Abstracts:

Giulio Codogni: Positivity of the Chow-Mumford line bundle for families of K-stable klt Fano varieties

The Chow-Mumford (CM) line bundle is a functorial line bundle defined on the base of any family of polarized varieties, in particular on the base of families of klt Fano varieties. It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable klt Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers.

In this talk, I will present a proof of the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming stability only for very general fibers. These results work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I will also present an application to the classification of Fano varieties. This is a joint work with Zs. Patakfalvi.

Soheyla Feyzbakhsh: Brill-Noether theory via wall-crossing

I will describe some of the recent applications of wall-crossing with respect to Bridgeland stability conditions in the Brill-Noether theory of higher rank vector bundles on smooth curves.

Giovanni Inchiostro: Wall crossing morphisms for moduli of stable pairs

Consider a moduli space M parametrizing stable pairs of the form (X, a_1 D_1+...+a_n D_n) with a_1,...,a_n positive rational numbers. Consider n positive rational numbers b_1,...,b_n with b_i <= a_i, and assume that the objects on the interior of M are pairs with K_X + b_1 D_1+...b_n D_n big. Then on the interior of M one can send a pair (X, a_1 D_1+...+a_n D_n) to the canonical model of (X, b_1 D_1+...+b_n D_n). If N is a moduli space of stable pairs with coefficients b_i this gives a set theoretic map from an open substack of M to N. We investigate when such a map can be extended to the whole M. Our main result is if the interior of M parameterizes klt pairs we can extend the map, up to replacing M and N with their normalizations. The extension does not exist if above we replace the word normalization with seminormalizaton instead. This is joint with Kenny Ascher, Dori Bejleri and Zsolt Patakfalvi.

Sönke Rollenske: Stratification of the moduli space of stable surfaces inspired by Hodge theory

Let M be a component of the moduli space of surfaces of general type and M its closure in the moduli space of stable surfaces. Green, Griffiths, Laza and Robles proposed a stratification of this space inspired by Hodge theory. I will explain the idea behind this stratification and investigate in an example what it does and does not distinguish in the boundary.

Cristiano Spotti: Geometric aspects of Kähler-Einstein metrics on klt pairs

In this talk I will discuss Kähler-Einstein metrics on klt pairs, focusing on the asymptotic of the metrics near the singularities and on examples of metric degenerations (based on joint works with M. de Borbon, P. Gallardo and J. Martinez-Garcia).

Roberto Svaldi: On the boundedness of elliptic Calabi-Yau varieties

One of the main goals in Algebraic Geometry is to classify varieties. The minimal model program (MMP) is an ambitious program that aims to realize this goal, from the point of view of birational geometry, that is, we are free to modify the structure of a given variety along closed subsets to improve its geometric features. According to the MMP, there are 3 building blocks in the birational classification of algebraic varieties: Fano varieties, Calabi-Yau varieties, and varieties of general type. One important question, that is needed to further investigate the classification process, is whether or not varieties in these 3 classes have finitely many deformation types (a property called boundedness). Our understanding of the boundedness of Fano varieties and varieties of general type is quite solid but Calabi-Yau varieties are still quite elusive. In this talk, I will discuss recent results on the boundedness of elliptic Calabi-Yau varieties, which are the most relevant in physics. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. This is joint work with C. Birkar and G. Di Cerbo.

Alan Thompson: Compactifications of the moduli space of K3 surfaces of degree 2

I will explain how to construct two different compactifications of the moduli space of K3 surfaces of degree two. The first uses techniques from the minimal model program. It is modular, meaning that its boundary points provide moduli for degenerate K3's, but its construction is implicit, so there is no geometric description of its boundary. The second is constructed explicitly, as a blow-up of an explicit compactification of the period domain (quotiented by an appropriate symmetry group). It has the opposite problems: it has a nice geometric description, but its boundary has no modular interpretation. I will then present a recent joint result, with V. Alexeev and P. Engel, which gives a morphism from the second compactification to the first, in effect solving the problems with both. If there is sufficient time remaining I will also sketch the proof of this result.

Poster session

There will be a poster session on Monday afternoon. If you plan to bring a poster, please inform us at moduli19hannover@gmail.com, so we can plan enough space for all posters.