Columbia Informal Complex Geometry
and PDE Seminar

Academic Year 2009-2010

Organizers: D.H. Phong, V. Tosatti

---

The seminar met on Thursdays from 2.40pm to 4.00pm in 520 Mathematics.

---

Schedule

• September 10
  Phong - Organizational Meeting

• September 17
  Jian Song - The Kähler-Ricci flow on complex surfaces - Reference

• September 24
  Valentino Tosatti - Lower boundedness of the K-energy on deformations of the Mukai threefold I - Reference

• October 1
  Valentino Tosatti - Lower boundedness of the K-energy on deformations of the Mukai threefold II - Reference

• October 8
  Gábor Székelyhidi - Holomorphic discs and the homogeneous complex Monge-Ampère equation - Reference

• October 15
  Jacob Sturm - Convergence of Riemannian Manifolds I - Reference

• October 29
  Jacob Sturm - Convergence of Riemannian Manifolds II - Reference

• November 5
  Valentino Tosatti - Quantization of the space of Kähler metrics - References 1 2

• November 12
  Tien-Cuong Dinh - Exponential estimates for plurisubharmonic functions and applications to dynamics - Reference

• November 19
  Gábor Székelyhidi - K-stability of constant scalar curvature Kähler manifolds - Reference

• December 10
  Valentino Tosatti - Kähler-Einstein metrics on Fano surfaces I - Reference

• January 28
  Valentino Tosatti - Kähler-Einstein metrics on Fano surfaces II - Reference

• February 4
  Gábor Székelyhidi - Okounkov bodies and test configurations - References 1 2

• February 11
  Jian Song - Regularity of plurisubharmonic upper envelopes in big cohomology classes - Reference

• February 18
  Jacob Sturm - A variational approach to complex Monge-Ampère equations - Reference

• February 25
  Phong - Convergence properties of the Yang-Mills flow on Kähler surfaces - References 1 2 3 4 5

• March 4
  Gábor Székelyhidi - The Bernstein problem for affine maximal hypersurfaces - Reference

• March 11
  Valentino Tosatti - Harmonic maps and rectifiability - Reference

• March 25
  Ovidiu Munteanu - Regularity of weakly harmonic maps - References 1 2 3

• April 1
  Nam Q. Le - The affine Plateau problem - Reference

• April 8
  Ben Weinkove (UC San Diego) - Contracting exceptional divisors by the Kähler-Ricci flow - Reference - Abstract

  We give a criterion under which a solution g(t) of the Kähler-Ricci flow contracts exceptional divisors on compact manifold and can be uniquely continued on a new manifold. This is a joint work with Jian Song.

• April 15
  Vincent Guedj (Marseille) - Viscosity solutions to degenerate complex Monge-Ampère equations - Abstract

  Degenerate complex Monge-Ampère equations on compact Kähler manifolds have been recently intensively studied using tools from pluripotential theory. We develop an alternative approach based on the concept of viscosity solutions. This approach works only for a rather restricted type of degenerate complex Monge-Ampère equations. Nevertheless, we prove that the local potentials of the singular Kähler-Einstein metrics constructed previously by the authors are continuous plurisubharmonic functions. They were previously known to be locally bounded. Another application is a lower order construction with a C⁰-estimate of the solution to the Calabi conjecture which does not use Yau's celebrated theorem. This is joint work with P.Eyssidieux and A.Zeriahi.

• April 22
  Sławomir Dinew (Jagiellonian University) - Hölder regularity for complex Monge-Ampère equations - Abstract

  In geometric analysis many arguments rely on a suitable regularity theory for the analyzed differential equations. Similarly to the solution of the Calabi conjecture often deriving suitable a priori estimates is in fact the heart of the matter. In the talk a regularity result for the complex Monge-Ampère equation will be presented. We will prove that any C^1,1 smooth plurisubharmonic solution u to the problem det(u_ij̄) = f with f strictly positive and Hölder continuous has in fact Hölder continuous second derivatives. For smoother f this follows form the classical Evans-Krylov theory yet in our case it cannot be applied directly. Instead we shall follow closely an idea of Xu-Jia Wang. Finally we shall discus how this particular regularity result can be used to justify an argument in the proof of uniqueness of metrics of constant scalar curvature by Chen and Tian.

  Yuan Yuan (Rutgers) - Kähler-Ricci flow on singular Calabi-Yau varieties

• April 29
  Adam Jacob - Nonabelian Hodge Theory - References 1 2 3

• May 6
  Jacob Sturm - The structure of spaces with Ricci curvature bounded below - Part I - References 1 2

  Jacob Sturm - The structure of spaces with Ricci curvature bounded below - Part II - References 1 2

• May 13
  Valentino Tosatti - Convergence of Einstein manifolds - References 1 2

• May 20
  Jacopo Stoppa (Cambridge) - Donaldson-Thomas invariants and hyperkähler metrics - Reference - Abstract

  I will briefly introduce Donaldson-Thomas invariants counting semistable sheaves on a Calabi-Yau threefold. I will then try to turn this into a differential geometry seminar, relying on the work of Gaiotto, Moore and Nietzke that connects DT invariants to some special hyperkähler metrics. I aim at pointing out one analogy between their metrics and some results I wrote down in this preprint.

---

Notes from the talks

• Valentino Tosatti - Lower boundedness of the K-energy on deformations of the Mukai threefold - PDF

• Valentino Tosatti - Kähler-Einstein metrics on Fano surfaces - PDF