Cohomology vanishing for tensor powers of tangent bundle on the flag variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:03:10Z http://mathoverflow.net/feeds/question/82253 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82253/cohomology-vanishing-for-tensor-powers-of-tangent-bundle-on-the-flag-variety Cohomology vanishing for tensor powers of tangent bundle on the flag variety Alexander Braverman 2011-11-30T06:56:47Z 2011-11-30T19:46:25Z <p>Let $X$ denote the flag variety of a semi-simple group $G$ (in characteristic 0) and let $T_X$ denote its tangent bundle. I would like to ask the following question(s):</p> <p>1) Is it true that for any $n\geq 0$ we have $H^i(X,T_X^{\otimes n})=0$ for $i>0$?</p> <p>2) More generally, let $\lambda$ be a dominant weight of $G$ and let $\mathcal O(\lambda)$ be the corresponding line bundle on $X$. Is it true that $H^i(X, T_X^{\otimes n}\otimes \mathcal O(\lambda))=0$ for $i>0$?</p> <p>When tensor powers of $T_X$ are replaced by symmetric powers, this is known to be true (for example it is proved in a paper of Kumar, Lauritzen and Thomsen).</p> http://mathoverflow.net/questions/82253/cohomology-vanishing-for-tensor-powers-of-tangent-bundle-on-the-flag-variety/82306#82306 Answer by Chuck Hague for Cohomology vanishing for tensor powers of tangent bundle on the flag variety Chuck Hague 2011-11-30T19:36:17Z 2011-11-30T19:46:25Z <p>I don't have a complete answer, but let me just note that your question 2) would be true if there is a Frobenius splitting of the cotangent bundle $T^*_{X^n}$ of $X^n$ which compatibly splits the diagonal copy $\Delta$ of $T^*_X$, the cotangent bundle of $X$. Since <code>$T^*_{X^n}$</code> is the cotangent bundle of the flag variety of the semisimple algebraic group $G^n$, there is a candidate for this splitting, namely the splitting of Kumar, Lauritzen, and Thomsen that you mention. I don't know, though, if their splitting compatibly splits $\Delta$; that is an interesting question.</p>