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Pamphlets

On this page I will post a series of short very informal pamphlets in Kahler geometry. Each pamphlet is dedicated to a particular paper in the field (or a small collection of related papers), and in some cases a brief introduction to a topic in the field, explaining the necessary background, and giving an outline of the main ideas. There is no original mathematics in this collection, and in certain instances concepts are deliberately over simplified to bring across the main points. I apologize in advance for the mistakes, misattributions and omissions – there will be periodic updates in which errors are hopefully corrected and new material added.

 

 

 

  1. sites.rutgers.edu/…0/cut-off-function.pdfSingular Kahler-Einstein currents. The theory of KE currents on singular varieties was developed by Eyssidieux, Guedj and Zeriahi in 2008 and has inspired a lot of work in the field. The point of view in these notes is taken mainly from Jian Song’s papers Riemannian geometry of Kahler Einstein currents I, II, III (part III coauthored with Sturm-Wang).
  2. Background for Scalar Curvature and Projective imbedding I contains material to prepare for reading this 2001 paper of S. Donaldson: “Scalar curvature and projective embeddings. I.” J. Differential Geom. 59 (2001)
  3. Weighted Monge Ampere Energy of Quasi-psh functions These are notes for the 2007 paper of Guedj-Zeriahi,”The sites.rutgers.edu/…0/cut-off-function.pdfweighted Monge-Ampère energy of quasiplurisubharmonic functions”J. Funct. Anal. 250 (2007), no. 2,
  4. Elliptic PDE and the Hodge Theorem These are notes based on Warner’s book which allows us to quickly prove the basic estimates from elliptic pde and then apply the results to prove the Hodge Theorem.
  5. Lempert Papers In these notes we summarize Lempert’s papers:  A) “La m ́etrique dsites.rutgers.edu/…0/cut-off-function.pdfe Kobayashi et la repr ́esentation des domaines sur la boule”, Bulletin de la S.M.F., tome 109, (1981), 427-474  B) “Intrinsic distances and holomorphic retracts”, Complex analysis and applications, Sofia, 1984  C) “Solving the degenerate complex Monge-Amp`ere equation with one concentrated singularity”, Math. Ann. 263, (1983) 515-532
  6. Berman KE implies K stable These are note on Berman’s paper “K-polystability of ℚ-Fano varieties admitting Kähler-Einstein metrics”. Invent. Math. 203 (2016)
  7. Coman Guedj Zeriahi Background This is some background material for reading  Coman, Dan; Guedj, Vincent; Zeriahi, Ahmed Extension of plurisubharmonic functions with growth control. J. Reine Angew. Math. 676 (2013)
  8. Donaldson Scalar II Background This is a discussion of  Donaldson, S. K. Scalar curvature and projective embeddings. II. Q. J. Math. 56 (2005), no. 3, 345–356.
  9. Teichmuller notes These notes discuss some preliminary ideas and include a proof of the uniformization theorem.
  10. Notes on the openness conjecture A brief outline of the main idea from Berndtsson,  “The openness conjecture and complex Brunn-Minkowski inequalities. Complex geometry and dynamics, 29–44,
    Abel Symp., 10, Springer, Cham, 2015.”

 

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