On real forms and birational transformations

This thesis is split into two parts: the first one, Part I, combines the first four papers that came out of my PhD, all on the topic of real forms. Two out of the four papers were written jointly with co-authors, namely one with Jérémy Blanc and Pierre-Marie Poloni, and the other one with Adrien Dubouloz. The second part, Part II, contains the fifth paper, on dynamical degrees. The introduction treats both parts together.
The papers are all available on the arXiv or can be accessed through the journals they were published in:
1. Anna Bot. Real forms on rational surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 24(3):1301–1326, 2023. Available at doi : 10.2422/2036-2145.202105_056 or
arXiv.2103.00909.
2. Anna Bot. A smooth complex rational affine surface with uncountably many real forms, accepted for publication at Ann. Sci. Éc. Norm. Supér. Available at arXiv.2105.08044.
3. Jérémy Blanc, Anna Bot and Pierre-Marie Poloni. Real forms of some Gizatullin surfaces and Koras-Russell threefolds. Publ. Mat., 67(2):851–890, 2023. Available at doi : 10.5565/publmat6722314 or arXiv.2108.12389.
4. Anna Bot and Adrien Dubouloz. Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms. Available at arXiv.2206.01713.
5. Anna Bot. The ordinal of dynamical degrees of birational maps of the projective plane. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 117-134. doi : 10.5802/cr-math.540. Also available at arXiv.2207.04408.
All parts rely on the intimate study of birational maps; though the objectives may change from one project to another, the tools from Birational Geometry are the same. This red thread strings all of the projects together.