User botong wang - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:45:07Z http://mathoverflow.net/feeds/user/4975 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85908/stability-condition-for-vector-bundles Stability condition for vector bundles Botong Wang 2012-01-17T15:19:59Z 2012-01-18T05:02:21Z <p>Let $X$ be a smooth projective variety over $\mathbb{C}$, and fix $A$ an ample divisor as the polarization. We say a vector bundle $E$ to be (semi-)stable, if for any proper subsheaf $F$ of $E$, $\mu(F)&lt;\mu(E)$ (resp. $\leq$). </p> <p>I guess it is not sufficient to check just subbundles $F$ of $E$. But is there a counterexample? Or more precisely, is there an example of $(X, A, E, F)$ satisfying the following conditions?</p> <p>(1)$X,A,E$ are as above, and $F$ is a proper subsheaf of $E$ breaking the stability condition, i.e., $\mu(F)\geq \mu(E)$.</p> <p>(2)There exists no vector bundle which break the stability condition, i.e., $\mu(F')&lt;\mu(E)$ for any subbundle $F'$ of $E$.</p> http://mathoverflow.net/questions/19635/gauge-theory-construction-of-moduli-of-vector-bundles Gauge theory construction of moduli of vector bundles Botong Wang 2010-03-28T16:37:16Z 2010-04-05T20:43:33Z <p>Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach. </p> <p>To classify all topologically trivial vector bundles is same as to classify all possible complex structures on the topologically trivial vector bundle $X\times \mathbb{C}^n$ module the ambiguity from the choice of basis. And a complex structure is determined by $\bar\partial=\bar\partial_0+\eta$, where $\bar\partial_0$ corresponds to the holomorphic structure as a trivial vector bundle, and $\eta$ is an anti-holomorphic $End(\mathbb{C}^n)$ valued one form, such that $d\eta+\eta\wedge\eta=0$. Let $M$ denote the set of all such $\eta$, and $G$ denote the gauge group with fiber $GL(\mathbb{C}^n)$, then $G$ acts on $M$. Now, set-theoretically the orbits will correspond to the set of isomorphism classes of rank $n$ topologically trivial holomorphic vector bundles, but the quotient topology seems rather bad (non-Hausdorff).</p> <p>My question is using this approach</p> <p>(1)Whether stability condition is reflected in this construction?</p> <p>(2)Can we obtain the same moduli space as using Mumford's GIT?</p> <p>(3)I guess this is a natural approach. So is there any article or book about this?</p> <p>The assumption of topologically trivial should not be necessary, just to simplify some notation. We only need to fix the underlying topological vector bundle.</p> http://mathoverflow.net/questions/18271/what-out-of-print-books-would-you-like-to-see-re-printed/20310#20310 Answer by Botong Wang for What out-of-print books would you like to see re-printed? Botong Wang 2010-04-04T16:13:41Z 2010-04-04T16:13:41Z <p>Kobayashi, "Differential geometry of complex vector bundles"</p> http://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers/19855#19855 Answer by Botong Wang for Polynomial representing all nonnegative integers Botong Wang 2010-03-30T17:41:18Z 2010-03-30T17:41:18Z <p>How about an alternative question: does there exist a polynomial $f\in\mathbb{Q}[x,y]$ with integer values at lattice points, and of degree at least two on each variable, such that for any prime $p$, the map $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ composing with $\mathbb{Z}\to\mathbb{F}_p$ is surjective? Or more specifically, does a degree two such polynomial exist? The last part shouldn't be too hard, but I don't know how to solve it.</p> http://mathoverflow.net/questions/19768/upper-half-plane-quotient-by-a-discrete-group Upper half plane quotient by a discrete group Botong Wang 2010-03-29T21:05:37Z 2010-03-30T11:57:43Z <p>I was reading Mehta and Seshadri's paper "Moduli of vector bundles on curves with parabolic structures".</p> <p>In the second paragraph, they wrote:</p> <p>"Suppose that $H$ mod $\Gamma$ has finite measure ($H$ is the complex upper half plane, and $\Gamma$ is a discrete group). Let $X$ be the smooth projective curve containing $H$mod$\Gamma$ as an open subset and $S$ the finite subset of $X$ corresponding to parabolic and elliptic fixed points under $\Gamma$."</p> <p>I am not sure about what parabolic and elliptic mean here. And why does such an $X$ exist? If I take a Riemann surface and remove several small balls from it, when is it biholomorphic to another Riemann surface removing several points?</p> http://mathoverflow.net/questions/19505/riemannian-geometry-introductory-text/19639#19639 Answer by Botong Wang for Riemannian Geometry Introductory Text Botong Wang 2010-03-28T17:03:32Z 2010-03-28T17:03:32Z <p>I like do Carmo's Riemannian geometry, which is more down-to-earth, and gives more intuition.</p> <p>Jurgen Jost's Riemannian geometry and geometric analysis is also a good book, which covers many topics including Kahler metric. </p> http://mathoverflow.net/questions/19635/gauge-theory-construction-of-moduli-of-vector-bundles Comment by Botong Wang Botong Wang 2010-04-02T20:52:32Z 2010-04-02T20:52:32Z Joel, that's right. Thanks for pointing out. http://mathoverflow.net/questions/19635/gauge-theory-construction-of-moduli-of-vector-bundles/19637#19637 Comment by Botong Wang Botong Wang 2010-04-02T20:51:23Z 2010-04-02T20:51:23Z Thanks, the book by Kobayashi is very helpful. http://mathoverflow.net/questions/19768/upper-half-plane-quotient-by-a-discrete-group Comment by Botong Wang Botong Wang 2010-03-31T21:28:33Z 2010-03-31T21:28:33Z I am still not sure whether I understand the words elliptic and parabolic or not. So an elliptic point means a point on $X$ with non-trivial stablizer, and a parabolic point means a point that is not in $X$, but comes from a natural completion of the quotient. I am not familiar with the automorphisms of the complex unit disk. Is there a book that I can look at and get familiar with these basic concepts? http://mathoverflow.net/questions/19768/upper-half-plane-quotient-by-a-discrete-group/19799#19799 Comment by Botong Wang Botong Wang 2010-03-31T21:19:11Z 2010-03-31T21:19:11Z Thanks for your answer too. I am not sure how is the Riemannian metric obtained? Do you just choose one which gives finite measure? Is there any relation with the conformal structure? http://mathoverflow.net/questions/19768/upper-half-plane-quotient-by-a-discrete-group/19771#19771 Comment by Botong Wang Botong Wang 2010-03-31T21:17:19Z 2010-03-31T21:17:19Z Thanks for you answer, the third paragraph is really helpful. What will be the space of all structures with parabolic ends in $S^2$ minus four points? http://mathoverflow.net/questions/12892/why-cant-subvarieties-separate/12896#12896 Comment by Botong Wang Botong Wang 2010-03-29T21:17:59Z 2010-03-29T21:17:59Z In Griffiths&amp;Harris, page 21, it is proved that a variety is irreducible if and only if its smooth locus is connected.