Jie Liu

Thesis and notes, etc.

• Positivity in algebraic geometry  [PDF]   (2023 @ UCAS, Beijing)
• Géométrie des variétés de Fano: sous-faisceaux du fibré tangent et diviseur fondamental  [abstract]  [note]  [PDF]  (Ph.D. thesis, 2018 @ LJAD, Nice, France)
  
  This thesis is devoted to the study of complex Fano varieties via the properties of subsheaves of the tangent bundle and the geometry of the fundamental divisor. The main results contained in this text are:
  (1) a generalization of Hartshorne's conjecture: a projective manifold is isomorphic to a projective space if and only if its tangent bundle contains an ample subsheaf
  (2) vanishing theorem on irreducible Hermitian symmetric spaces of compact type \(M\) and its applications to the stability of the tangent bundles of smooth complete intersections in \(M\) and effective restriction problem;
  (3) Bogomolov-type inequality for Fano manifolds of Picard number one and its applications to effective nonvanishing problem, anticanonical geometry of Fano manifolds and the calculation of Seshadri constants of the anticanonical divisors of Fano manifolds with large index;
  (4) geometry of smooth Moishezon manifolds with Picard number one.
• Techniques \(L^2\) en géométrie complexe  [abstract]  [PDF] (Master's thesis, 2015 @ IRMAR, Rennes, France)
  
  The main aim of this thesis is to review the proof of Oshawa-Takegoshi extension theorem and its applications in complex geometry, including Siu's invariance of plurigenera, semi-continuity theorem of Demailly-Kollár and strong openness conjecture and so on.

For students

Below I collect some general adivices from other mathematicians to students, especially for those who want to pursue a PhD degree in mathematics.

• To the reader: Preface to An Introduction to Riemannian Geometry  (by Hung-Hsi Wu, in Chinese only)
• For potential Ph.D. students (by Ravi Vakil)
• Career advice (by Terence Tao)
• Advice for Ph.D. students in math (by Bjorn Poonen)
• Advice for the Young Scientist (by John Baez)

Useful Links

• Upcoming Conferences in Algebraic Geometry (maintained by Ravil Vakil)
• My entry in the Mathematical Genealogy Project   [PDF]