University of Michigan
Department of Mathematics
 Topics in Algebraic Geometry
Winter 2008
Mondays 3-4pm, 2866 East Hall

This semester, we will be looking at Log Geometry.

Schedule of Talks
Date Speaker Topic Abstract
Monday January 7th Not meeting this week Not meeting this week
Monday January 14th Radu Laza Introduction to Log Geometry
Monday January 21st Not meeting this week Martin Luther King Day
Monday January 28th Mircea Mustaţă The definition of a log structure
Monday February 4th Meeting cancelled
Monday February 11th Howard Thompson TBD
Monday February 18th Kevin Tucker TBD
Monday March 10th Kyle Hofmann Logarithmic differentials We'll discuss the definition and basic properties of differentials for logarithmic schemes. These are mostly analogous to ordinary differentials. In some cases they are identical to the ordinary differentials, and we'll give a criterion for when this is so. But they aren't always identical to the ordinary differentials: We'll compute the relative differentials for a morphism of affine toric varieties, and as a consequence, it will turn out that the map x -> x^n is logarithmically unramified when the characteristic of the ground field does not divide n. We'll also describe the case of a normal crossing divisor.
Monday March 17th Kyungyong Lee Smoothings of normal crossing varieties In deformation theory it is often useful to consider only deformations that are compatible with a log structure. I will talk about the theorem saying that a normal crossing variety has a log structure of semistable type if and only if it is d-semistable. I'll focus on examples.
Monday March 24th Yogesh More Log Smoothness We will define the notion of log smoothness, and illustrate it with examples.
Monday March 31st Eugene Eisenstein TBA TBA
Monday April 7th Karl Schwede Moduli of curves from the point of view of log geometry. We will discuss the paper Log smooth deformation and moduli of log smooth curves by Fumiharu Kato. A reprint can be found here http://www.math.kyoto-u.ac.jp/~kato/Data/moduli.pdf.
Monday April 14th Howard Thompson Log smooth deformation theory We will discuss the paper Log smooth deformation theory by Fumiharu Kato in light of his paper Functors of log Artin rings.

Related papers and other references

  • Lectures on logarithmic algebraic geometry, Arthur Ogus. This manuscript of a book will be our main source.
  • A list of links compiled by Howard Thompson, Click Here
    Some of these are also directly linked below.
  • Logarithmic structures of Fontaine-Illusie, Kazuya Kato. The foundational paper.
  • Toric Singularities, Kazuya Kato. Links log geometry with toric varieties.
  • Toroidal crossings and logarithmic structures, Stefan Schroer and Bernd Siebert. Suggested as a place to start if interested in string theory.
  • Affine manifolds, log structures, and mirror symmetry, Mark Gross and Bernd Siebert. A survey article on links with mirror symmetry. Possibly another good place to start. Also see Mirror Symmetry via Logarithmic Degeneration Data I and Mirror Symmetry via Logarithmic Degeneration Data II
  • Log Smooth Deformation Theory, Fumiharu Kato. Links with deformation theory.
  • Functors of log Artin rings , Fumiharu Kato. Further links with deformation theory.
  • Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Yujiro Kawamata and Yoshinori Namikawa.
  • Semistable degenerations and period spaces for polarized K3 surfaces, Martin Olsson. An application to K3 surfaces.
  • Logarithmic Embeddings and Logarithmic Semistable Reductions, Fumiharu Kato. A criterion for logarithmic embeddings is given.
  • Reference cards on log schemes, by Maurizio Cailotto

    For more information, or to volunteer to give a talk, contact Karl Schwede or Radu Laza.