

Nilmanifolds with invariant complex structures have proved to be a source of many examples of compact complex manifolds with interesting geometric properties. For instance, Hasegawa used rational homotopy theory to prove that non-toral nilmanifolds cannot admit any Kähler metric. However, there are complex nilmanifolds having symplectic forms which are compatible with their complex structure. There are also nilmanifolds admitting strong Kähler with torsion, astheno-Kähler or, more generally, generalized Gauduchon structures. Moreover, balanced Hermitian metrics exist on some nilmanifolds giving rise to solutions of the Hull-Strominger system. Invariant complex structures on nilmanifolds have been described up to complex dimension 3, the Kodaira-Thurston manifold and the Iwasawa manifold being well-known examples in dimension 2 and 3, respectively. In this talk we present recent advances in the complex geometry of nilmanifolds in higher dimensions, with special attention to complex dimension 4. Indeed, in this dimension one can get a general description of invariant complex structures, which allows to clarify several aspects of the geometry of compact complex manifolds of dimension $\geq 4$. More concretely, we will show the existence of: compact complex nilmanifolds admitting both a balanced metric and an astheno-Kähler metric, a family of complex nilmanifolds with infinitely many real homotopy types, and (pseudo-)Hermitian nilmanifolds with interesting deformation properties. Joint work with Adela Latorre and Raquel Villacampa.