Past Event

ALPE took place in Toulouse on the 10-11 of June, 2024.

Programme

List of talks and abstracts.

Titles and abstracts

• Michele Bolognesi - Université de Montpellier Finite dimensional transcendental motives and rationality of cubic fourfolds ▼
  In this seminar, which takes inspiration from more or less recent works with C. Pedrini, I will introduce the transcendental motive of a cubic hypersurface in P^5, and I will briefly relate this object to the birational geometry of the cubic. Using a bit of intersection theory on the moduli space of cubic fourfolds, I will show how it is possible to find an infinity of curves that parametrize cubics with finite dimensional Chow motive, and how these curves are in some sense ubiquitous within certain loci of special cubics inside the module space. Everything will be preceded by a broad introduction to the classical aspects of the geometry of cubic fourfolds, their cohomology and Hodge theory.
• Matt Booth - Lancaster University Nonsmooth Calabi—Yau algebras ▼
  Calabi—Yau dg algebras, and their many-object cousins Calabi—Yau dg categories, are the analogues of Calabi—Yau manifolds in noncommutative derived geometry. To give a quick definition, a CY dga is a smooth dga whose dualising complex is a shift of the diagonal bimodule. I'll talk about some recent work in progress, joint with Joe Chuang and Andrey Lazarev, about the sort of objects one gets when dropping smoothness from this definition. In particular, we show that nsCY dgas are Koszul dual to symmetric Frobenius coalgebras; we also show similar statements for Gorenstein vs. Frobenius and smooth vs. proper. As an application we derive a new characterisation of Poincaré duality spaces.
• Emma Lepri - University of Glasgow L-infinity liftings of semiregularity maps and deformations ▼
  After a brief introduction to the semiregularity maps of Severi, Kodaira and Spencer, and Bloch, I will focus on the Buchweitz-Flenner semiregularity map and on its importance for the deformation theory of coherent sheaves. The subject of this talk is the construction of a lifting of each component of the Buchweitz-Flenner semiregularity map to an L-infinity morphism between DG-Lie algebras, which allows to interpret components of the semiregularity map as obstruction maps of morphisms of deformation functors. As a consequence, we obtain that the semiregularity map annihilates all obstructions to deformations of a coherent sheaf on a complex projective manifold. Based on a joint work with R. Bandiera and M. Manetti.
• Sveta Makarova - Universität Duisburg-Essen Grothendieck duality for algebraic stacks ▼
  There is a theory of Grothendieck duality developed for Deligne-Mumford stacks, however in some applications it is desirable to consider morphisms between stacks that are not Deligne-Mumford. I will tell about an adaptation to infinity-categories of the formal story of Neeman's Grothendieck duality for tensor triangulated categories, and then explain how to use these categorical results to obtain geometric statements. I will apply this result to determinantal line bundles on moduli of sheaves on stacky curves. Namely, I will explain how these line bundles depend on the determinant of the vector bundle defining them, and point out where Grothendieck duality is needed.
• Augustinas Jacovskis - University of Luxembourg Categorical Torelli for double covers ▼
  Consider a threefold double cover X of (weighted) projective space, ramified in a canonically polarised surface Z. In this talk I'll describe a semiorthogonal decomposition of the mu_2-equivariant Kuznetsov component of X, and show that it contains a copy of Db(Z). This gives a relationship between the K-theory of the equivariant Kuznetsov component, and the primitive cohomology of Z. Using this relationship and classical Torelli theorems for hypersurfaces in (weighted) projective space, I'll show that for certain classes of prime Fano threefolds, an equivalence of Kuznetsov components implies that they're isomorphic. This is joint work with Hannah Dell and Franco Rota.