| Sbornik: Mathematics |
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Stable vector bundles and the Riemann–Hilbert problem on a Riemann surface I. V. Vyuginab, L. A. Dudnikovac a Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia b Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia c National Research University Higher School of Economics, Moscow, Russia
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     § 1. Statement of the problem and basic definitionsConsider a compact Riemann surface $X$ of genus $g$ and a representation
$$
\begin{equation}
\chi\colon \pi_1(X\setminus D,z_0)\longrightarrow \operatorname{GL}_p(\mathbb{C}),
\end{equation}
\tag{1.1}
$$
where $D=\{a_1,\dots,a_n\}\subset X$ is a divisor consisting of $n$ points. Let $\gamma_i$ be the generators of $\pi_1(X\setminus D,z_0)$ corresponding to loops around the points $a_i$, and let $\delta_{2j-1}$ and $\delta_{2j}$ be the generators of $\pi_1(X\setminus D,z_0)$ corresponding to the A- and B-cycles around the $j$th handle. The representation (1.1) can be defined using the set of matrices corresponding to the generators: $G_i=\chi(\gamma_i)$ and $H_j=\chi(\delta_j)$, $i=1,\dots,n$, ${j=1,\dots,2g}$. The generators satisfy the relation
$$
\begin{equation*}
G_1\dotsb G_n H_1H_2H_1^{-1}H_2^{-1}\dotsb H_{2g-1}H_{2g}H_{2g-1}^{-1}H_{2g}^{-1}=I,
\end{equation*}
\notag
$$
which follows from the null-homotopy of the corresponding loop:
$$
\begin{equation*}
\gamma_1\dots\gamma_n\delta_1\delta_{2}\delta_{1}^{-1}\delta_{2}^{-1} \dots\delta_{2g-1}\delta_{2g}\delta_{2g-1}^{-1}\delta_{2g}^{-1}\sim \mathrm{id}
\end{equation*}
\notag
$$
on $X\setminus D$. Let $E$ be a holomorphic vector bundle on $X$ of degree $\deg E$ and rank $\operatorname{rk} E$. This vector bundle is defined by a covering $\{ U_{\alpha}\}$ of the surface $X=\bigcup_{\alpha}U_{\alpha}$ by open sets $U_{\alpha}$ and holomorphically invertible transition functions
$$
\begin{equation*}
g_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\to \operatorname{GL}_p(\mathbb{C}), \qquad U_{\alpha}\cap U_{\beta}\ne \varnothing,
\end{equation*}
\notag
$$
which satisfy the cocycle condition. We formulate the following definitions (for more details, see [1] and [2]). Definition 1. Let $E=(\{U_{\alpha}\}, \{ g_{\alpha\beta}\})$ be a bundle such that all transition functions $g_{\alpha\beta}$ have a block upper triangular form with upper left square blocks $g_{\alpha\beta}'$. A subbundle $E'$ of $E$ is a bundle over the same base given by the data corresponding to the upper left block of the data of the original bundle. Definition 2. The slope $\mu(E)$ of a bundle $E$ is the ratio of the degree of the bundle to its rank: $\mu(E)=\deg(E)/\operatorname{rk}(E)$. The slopes of subbundles are defined similarly. Definition 3. All subbundles of $E$, apart from the empty bundle and $E$ itself, are called proper. A bundle $E$ is called semistable if the slope of each of its proper subbundles $E'$ is not greater than the slope of $E$: If strict inequalities hold for all proper subbundles, then the bundle $E$ is called stable. 