Title: Stacks at the service of resolution of singularities

Abstract: After discussing the process of blowing up, I'll report on joint work with Temkin and Wlodarczyk, with Quek, and with Schober, using stacks to give resolution algorithms. These include Deligne--Mumford stacks (with weighted blowups) and Artin stacks (with multi-weighted blowups).

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Title: Degenerations and compact moduli of K3 surfaces with an involution

Abstract: We describe degenerations and compact moduli of K3 surfaces with a non symplectic involution. This is a joint work with Philip Engel.

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Title: Slope inequalities for irregular fibrations over curves

Abstract: Twenty years ago Rita Pardini proved Severi Inequality for irregular surfaces using a clever use of Slope Inequality for fibred surfaces. A slight modification of her proof works well in higher dimensions but unfortunately higher dimensional slope inequalities are known only in some special cases, closely related with stability of fibres. Since then, higher dimensional (Clifford-)Severi inequalities have been obtained using different techniques.
In this talk I will go back to the original Pardini's strategy and show how Slope Inequalities and Clifford-Severi inequalities are in fact equivalent for varieties of maximal Albanese dimension. This fact allows us to give sharp lower bounds of different classes of such fibrations and study the limit cases. We will also study the cases of non-maximal Albanese dimension varieties where this equivalence does not hold in general. The main new technical tools are a continuous version of the Xiao's method and the introduction of new continuous invariants for irregular fibrations.

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Title: Manifolds woth vanishing Chern classes and some questions by Severi/Baldassarri

Abstract: We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only manifolds with vanishing Chern classes. We exhibit Hyperelliptic Manifolds which are not Abelian varieties (nor complex tori) and whose Chern classes are zero not only in integral homology, but also in the Chow ring.
We prove moreover that the Bagnera de Franchis manifolds (T/G where G is cyclic) have topologically trivial tangent bundle.
Motivated by a more general question addressed by Mario Baldassarri in 1956, we discuss the Hyperelliptic Manifolds, the Pseudo- Abelian Varieties introduced by Roth, and we introduce a new notion, of Manifolds Isogenous to a k-Torus Product: the latter have the last k Chern classes trivial in rational homology and vanishing Chern numbers. We show that the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: in dimension 2 these are the surfaces with KX nef and c2(X) = 0.
One may ask whether a similar picture holds also in higher dimension, and we discuss manifolds whose tangent (resp. cotangent bundle) has a trivial summand.

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Title: On the moduli part of a klt-trivial fibration

Abstract: Let f: (X,B)->Y be a fibration such that the log-canonical divisor K+B is trivial along the fibres of f.
The canonical bundle formula is a way of expressing K+B as the pullback of the sum of three divisors: the canonical divisor on Y; a divisor, called discriminant, carrying informations on the singular fibres; a divisor called moduli part keeping track of the birational variation of the fibres and having positivity properties.
Let T be a connected, reduced but not necessarily irreducible divisor of Y.
In this talk I will give a geometric interpretation of the semiampleness of the restriction of the moduli part to T and give some results towards the semiampleness of the restriction.

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Title: A new proof of a theorem of Gordan and Noether

Abstract: In a fundamental paper of 1876 Gordan and Noether fixed Hesse’s claim by showing that a complex projective hypersurface V(F) ={F=0} of the projective space of dimension n<4 is a cone if and only if the determinant of the Hessian of F is zero. One of the applications is a Lefschetz-type result for Standard Gorenstein Artinian Algebras (SAGA) of codimension less than 5.
Here we show that the logical line of this implication can be reversed. We give a direct geometric and elementary proof of the Lefschetz property and then we deduce Gordan-Noether theorem using Maculay’s theory. Some application of our method are also given. This work is a collaboration with Davide Bricalli and Filippo Favale.

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Title: Proper Period Matrix Representations at Infinity

Abstract: The problem of generalizing the Satake-Baily-Borel compactification and Borel’s extension theorem to arbitrary period mappings raises questions about the global behavior of period mappings at infinity. I will discuss one of those questions, and what the answer tells us about the motivating problem.

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Title: Stable surfaces with small invariants

Abstract: Surfaces of general type with particular invariants and their moduli spaces have been studied for over a century. Nowadays the Gieseker moduli space of surfaces of general type is known to admit a natural compactification in the moduli space of stable surfaces. I will survey some results of a long running effort to understand stable surfaces with $K^2 = 1$ and their moduli, started in collaboration with Rita Pardini and Marco Franciosi and continued partially in collaboration with Julie Rana and Stephen Coughlan.

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Title: Effective characterization of quasi-abelian surfaces

Abstract: We provide a characterization theorem for quasi-abelian surfaces, extending to the logarithmic setting results of Enriques and Chen--Hacon. This is a joint work with M. Mendes Lopes and R. Pardini.

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Title: A topological characterization of hyper-Kähler fourfolds of Hilb^2(K3) type

Abstract: There are two known deformations types of hyper-Kähler (HK) fourfolds, namely Hilb^2(K3) (Beauville, Fujiki) and the generalized Kummer variety K_2(A) (Beauville). It is however still unknown whether there are other topological types or deformation types of HK fourfolds. Some strong topological restrictions on HK fourfolds are known by work of Beauville, S. Salamon, Verbitsky and Guan. In this talk I will sketch the proof of the following result
Thm. A hyper-Kähler fourfold X is a deformation of Hilb^22(K3) if and only if it has two integral degree 2 cohomology classes satisfying the conditions l^4=0, m4=0, l^2m2=2. In particular, a HK fourfold which is homeomorphic to Hilb^2(K3) is a deformation of Hilb^2(K3).
This is joint work with Debarre, Huybrechts and Macrì.