5
$\begingroup$

I am looking for a proof for the following statement which seems is due to Edward Witten

*

Take the Teichmuller space $\mathcal T_{g,n}$ of complex structures of a surface of genus $g$ with $n$ punctured then there exists a vector bundle $\pi:E\to \mathcal T_{g,n}$ such that the fibers $\pi^{-1}(J)=H^0(M_J,L^{\otimes k})$ are the complex vector space of holomorphic sections where $L\to M_J$ is a holomorphic line bundle and $M_J$ is the moduli space of semi-stable holomorphic vector bundles of rank $N$ on the surface $S$ (with complex structure $J$ )with vanishing first Chern class and trivial determinant.

*

Definition of Semi stability of a holomorphic vector bundle:

Following Mumford and Takemoto, we say that a holomorphic vector bundle $E$ over a compact K\"ahler manifold $(M, \omega)$ is $\omega$-stable (resp. $\omega$-semi-stable) if, for every subsheaf $\mathcal F$ of $\mathcal O(\mathcal E)$ with $rank\; \mathcal F < rank\; \mathcal E$, we have

$$\mu(\mathcal F)<\mu (\mathcal O(\mathcal E))\;\; \; \; (resp\; \; \mu(\mathcal F)\leq \mu (\mathcal O(\mathcal E)))$$

Let $\mathcal E$ be a coherent analytic sheaf over a compact K\"ahler manifold $(M, \omega)$ of dimension $n$. We define the degree of $\mathcal E$, by

$$\text{deg}\; \mathcal E=\int_M c_1(\mathcal E)\wedge \omega^{n-1}$$

For a torsion-free sheaf $\mathcal F$ of rank $r$ on $X$ we set $$ deg \mathcal F:=deg (\Lambda^r \mathcal F )^{\vee\vee}$$ Note that, for an $\mathcal O_X$-module $\mathcal G$ we denote by $\mathcal G^\vee$ the dual sheaf $Hom_{\mathcal O_X} (\mathcal G, \mathcal O_X)$.

We define the slope of $\mathcal E$, to be $$\mu(\mathcal E)=\frac{deg \mathcal E}{rk \mathcal E}$$

As remark :A coherent sheaf $\mathcal E$ over a smooth projective variety $M$ always admits a finite locally free resolution. $0\longrightarrow \mathcal E_n\longrightarrow \mathcal E_{n-1}\longrightarrow\cdots\longrightarrow \mathcal E_0\longrightarrow \mathcal E\longrightarrow 0$. So we define the determinant of $\mathcal E$ to be $$\textrm{det}(\mathcal E)=\otimes\textrm{det}({\mathcal E}_i)^{(-1)^i}$$ Note that this definition is independent of the choice of a resolution. We define the first Chern class $c_1(\mathcal E)=c_1(\det \mathcal E)$.

If $\mathcal F$ be torsion free then the notion of determinant bundle is coincide with classical notion $det\mathcal F=(\wedge^p\mathcal F)^{\vee\vee}$

$\endgroup$
  • $\begingroup$ I would think look somewhere in Jorgen Andersen's papers. $\endgroup$ – AHusain Jun 7 '17 at 5:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy