ICM 2026 Invited Section Lecture – Jorge Vitório Pereira

Value of the Information & Strength of the Argument

The information value is exceptional, offering insight into a deeply intricate area of mathematics where several subfields (birational geometry, deformation theory, and differential geometry) converge. The argumentation is extremely solid, built on precise definitions and rigorous proofs. The speaker carefully navigates from established classical results to very recent breakthroughs, explaining the logical necessity of each step (e.g., restricting to canonical singularities, using uniruledness to gain rational curves, and employing positive characteristic to gain extra derivations). The clear exposition of the logic behind the classification theorems demonstrates a robust and convincing mathematical narrative.

Scientific Rigor, Source Quality, Title Accuracy

Scientific rigor is exemplary. The speaker meticulously credits prior authors (Reeb, Brunella, Cerveau, do Nascimento, Touzet, Druel, Miyaoka, Campana, etc.) and clearly separates proven theorems from conjectures. The sources are named explicitly without URLs in the talk’s description, which is standard for mathematical presentations. The title accurately reflects the content, as the entire lecture is dedicated to foliations with numerically trivial canonical bundle. A high degree of technical sophistication is maintained throughout, targeting an expert audience.

The title perfectly matches the content: an invited section lecture at ICM 2026 focused on holomorphic foliations with numerically trivial canonical bundle on projective manifolds.

9/10

High-quality invited lecture by a leading expert, presenting a rigorous survey of advanced research in algebraic geometry. The presentation is structured, credits prior work accurately, and demonstrates a high level of technical depth and mathematical sophistication.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction to holomorphic foliations, definitions of tangent sheaf, conormal sheaf, and Frobenius theorem.
• 04:00Definition of numerically trivial canonical bundle and motivations from projective space and Poisson geometry.
• 08:00Classical classification results on projective spaces, including Cerveau-do Nascimento theorem for degree-two foliations on P3.
• 12:00Holomorphic Poisson geometry and its role in constructing examples of foliations with trivial canonical bundle.
• 17:00Touzet's classification of smooth codimension-one foliations on compact Kähler manifolds.
• 23:00Presentation of joint work with Loray and Touzet: canonical singularities and the smoothness result on non-uniruled manifolds.
• 29:00Deformation theory of rational curves and the role of Birkhoff-Grothendieck splitting in controlling the foliation.
• 35:00Reduction to positive characteristic to exclude transverse Lie algebras and prove the existence of closed rational one-forms.
• 39:00Current developments in the minimal model program for foliations and open problems in higher codimensions.

• Cerveau & do Nascimento (1996) classification of degree-two foliations on P3 — Mentioned as a celebrated result classifying codimension-one foliations with trivial canonical bundle on P3.
• Touzet (2008) classification of smooth codimension-one foliations — Mentioned as extending the classification to compact Kähler manifolds using differential geometric methods.
• Druel (1999) study of Poisson structures on projective threefolds — Mentioned in the context of pin pointing the birth of holomorphic Poisson geometry and its connection to canonical bundle computations.
• Miyaoka's generic semi-positivity theorem — Mentioned as providing key techniques used in the deformation theory and positive characteristic arguments.

• Touzet's classification of smooth foliations — Agrees with the presented framework for the smooth case, which is later extended to canonical singularities.
• Druel's work on Poisson structures — Aligns with the result that Poisson structures give rise to foliations with numerically trivial canonical bundle.
• Spicer and Caccini on MMP for foliations — Provides the birational geometry foundations needed to understand the canonical singularities case mentioned.