Weight filtration and Hodge theory for tropical varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:39:15Z http://mathoverflow.net/feeds/question/52681 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52681/weight-filtration-and-hodge-theory-for-tropical-varieties Weight filtration and Hodge theory for tropical varieties Jeffrey Giansiracusa 2011-01-20T19:41:15Z 2012-07-09T14:38:59Z <p>Many concepts is algebraic geometry have tropical analogues. </p> <p><strong>Question:</strong> Is there an analogue of the weight filtration or Hodge filtration for tropical varieties?</p> <p>A tropical curve ends up being essentially a metric graph. The tropical genus is the first Betti number of the graph. There is a period mapping (analogous to the classical Abel-Jacobi period map) from the moduli space of tropical curves of genus $g$ to the space <code>$GL_g(\mathbb{R})/O_g(\mathbb{R})$</code>. Can this period map be interpreted as a classifying map for variation of tropical Hodge structure?</p> http://mathoverflow.net/questions/52681/weight-filtration-and-hodge-theory-for-tropical-varieties/54486#54486 Answer by Eric Katz for Weight filtration and Hodge theory for tropical varieties 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/52681/weight-filtration-and-hodge-theory-for-tropical-varieties/101779#101779 Answer by David Speyer for Weight filtration and Hodge theory for tropical varieties David Speyer 2012-07-09T14:38:59Z 2012-07-09T14:38:59Z <p>Itenberg, Kazarkov, Mikhalkin and Zharkov have formulated a definition of tropical $H^{p,q}$ and presented it at a number of conferences, although I don't think that there is a preprint yet. You can watch Zharkov's talk on this from 2009 <a href="http://www.msri.org/web/msri/online-videos/-/video/showVideo/3839" rel="nofollow">here</a>. </p> <p>Their definition is restricted to the case that <code>$\mathrm{Trop} \ X$</code> locally looks like a tropical linear space. (For example, if $X$ is a curve then, at a vertex of degree $d$, the directions of the incoming edges must span a space of dimension $d-1$ and the unique relation between the minimal lattice vectors on these directions must be that their sum is $0$.) This can be thought of as a tropical "smoothness" condition.</p> <p>Roughly speaking, $H^{0,q}$ is related to the topological cohomology of <code>$\mathrm{Trop} \ X$</code> which should, in this context, be viewed as the cohomology of the sheaf of locally constant functions on <code>$\mathrm{Trop} \ X$</code>. $H^{p,q}(X)$ is related to the cohomology of a sheaf on <code>$\mathrm{Trop} \ X$</code> which is related to the degree $p$ part of Orlik-Solomon algebras of the matroids locally describing the relevant tropical linear spaces. (In other words, to $H^p$ of the corresponding hyperplane arrangement complement.)</p> <p>Until a preprint arrives, the best reference seems to be the papers of Kristin Shaw, a student of Mikhalkin. See Section 2.2 of <a href="http://www.math.toronto.edu/shawkm/theseShaw.pdf" rel="nofollow">her thesis</a> for a good start.</p>