## Trascendence, Diophantine Approximation, and Localpositivity

#### Basic Course

Lecturer

##### Alex Küronya
University of Frankfurt
##### Stefano Urbinati
MAT/03
Freq. 4 / Ass. 2
###### Period
28/02/2022 – 11/03/2022
16 hours
##### Program

The topic of the course is the area between the interconnected fields of the study of transcendental numbers, Diophantine approximation on algebraic varieties, and local positivityof line bundles on projective varieties. We will roughly cover the basics of all three subjects,while following the train of thought how positivity of line bundles contributes to our understanding of transcendence theory and Diophantine approximation. Since birational geometryhas been evolving fast in the last decades, it is worth considering how new developments inthe area — especially the recently arising asymptotic theory — reflect on our understanding of these parts of number theory. Most of the course will have a two-tier approach, with its main thread being an advanced undergraduate affair, but providing information on the level of current research all the way through. For the first two-thirds of the course a basic knowledge of number theory andalgebraic geometry on the level of a decent first course is required, but not necessarily more. For the last part a working knowledge of birational geometry will be quite useful. For time and pedagogical reasons not all results will be proven in full detail, the emphasis will be rather on techniques, examples, and most importantly on the connection between number theory and local positivity. We will certainly discuss the cornerstones of transcendence theory and Diophantine approximation, but often they will serve as a compass and we will not necessarily care about all the details.

1. Transcendental numbers
• Define algebraic and transcendental numbers and recall their basic properties. We will discuss Liouville’s criterion and example, the transcendence proofs ofπandewhere the idea of certain auxiliary polynomials make their first entrance.
• Consider geometrically defined sequences from birational geometry whose algebraicity/transcendence is open, but has/would have strong structural consequences.
• Define semi-algebraic sets and periods in the sense of number theory, see their firstproperties, interesting examples, and the possible connection to volumes of line bundles.
• Review some milestones in transcendence theory (Lindemann– Weierstraß, Gelfond–Schneider, Baker) and analyze the structure of their proofs.
2. Diophantine approximation on the line
• Describe the Diophantine approximation problem on the line and on an algebraicvariety, prove some first results.
• State and prove Liouville’s Theorem, while carefully analyzing its strategy.
• Explain Dyson’s Lemma and its generalization by Esnault–Viehweg et al.
• Study the statement and the proof of the Thue–Siegel–Roth Theorem, and discussthe connection to positivity of line bundles.
3. Positivity of line bundles
• Introduce various notions of positivity (ample, nef, big, etc.), numerical equivalence ofline bundles/divisors, first appearance of convex geometry: cones of positive divisors.
• Discuss the asymptotic approach to invariants of line bundles: volume, asymptotic base loci. Emphasize that asymptotic invariants have better formal properties, often respect numerical equivalence.
• Summarize known results on arithmetic properties of volumes of line bundles, incase of interest explain the connection with the Riemann– Roch Theorem and Zariski decomposition.
• Define local positivity and Seshadri constants as a way to measure it. Describe theformal properties of Seshadri constants with an eye on Diophantine approximationconstants.
• Reappearance of convex geometry: Study Newton–Okounkov bodies of line bundlesand their connection to (local) positivity.
• Discuss arithmetic properties of volumes in the light of Newton– Okounkov theory,and the question of semi-algebraicity of Newton– Okounkov bodies.
4. Diophantine approximation in higher dimensions
• Recall the Diophantine approximation problem, state and discuss the Schmidt Sub-space Theorem, and the theorems of Faltings–W ̈ustholz and McKinnon–Roth.
• Study multiplicative filtrations on sections rings, order of vanishing, concave trans-form, theβ-invariant, mention the equivalent description of Evertse using weights.
• Discuss the relationship between local positivity and Diophantine approximation inthe given setting and prove the McKinnon–Roth Theorem via Faltings–W ̈ustholz.
• If time permits, play with semistability of filtered vector spaces/weights, and give asketch of the proof of the Faltings–W ̈ustholz approximation theorem.

As far as taking the course for credit goes, there will be an ample supply of exercises and more challenging problems, so that obtaining a grade based on homework should be unproblematic. Parts 1-2-3 are likely to be suitable for students with at least one number theory and algebraic geometry course as a background. During Part 4 I will move more freely and use more advance machinery, while still focusing on conveying the relevant ideas. The level of detail and to some extent the material presented will depend on the background and mathematical interest of the audience.

Seminary