1.1. Connection on a bundleDefinition 4. Let $\tau_X^*$ denote the cotangent bundle of $X$, let $f$ be a holomorphic function on $X$, let $s$ be a holomorphic section of $E$, and let $U$ be an open set in an atlas of $X$. A connection $\nabla$ on a bundle $E$ over a complex manifold $X$ is a $\mathbb{C}$-linear map satisfying the identity In an open set $U_{\alpha}$ a connection $\nabla$ is determined by 1-forms $\omega_\alpha$ satisfying Definition 5. Let $\omega_{\alpha}$ and $\omega_{\beta}$ be connection forms defined in $U_{\alpha}$ and $U_{\beta}$, respectively. Then the following gauge condition holds for the transition functions $g_{\alpha\beta}$ such that $U_{\alpha}\cap U_{\beta}\ne \varnothing$: Definition 6. A logarithmic (Fuchsian) connection on $X$ is a connection that is given by forms $\omega_k=\mathcal{B}_k(z)/(z -a_k)\,dz$ in a neighbourhood of all singular points $z=a_k$, where $\mathcal{B}_k(z)$ is a matrix function holomorphic at $z=a_k$. In what follows we consider pairs $(E,\nabla)$ consisting of a holomorphic bundle $E$ on a Riemann surface $X$ and a logarithmic connection $\nabla$ with singularity divisor $D=\{a_1,\dots ,a_n\}\subset X$. A monodromy representation (1.1) is defined in an obvious way for the connection $\nabla$. Definition 7. A subbundle $E'$ of a bundle $E$ is said to be stabilized by a connection $\nabla$ if the map (1.2) takes sections of the subbundle $E'$ to sections of ${E'\otimes \mathcal{ O}(\log D)}$, where $\mathcal{O}(\log D)$ denotes the sheaf of meromorphic 1-forms with simple poles at points in $D$. This is equivalent to the condition that the subspace of the fibre that corresponds to $E'$ is invariant under the action of the monodromy operators. Definition 8. A semistable pair consists of a bundle $E$ and a connection $\nabla$ such that for any proper subbundle $E'$ stabilized by $\nabla$, If strict inequalities hold, then the pair $(E,\nabla)$ is called stable. 1.2. Local theory of Fuchsian singular pointsLet $\mathcal{L}$ denote the space of horizontal sections of a connection in a neighbourhood of a singular point $z=a_k$. Definition 9. Eigenvalues $\beta_k^j$, $j=1,\dots,p$, of the residue matrix $\mathcal{B}_k(0)$ of the connection form $\omega_k$ in a neighbourhood of a singular point $z=a_k$ (see Definition 6) are called exponents of the connection at $a_k$ (we have $p=\operatorname{rk} E$). A basis $Y(z)$ of horizontal sections of the connection $\nabla$ in a neighbourhood of a Fuchsian singular point $z=a_k$ has the form Here $U_k(z)$ is a matrix that is holomorphically invertible at $z=a_k$, $\Lambda_k=\operatorname{diag}(\lambda_k^1,\dots,\lambda_k^p)$ is a diagonal integer valuation matrix and where $G_k$ is the monodromy matrix at $z=a_k$. The matrix $E_k$ is chosen so that its eigenvalues $\rho_k^j$, $j=1,\dots,p$, satisfy
$$
\begin{equation}
0\leqslant\operatorname{Re} \rho_k^j<1, \qquad j=1,\dots,p.
\end{equation}
\tag{1.7}
$$
The matrices $S_k$ conjugate $G_k$ in such a way that the matrix is holomorphic at $z=a_k$. Then $\{Y_k(z)=Y(z)C_k^{-1}\}$ is called an associated basis at $z=a_k$. An associated basis always exists (see [1]). Integer diagonal elements $\lambda_k^j$ of the matrix $\Lambda_k$ are called valuations at $z=a_k$, and the following relations hold:
$$
\begin{equation*}
\beta_k^j= \lambda_k^j+\rho_k^j, \qquad k=1,\dots,n, \quad j=1,\dots,p.
