Samouil Molcho

Email: samouil.molcho (at) mail (dot) huji (dot) ac (dot) il


I am a postdoc at the Hebrew University of Jerusalem working in algebraic geometry, under Michael Temkin . My advisor was Dan Abramovich .


Mailing Address:
Einstein Institute of Mathematics
Edmond J. Safra Campus
The Hebrew University of Jerusalem
Givat Ram. Jerusalem, 9190401, Israel
Fax. 972-2-5630702

Other Information

My CV
My Research Statement

Research Interests

Logarithmic and Tropical Geometry; Gromov-Witten Theory and Enumerative Geometry; Moduli Theory; Manifolds with Corners.

My research focuses on using techniques of logarithmic geometry to understand geometric problems, particularly problems that arise from moduli theory. Logarithmic geometry is a beautiful theory developed by Fontaine, Illusie and Kato, which lies on the crossroads between algebraic, tropical and Berkovich geometry. One of the guiding principles of logarithmic geometry is that many singular spaces are smooth in the logarithmic context, and can be studied very effectively using techniques that are straightforward analogues of the techniques used to study smooth spaces. My early interests were in using logarithmic techniques to study degenerate objects in differential geometry, such as manifolds with corners, and on the algebro-geometric side in degenerations of stable maps and enumerative invariants. Recently I have become interested in log abelian varieties -- in particular log Picard varieties. These are a kind of degeneration of abelian varieties that exist in the logarithmic setting, but cannot exist in the world of classical schemes. In a sentence, they are honest algebraic spaces in the category of log schemes. These spaces have rich structure and some striking properties, but their geometry is mysterious and remains still largely undeveloped. My Oberwolfach report on the Log Picard variety can perhaps give a flavour.

Papers

1. Logarithmic Geometry and Manifolds with Corners
joint with W.D. Gillam 		link
2. Localization for Logarithmic Stable Maps
joint with E. Routis 		link
3. Logarithmic Stable Toric Varieties and Their Moduli
joint with K. Ascher		 link
4. Moduli of Morphisms of Logarithmic Schemes
Appendix C in J. Wise's paper 		link
5. A Theory of Stacky Fans
with W.D. Gillam 		link
6. Universal Weak Semistable Reduction
link
7. The Logarithmic Picard Group and its Tropicalization
with J. Wise 		link
8. Logarithmically Regular Maps
with M. Temkin 		link
The following papers are complete, but my collaborators and I had wanted to revisit in the future to make additions. Specifically, we had wanted to explain the differential geometric side of 9., that is, the connection of the differential geometric realization of the Chow quotient to the moduli space of flow lines of a Morse function. In 10. we wanted to tie in the results with more familiar versions of tropicalization. Since a revision has not happened in a while, and the ideas in the papers have been useful to me in other contexts, I post them here in case someone else finds them helpful as well.
9. Stable Logarithmic Maps as Moduli Spaces of Flow Lines
joint with W.D. Gillam		 link
10. Tropicalizing the Moduli Space of Broken Toric Varieties
joint with J. Wise		 link

In preparation

1. The Logarithmic Deligne Pairing
joint with M. Ulirsch, J. Wise
2. Crepant Resolutions and the Log McKay correspondence
joint with G. Liu
3. Log Picard and Neron Models
joint with D. Holmes, G. Orrechia, T. Poiret, J. Wise
4. Compactifications of the Universal Tropical Jacobian
joint with M.Melo, M. Ulirsch, F. Viviani J. Wise