IMSA & ICMS Joint Event: Atoms, Hodge Theory and MMP

Roy Magen, ICMS: Computing mixed Hodge numbers

After giving a brief introduction to mixed Hodge structures and associated numerical invariants, I will compute some examples for quartic 3-folds and sextic double solids. This will have applications to singularity theory and rationality problems which will be explored further in Erik Paemurru's talk.

Aleksandr Novikov, ICMS: Semiorthogonal Decompositions in Algebraic Geometry

This is a survey talk on derived categories in algebraic geometry. I will review the formalism of semiorthogonal decompositions (SODs), focusing on the underlying adjunction functors and admissibility conditions that make them work. We will look at standard examples, including exceptional collections, and discuss how mutations of these components naturally give rise to braid group actions. A main focus will be how SODs connect — or fail to connect — to classical geometric invariants. We will look at cases where the classical geometry breaks down, leaving behind non-commutative components or phantom categories that are invisible to standard additive invariants. I will finish with a discussion of some recent attempts and ongoing strategies to recover geometric meaning from these abstract triangulated structures.

Jérémy Guéré, University of Grenoble: New Atomic Invariants

In the first lecture, I will review the construction of atoms, beginning with an overview at the formal level before addressing the technical difficulties that necessitate the use of non-Archimedean fields. I will also discuss the behavior of the Hodge structure under Iritani's blow-up formula. In the second lecture, I will introduce the new atomic invariant and provide the proof for the following theorem: if a smooth complex cubic fourfold is rational, then its primitive cohomology is isomorphic, as a Hodge structure, to the shifted middle cohomology of a projective K3 surface. The proof relies on explicit computations for surfaces that I will present.

Giovanni Neto, ICMS Sofia: Flag Varieties as GKM Spaces and Equivariant Gromov–Witten Theory

In this talk, we present flag varieties as GKM spaces and use their Lie-theoretic structure to describe the associated combinatorial data explicitly. This perspective enables computations in equivariant cohomology, including the equivariant Gromov–Witten invariants. This will provide simpler atoms computations and lead to solving problems in G-equivariant birational geometry. This is based on ongoing joint work with Ludmil Katzarkov.

Pedro Muniz, ICMS Sofia: The Cohomology of Solvmanifold SYZ Mirrors

In this talk, we provide necessary and sufficient conditions for the existence of non-Kähler SYZ mirror pairs, in the sense of Lau, Tseng, and Yau, for solvmanifolds, following a construction proposed by Bedulli and Vannini. We further investigate the relationship between the cohomology of these pairs and, time permitting, propose new cohomological frameworks tailored to this setting. This is joint work in progress with L. Cavenaghi, L. Grama, and L. Katzarkov.

Ludmil Katzarkov, University of Miami, ICMS