most recent 30 from http://mathoverflow.net 2013-05-23T07:06:24Z http://mathoverflow.net/feeds/user/12761 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113846/how-to-recover-toric-invariants-tropically/114203#114203 Eric Katz 2012-11-23T01:47:43Z 2012-11-23T01:47:43Z

<p>One has to be precise when one talks about the tropicalization of a toric variety. Usually, one takes tropicalizations of subvarieties of an algebraic torus. To get around this constraint, there is a notion of extended tropicalization of subvarieties of a toric variety. Although I haven't checked the definitions, the extended tropicalization of a toric variety should contain exactly the same information as the fan.</p> <p>The question should perhaps go the other way. There are lots of recipes for reading geometric invariants of toric variety from the fan which is a combinatorial object. Given a subvariety of a toric variety, can one read off geometric invariants of the subvariety from the tropicalization. The case of the toric variety is the prototypical example, but maybe other varieties act like toric varieties in this sense.</p>

http://mathoverflow.net/questions/60251/generalizations-of-drinfeld-symmetric-space-drinfeld-homogeneous-space-drinfe Eric Katz 2011-04-01T03:09:49Z 2011-04-01T03:09:49Z

<p>Are there natural generalizations of the Drinfeld symmetric space? For $\mathbb{K}$, a non-Archimedean local field, the Drinfeld symmetric space can be defined as the complement of all $\mathbb{K}$-rational hyperplanes in $\mathbb{P}^r_{\overline{\mathbb{K}}}$. One can generalize it in the following direction: let $X$ be some variety defined over $\mathbb{K}$ and $L$ some line bundle on $X$ defined over $\mathbb{K}$; consider the complement of the union of the zero loci of sections $L$ defined over $\mathbb{K}$. An example of this would be to consider $X=G/H$, a homogeneous space and $L$, a line bundle defined by a representation of $G$. Even more concretely, one can consider a flag variety or a Grassmannian and the polarization induced by its Plucker embedding.</p> <p>This seems like a natural thing to consider. Moreover, the analogs of the Drinfeld's symmetric space's connections to Bruhat-Tits theory and to degenerations of $\mathbb{P}^r$ might be interesting. </p> <p>Has any such thing appeared in the literature? </p>

http://mathoverflow.net/questions/54860/when-is-the-quotient-of-a-tropical-curve-also-a-tropical-curve/54916#54916 Eric Katz 2011-02-09T19:20:44Z 2011-02-10T06:23:54Z

<p>A good place to start would be to look at covering spaces. A covering space of a topological space X is a space that is locally isomorphic to X but globally unwinds some of the topology. A good reference is Hatcher's Algebraic Topology notes (section 1.3 has what you'd need in the first couple of pages). Sometimes X is the quotient of its covering by a group action. This property is called normality and there is a nice group theoretic condition for it. As a fun exercise, before you even begin, try to find some graphs Y with group actions such that Y/G is the figure 8.</p> <p>In a certain sense, covering spaces are the "easiest" types of quotients. This is because the action is properly discontinuous. For this reason, the quotient inherits the tropical structure of the original graph. For more exotic group actions, you may want to look at Matthias Herold's preprint: "Tropical orbit spaces and the moduli spaces of elliptic tropical curves." There should be some clues in that paper for how to work with graphs with finite leaves.</p>

http://mathoverflow.net/questions/53306/what-can-we-learn-from-the-tropicalization-of-an-algebraic-variety/54490#54490 Eric Katz 2011-02-06T04:01:26Z 2011-02-06T04:01:26Z

