User botong wang - MathOverflow

Stability condition for vector bundles

2012-01-17T15:19:59Z

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 $E$, $\mu(F)<\mu(E)$ (resp. $\leq$).

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?

(1)$X,A,E$ are as above, and $F$ is a proper subsheaf of $E$ breaking the stability condition, i.e., $\mu(F)\geq \mu(E)$.

(2)There exists no vector bundle which break the stability condition, i.e., $\mu(F')<\mu(E)$ for any subbundle $F'$ of $E$.

Gauge theory construction of moduli of vector bundles

2010-03-28T16:37:16Z

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.

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).

My question is using this approach

(1)Whether stability condition is reflected in this construction?

(2)Can we obtain the same moduli space as using Mumford's GIT?

(3)I guess this is a natural approach. So is there any article or book about this?

The assumption of topologically trivial should not be necessary, just to simplify some notation. We only need to fix the underlying topological vector bundle.

Answer by Botong Wang for What out-of-print books would you like to see re-printed?

2010-04-04T16:13:41Z

Kobayashi, "Differential geometry of complex vector bundles"

Answer by Botong Wang for Polynomial representing all nonnegative integers

2010-03-30T17:41:18Z

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.

Upper half plane quotient by a discrete group

2010-03-29T21:05:37Z

I was reading Mehta and Seshadri's paper "Moduli of vector bundles on curves with parabolic structures".

In the second paragraph, they wrote:

"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 $\Gamma$."

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?

Answer by Botong Wang for Riemannian Geometry Introductory Text

2010-03-28T17:03:32Z

I like do Carmo's Riemannian geometry, which is more down-to-earth, and gives more intuition.

Jurgen Jost's Riemannian geometry and geometric analysis is also a good book, which covers many topics including Kahler metric.

Comment by Botong Wang

2010-04-02T20:52:32Z

Joel, that's right. Thanks for pointing out.

Comment by Botong Wang

2010-04-02T20:51:23Z

Thanks, the book by Kobayashi is very helpful.

Comment by Botong Wang

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?

Comment by Botong Wang

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?

Comment by Botong Wang

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?

Comment by Botong Wang

2010-03-29T21:17:59Z

In Griffiths&Harris, page 21, it is proved that a variety is irreducible if and only if its smooth locus is connected.