Conference on Hodge theory, birational geometry and atoms

Room: SEM_SALA | Type: ATIVIDADES_SOCIAIS

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 explicitly describe the associated combinatorial data. This perspective enables computations in equivariant cohomology, including the equivariant Gromov–Witten
invariants. This will provide simpler atoms computations and lead to solve problems in G-
equivariant birational geometry. *ongoing joint work with Ludmil Katzarkov

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Rational Cubic Fourfolds and Their Relation to K3 Surfaces

I will briefly review the construction of atoms, specifically addressing certain technical difficulties regarding the behavior of Hodge structures under Iritanis blow-up formula. Subsequently, I will introduce a new atomic invariant and provide a proof for the following theorem: if a smooth complex cubic fourfold is rational, then its primitive cohomology is isomorphic - as a rational Hodge structure - to the shifted middle cohomology of a projective K3 surface.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Room: SEM_SALA | Type: ATIVIDADES_SOCIAIS

Quantum cohomology and irrationality of Gushel-Mukai fourfolds

Gushel-Mukai fourfolds behave very similarly to cubic fourfolds: they are Fano manifolds of K3-type, one can associate to them a (general) IHS manifold, their rationality is conjecturally controlled by their cohomology. In this talk, I will explain how one can get the
irrationality of very general Gushel-Mukai fourfolds through the theory of atoms. This will be done, as for cubics, by computing the small quantum cohomology of Gushel-Mukai fourfolds. Moreover, thanks to a suitable deformation of the small quantum cohomology ring, we will also deduce that a rational Gushel-Mukai fourfold has the same rational cohomology as some K3 surface. This is a joint work with L. Manivel and N. Perrin.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Recent progresses in the theory of atoms

In this talk we report on some new results related to the theory of atoms obtained jointly with L. Katzarkov and M. Kontsevich, which will appear in the '9.5 lectures in the theory of atoms' we are working together.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Room: SEM_SALA | Type: ATIVIDADES_SOCIAIS

Atom applications and MMP

In this talk we introduce the theory of Atoms. Applications and connection with MMP will be discussed.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Degeneracy loci of Poisson fivefolds

Let $X$ be a Fano manifold of dimension five and let $\sigma\in H^0(\X,\wedge^2 T_{X})$ be a holomorphic Poisson structure. We prove that the rank zero locus $D_0(\sigma)$ has an irreducible component of dimension at least $1$ and that the rank at most two locus $D_2(\sigma)$ has an irreducible component of dimension at least $3$. This confirms a conjecture by Bondal in dimension five. The proof uses: (i) an algebraic integrability criterion for codimension one foliations on weak Fano manifolds; (ii) a lemma of Gualtieri--Pym on degeneracy loci of ample Poisson modules; and (iii) a generalization of a result by Esteves-Kleiman on the zero loci of Pfaff fields along invariant subvarieties.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Room: SEM_SALA | Type: ATIVIDADES_SOCIAIS

Spectral decomposition of nc-Hodge structures and F-bundles

Non-commutative Hodge structures, which generalize classical Hodge structures for non-commutative spaces, consist of de Rham and Betti data, encoded as Stokes structures. These structures naturally arise in mirror symmetry in the context of the Gamma conjectures. In this talk, I will first discuss the behavior of Betti data under the topological Laplace transform, relating this to the spectral decomposition of non-commutative Hodge structures. I will then talk about the formal and non-Archimedean analogues of nc-Hodge structures, known as F-bundles, focusing on the spectral decomposition theorem for maximal F-bundles and its connection to Hodge atoms.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Categorical atoms and a view on birational classification of surfaces

In a recent work with Julia Schneider and Evgeny Shinder we constructed canonical "atomic" semi-orthogonal decompositions for derived categories of surfaces. Components of these decompositions are viewed as atoms and provide birational invariants of surfaces. I will explain how to compare atoms and compute some examples. Then I will review birational classification of surfaces over an arbitrary perfect field in terms of atoms, which appears to be uniform and concise. These results are mostly known due to works by many people, but some are new: for example, we show that two birational minimal del Pezzo surfaces of degree 4 are isomorphic.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Room: SEM_SALA | Type: ATIVIDADES_SOCIAIS

Finite simple subgroups of the real Cremona group of rank three

Very little is known about the classification of finite subgroups of Cremona in dimension three. It is natural to start with the case of simple groups, and this step was achieved by Prokhorov in 2009 over the field of complex numbers. In the work I will present, we show that the only non-cyclic finite simple subgroups of the real Cremona group of rank three are A5 and A6. This is a joint project with I. Cheltsov and Y. Prokhorov.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Blowups, Gale duality and moduli spaces

In this talk, we discuss the birational geometry of blowups of projective spaces at points in general position. For that, we explore Gale duality, a correspondence between sets of $n=r+s+2$ points in projective spaces $\mathbb{P}^s$ and $\mathbb{P}^r$. For small values of $s$, this duality has a remarkable geometric manifestation: the blowup of $\mathbb{P}^r$ at $n$ points can be realized as a moduli space of vector bundles on the blowup of $\mathbb{P}^s$ at the Gale dual points.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Room: SEM_SALA | Type: ATIVIDADES_SOCIAIS

SYZ for almost abelian Lie groups

In this talk, we discuss SYZ mirror symmetry in the non-Khler context, in the sense of Lau-Tseng-Yau. We construct SYZ dual pairs
for a family of solvmanifolds called almost abelian, and we explore applications related to dualities between Bott-Chern and Tseng-Yau
cohomologies. This is joint work in progress with L. Cavenaghi, L.
Katzarkov, and P. Muniz.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA

Rationally inequivalent points on complete intersections

Let X in P^{n+k} be a very general complete intersection of multidegree (d_1,...,d_k). Inspired by work of Voisin, in 2021 Chen--Lewis--Shen conjectured that the set of points rationally equivalent to and different from a fixed point x in X has dimension at most 2n - \sum (d_i-1). The authors proved this conjecture for hypersurfaces (k=1) and Riedl--Yang proved it when \sum (d_i-1) \leq 2n or \sum (d_i-1) \geq 2n+2. I will explain how to generalize the technique of Riedl--Yang to tackle the full conjecture as well as to obtain finer information about rational equivalence of points on complete intersections.

Room: Auditório 2 - Ricardo Mañé | Type: PALESTRA_PLENARIA