<p>For general subvarieties of an algebraic torus, the tropicalization knows about the class of the subvariety in a suitable toric compactification of the algebraic torus. So you can compute intersection products of subvarieties of a torus tropically. See the recent preprint of Osserman-Payne for the state of the art.</p> <p>For subvarieties X that are schon (which is a natural smoothness condition), you can say a lot more. There is a natural dualizing complex $\Gamma_X$ which maps to Trop(X) whose homology reflects the lowest bit of the weight filtration on X. From this fact, you can get the natural generalization of $g(X)\geq b_1(\Gamma)$ (it is not in general true that $g(X)\geq b_1(\operatorname{Trop}(X))$ because the tropicalization map may have disconnected fibers (as was pointed out by Speyer)). </p> <p>There are two special cases where you can say a lot more: when $X$ is a schon hypersurface and when Trop(X) is smooth (smoothness here means Trop(X) is locally modeled on matroid fans). In this case, you can say things about the Hodge numbers of X. See my paper with Stapledon for details. Warning: the results of that paper require compactifying X by completing the algebraic torus to a toric variety. We have a sequel in the works that will use more sophisticated Hodge theory to get around that problem.</p>

http://mathoverflow.net/questions/52681/weight-filtration-and-hodge-theory-for-tropical-varieties/54486#54486 Eric Katz 2011-02-06T03:44:05Z 2011-02-06T03:44:05Z

<p>I'm not quite sure if there's a useful notion of Hodge/weight filtration on a tropical variety. If we look at tropical varieties that are tropicalization of algebraic varieties over a non-Archimedean field, the topology of the tropical variety is related to the lowest weight bit of the weight filtration. I don't know how that bit is naturally filtered any further. </p> <p>The question I was curious about in my research statement is: is there a combinatorial way to encode higher bits of the weight filtration? I suspect that they can be expressed as a complex of sheaves on the tropical variety. </p> <p>I think there's probably a precise way of formulating your statement about tropical curves as "the Abel-Jacobi map commutes with tropicalization for totally degenerate curves." For details, look at p.19 of my paper with David Helm. I'm not sure if there's a natural way to tropicalize the period domain, but that'd be a fun question to address.</p>

http://mathoverflow.net/questions/113846/how-to-recover-toric-invariants-tropically/114203#114203 Eric Katz 2012-12-16T06:29:30Z 2012-12-16T06:29:30Z

For extended tropicalizations, see: <a href="http://arxiv.org/abs/0805.1916" rel="nofollow">arxiv.org/abs/0805.1916</a> The Todd class of a toric variety can certainly be formulated in terms of combinatorial data. But it might be quite difficult to make sense of the Todd class of a more general tropical variety. The most simple tropical varieties after tropicalizations of toric varieties are locally matroidal varieties (see papers of Mikhalkin, my paper with Stapledon for details, or Dustin Cartwright's recent paper on the Grobner stratification for details). One can probably talk about the chern classes of their tangent bundles...

http://mathoverflow.net/questions/109941/publication-and-career-as-a-fresh-ph-d/109948#109948 Eric Katz 2012-10-18T03:15:58Z 2012-10-18T03:15:58Z

Just a thought: it's often hard to judge the strength of your own results. Often while writing a dissertation, one has a lot of ideas that they can't fit in or don't have time to work on but which seem easy because he/she has developed a big machine. These seem less easy to people who have not written that same dissertation. I recommend to recent Ph.D.'s that they turn those ideas into papers. Chances are that they are going to make lots of mistakes in trying to publish their results and may have a turnaround as long as a year on papers. So it's worth writing some small papers early on.

http://mathoverflow.net/questions/55018/tropical-homological-algebra Eric Katz 2011-02-15T19:55:33Z 2011-02-15T19:55:33Z

An interesting couple of question along these lines is: 1. can one interpret the Baker-Norine rank of a linear system on a metric graph as the rank of a tropical module? 2. are there higher cohomology analogs of the Baker-Norine rank. The Baker-Norine rank makes sense from the point of view of deformation theory, but there should be a way of doing tropical homological algebra to make it natural.

http://mathoverflow.net/questions/53306/what-can-we-learn-from-the-tropicalization-of-an-algebraic-variety/54490#54490 Eric Katz 2011-02-06T04:06:26Z 2011-02-06T04:06:26Z

To JSE: higher dimensional analogs of Matt Baker's specialization lemma should hold. I've worked out some special cases for surfaces using tropical intersection theory. I think the right approach to the higher-dimensional analog should use rigid analytic geometry.