User roman - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:43:10Z http://mathoverflow.net/feeds/user/24508 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114557/cotangent-to-flags-as-a-quiver-variety cotangent to flags as a quiver variety Roman 2012-11-26T17:14:53Z 2013-03-19T12:43:26Z <p>It is easy to realize cotangent space to the flag variety $Fl=SL_n/B$ as a Nakajima quiver variety: consider the finite quiver of type A, the dimension vectors v=(1,2,...,n-1), w=(0,...,0,n); an appropriate stability condition (polarization) amounts to the condition that the arrow from the i-dimensional space to the (i+1)-dimensional one is injective, and we end up with a complete flag in the n-space, the arrows in the opposite direction giving a cotangent vector.</p> <p>Now, if I understand correctly, the other stability conditions (of which there is n!) should produce quiver varieties which are also isomorphic to $T^*(Fl)$. How to see this, preferably using equally explicit linear algebra? Is it explained in the literature?</p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/116339#116339 Answer by Roman for Euler-Maclaurin formula and Riemann-Roch Roman 2012-12-14T02:25:23Z 2012-12-14T02:25:23Z <p>I was about to post the same question and came across yours. I wasn't aware of the "toric" direction here that other people have referred to, but I know a pretty answer in the particular case when $X$ is the flag variety of a semi-simple algebraic group. In this case RR reduces to saying that $\chi(F)={\mathrm{const}} \int ch(F\otimes L^{-1})$ where $L$ is the square root of the canonical class and the constant is explicit. So in this case at least multiplication by Todd does amount to shift (by half-forms) as in Euler-Maclaurin formula. Furthermore, in this form the formula has a very short proof via characteristic $p$, deducing it from the fact that $Fr_*(L)=L^{p^d}$, $d=\dim(X)$.</p> <p>[Well, in fact it also follows from Weyl dimension formula which of course has many other proofs, I just happen to like this char p proof.] It would be cool to have a proof of the general case along these lines. Something related has been done by Pink and Rossler, arXiv:0812.0254. </p> http://mathoverflow.net/questions/115217/stable-maps-of-sheaves-on-a-curve stable maps of sheaves on a curve Roman 2012-12-02T23:54:21Z 2012-12-02T23:54:21Z <p>There is a notion of (semi)stability for a pair consisting of a vector bundle on a smooth projective algebraic curve and its section, so the moduli space of (semi)stable pairs can be constructed and studied. How to spell out a generalization to more general pairs of coherent sheaves on a curve with a map between them?</p> http://mathoverflow.net/questions/114926/representation-theory-of-cartan-type-simple-algebra-in-positive-characteristic representation theory of Cartan type simple algebra in positive characteristic Roman 2012-11-29T20:30:03Z 2012-12-01T11:12:25Z <p>This is a general question about representation theory of finite dimensional simple Lie algebras of Cartan type over algebraically closed fields of positive characteristic (vector fields on Frobenius neighborhood of a point on a smooth variety preserving an appropriate tensor field). Has the basic picture of representations been worked out? Classification, dimensions, character formulas?</p> <p>Another questions is if their cohomology is understood. </p> http://mathoverflow.net/questions/99717/blow-ups-of-secant-varieties/99802#99802 Answer by Roman for blow-ups of secant varieties Roman 2012-06-16T18:11:34Z 2012-06-16T18:11:34Z <p>Can you give an example when X_k isn't smooth? -- thanks</p> http://mathoverflow.net/questions/114926/representation-theory-of-cartan-type-simple-algebra-in-positive-characteristic Comment by Roman Roman 2012-11-30T20:32:58Z 2012-11-30T20:32:58Z Thanks for the references! http://mathoverflow.net/questions/114926/representation-theory-of-cartan-type-simple-algebra-in-positive-characteristic/114973#114973 Comment by Roman Roman 2012-11-30T20:31:42Z 2012-11-30T20:31:42Z Thank you!! Is there a survey where some of this (representation theory, not the classification of Lie algebras) is explained? http://mathoverflow.net/questions/114557/cotangent-to-flags-as-a-quiver-variety Comment by Roman Roman 2012-11-28T00:50:45Z 2012-11-28T00:50:45Z Of course I am considering a generic $\theta$ in order to get cotangent bundle which is nonsingular variety. For example $\theta=(1,...,1)$. http://mathoverflow.net/questions/114557/cotangent-to-flags-as-a-quiver-variety Comment by Roman Roman 2012-11-27T02:13:57Z 2012-11-27T02:13:57Z I just mean the usual $\theta$ stability, as defined for example in Ginzburg's notes cited