The theory of Higgs bundles over smooth complex projective varieties is a crossroad of algebraic geometry, complex differential geometry and mathematical physics of course.

In this talk I will not go neither into the history of these very interesting mathematical objects nor their applications in physics. Instead I will try to explain what are the "ingredients" which one involves in this definition; to be clear, I will attempt to explain the geometric origins and the meanings of these "ingredients".

If the time permits, I will introduce some notion of "positivity" for Higgs bundles, which are the main topics of my Ph.D. thesis.

• Biswas, Bruzzo, Capasso, Graña Otero, Gurjar, Hernández Ruipérez, Lanza, Lo Giudice, Peragine - various papers
• S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press.
• R. K. Lazarsfeld (2004) Positivity in algebraic geometry. I and II, Springer-Verlag.
• C. Okonek, M. Schneider, H. Spindler (1988) Vector Bundles on Complex Projective Spaces. With an Appendix of S. I. Gel'fand, Birkhäuser.
• C. T. Simpson - Higgs bundles and local systems , Pubblications Mathématiques de l'I.H.É.S., 75 (1992) 5-95.