Why study Bogomolov's T-Stability

Bogomolov introduced the notion of $T$-stability. I know that such stability does not sit in the category of canonical metrics on vector bundles. We know that if a vector bundle admits a Hermitian-Einstein metric then it is $T$-stable but Bogomolov showed that the converse holds true if $H^{1,1}(M, C)$ is one dimensional, or if $ω$ represents an integral class and $Pic(M)/Pic^0(M) = \mathbb Z$, where here $Pic^0(M)$ denotes the subgroup of the Picard group $Pic(M)$ consisting of holomorphic line bundles with vanishing first Chern class.

My question is

1. -Why is the study of such $T$-stability important? What are the known applications of $T$-stability?.

2. -Can we remove such additional assumptions and get $T$-stability to be equivalent with Mumford's stability? Is there any counterexample known?

Bogomolov $T$-stability defined as follows

Definition:Let $S$ be a torsion-free coherent analytic sheaf over a compact complex manifold $M$. A weighted flag of $S$ is a sequence of pairs $\mathcal F = \{(S_i, n_i); 1\leq i \leq k\}$ consisting of subsheaves $$S_1 \subset S_2 \subset \cdots \subset S_k \subset S$$ with $0 < rank S_1 < rank S_2 < \cdots < rank S_k < rank S$

and positive integers $n_1, n_2, \cdots , n_k$. We set $$r_i = rank S_i , r = rank S$$ To such a flag $\mathcal F$ we associate a line bundle $\mathcal T_{\mathcal F}$ by setting $$\mathcal T_{\mathcal F} =\prod_{i=1}^k\left((\det S_i)^r(\det S)^{-r_i}\right)^{n_i}$$

We say that $S$ is $T$-stable if, for every weighted flag $\mathcal F$ of $S$ and for every flat line bundle $L$ over $M$, the line bundle $\mathcal T_{\mathcal F} \otimes L$ admits no nonzero holomorphic sections. We say that $S$ is $T$-semistable if, for every weighted flag $\mathcal F$ of $S$ and for every flat line bundle $L$ over $M$, every nonzero holomorphic section of the line bundle $\mathcal T_{\mathcal F} \otimes L$ (if any) vanishes nowhere on $M$.

Gieseker stability: Let $L$ be an ample line bundle with $c_1(L) =\beta$ on a smooth projective variety $X$ of dimension $g$. A torsion-free sheaf $E$ is Gieseker stable (respectively Gieseker semistable) with respect to $\beta$ if for each subsheaf $F$ we have $$P(F) < P(E) \;\; \;\; (P(F) \leq P(E))$$ where $P(E) = \frac{\chi(E \otimes L^n)}{r(E)}$, is the reduced Hilbert polynomial.

Using Riemann-Roch theorem we have

$$P(E\otimes \mathcal O_X(1)^{\otimes m})=rk E\frac{m^n}{n!}+(deg E-rk E\frac{deg K}{2})\frac{m^{n-1}}{(n-1)!}+\cdots$$

where $K$ is the canonical divisor. From this it follows that $$slope\; stable \Longrightarrow Gieseker\; stable \Longrightarrow Gieseker\; semistable \Longrightarrow slope \; semistable$$

1. What is the relation between Bogomolov T-stability and Gieseker stability? Under which condition they are equivalent?

Definition of Slope stability: 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 $det\mathcal F=(\wedge^p\mathcal F)^{\vee\vee}$

• We have the following theorem which says that Mumfords stability is equivalent with Bogomolov $T$-stability. Let $E$ be a holomorphic vector bundle over a compact Riemann surface $M$. Then $E$ is stable (resp. semi-stable) if and only if it is T-stable (resp. T-semi-stable). – user21574 Jun 5 '17 at 23:27