Mathematics

Description:
In the first part of the course we will cover the basic theory of compact complex manifolds, in particular those admitting a strongly compatible Riemannian metric (i.e. a Kähler metric), especially in the case of vanishing Ricci curvature (i.e. Calabi-Yau manifolds). In the second part we will study a remarkable class of submanifolds of Calabi-Yau manifolds, known as special Lagrangian submanifolds, following ideas of Thomas, Yau, Joyce and Li.

Contents:
Complex manifolds. Hermitian and Kähler metrics. Aubin-Calabi-Yau Theorem. Deformed Hermitian Yang-Mills connections. Special Lagrangian submanifolds and the Thomas-Yau Conjecture.

References:

Kähler geometry:

• G. Székelyhidi,  An introduction to extremal Kähler metrics. Graduate Studies in Mathematics, 152. American Mathematical Society, Providence, RI, 2014. xvi+192 pp. ISBN: 978-1-4704-1047-6
• G. Tian, Canonical metrics in Kähler geometry. Notes taken by Meike Akveld. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2000. vi+101 pp. ISBN: 3-7643-6194-8

Special Lagrangians:

• D. Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62.
• Y. Li, Thomas-Yau conjecture and holomorphic curves. EMS Surv. Math. Sci. (2025), published online first DOI 10.4171/EMSS/96.