\end{equation*}
\notag
$$
According to Levelt’s theory (see [1]), in a neighbourhood of a singular point of the connection there exists a basis of horizontal sections $Y_k(z)$ such that and the matrix $E_k$ is upper triangular. The local monodromy matrix at $z=a_k$ is upper triangular in this basis. Such a basis is called a Levelt basis. A weak Levelt basis at $z=a_k$ is a basis that is the union of Levelt bases of all eigensubspaces of the local monodromy at $z=a_k$. Levelt and weak Levelt bases are associated, that is, in these bases the solution can be written in the form (1.5) and the valuations assume all of their values taking multiplicities into account, or, equivalently, condition (1.8) is satisfied. 1.3. Constructing a bundle with connection from local dataLet the following objects be fixed:
These data define uniquely a pair $(E^{\Lambda,S},\nabla^{\Lambda,S})$ consisting of a holomorphic vector bundle $E^{\Lambda,S}$ on $X$ and a logarithmic connection $\nabla^{\Lambda,S}$ with singularity divisor $D$ and monodromy (1.1). This construction was described in detail in [1], Lecture 8. We recall it briefly. On the punctured surface $X\setminus D$ there exists a unique pair consisting of a holomorphic bundle and a holomorphic connection with prescribed monodromy (1.1), which can be defined using constant transition functions and zero connection forms. To construct the pair $(E^{\Lambda,S},\nabla^{\Lambda,S})$ we extend this unique holomorphic bundle with holomorphic connection and prescribed monodromy over $X\setminus D$ to the points in the divisor $D$. We cover the Riemann surface $X$ by neighbourhoods $O_1,\dots,O_n$ of singular points $a_1,\dots,a_n$ and a finite number of open sets $\{U_{\alpha}\}$ covering $X\setminus D$, so that the $U_\alpha$ do not contain points of $D$. To extend the pair to a singular point $a_k$, consider the neighbourhood $O_k$ of $a_k$ and an open set $U_{\alpha}$ containing $a_k$ in its closure, such that the intersection $O_k\cap U_{\alpha}$ is connected and simply connected. The transition function between $U_{\alpha}$ and $O_k$ is defined by
$$
\begin{equation*}
g_{k\alpha}(z)=(z-a_k)^{\Lambda_k}(z-a_k)^{E_k}S_k^{-1},
\end{equation*}
\notag
$$
where the matrix $E_k$ (see formula (1.6)) has eigenvalues $\rho_k^j$ satisfying condition (1.7). On the open set $O_k$ the connection form looks like
$$
\begin{equation*}
\omega_k=\frac{\Lambda_k+(z-a_k)^{\Lambda_k} E_k(z-a_k)^{-\Lambda_k}}{z-a_k}\, dz.
\end{equation*}
\notag
$$
The pair constructed, consisting of a holomorphic bundle over $X$ and a logarithmic connection $\nabla$, will be denoted by $(F^{\Lambda,S},\nabla^{\Lambda,S})$, or simply $(F,\nabla)$. It was shown in [1] that each holomorphic bundle $F$ with logarithmic connection $\nabla$, singularity divisor $D$ and prescribed monodromy can be constructed from some admissible set of matrices $\Lambda$ and $S$ (a set is admissible if the matrices (1.8) are holomorphic at $a_k$). Next we describe the procedure for the analytic continuation of an associated basis at a singular point $a_k$ to a nonsingular point $z_0$. Consider the associated basis of horizontal sections of the connection at the point $z=a_k$: We connect $a_k$ with $z_0$ by a path between the neighbourhood $O_k$ and $z_0$. We have the following formula for the basis of horizontal sections at $z_0$
$$
\begin{equation*}
g_{\alpha k}Y_k=S_k(z-a_k)^{-E_k}(z-a_k)^{-\Lambda_k}(z-a_k)^{\Lambda_k}(z-a_k)^{E_k}=S_k,
\end{equation*}
\notag
$$
where the index $\alpha$ refers to the open set $U_{\alpha}$ containing $a_k$ on its boundary. Note that the basis does not change upon analytical continuation from $U_{\alpha}$ to $z_0$ along the distinguished path. This follows from the construction of bundles using the coordinate description (see [1]). The degree of a bundle $F$ with logarithmic connection $\nabla$ and singular points $a_1,\dots,a_n$ is given by the formula
$$
\begin{equation*}
\operatorname{deg} F= \sum_{k=1}^n \operatorname{tr} \mathcal{B}_k(0)=\sum_{k=1}^n\sum_{j=1}^p(\lambda_k^j+\rho_k^j).
\end{equation*}
\notag
$$
The proof can be found, for example, in [3], Appendix B. 1.4. The Harder–Narasimhan filtrationThis is an increasing filtration of subbundles $E_i$ of the bundle $E$ such that the successive quotients $E_i/ E_{i-1}$ are semistable, their slope decreases with $i$, and $E_1$ is the subbundle with the greatest slope of all subbundles. The existence and uniqueness of this filtration were proved in [4]. Definition 10. The Harder–Narasimhan filtration of a bundle $E$ is a sequence of bundles with the following properties: We refer to $\mu_j$ as the slope of the $j$th associated quotient. § 2. Counterexample to a weak version of the Riemann–Hilbert problemIn this section we give an example of a representation that cannot be realized as the monodromy representation of a semistable pair consisting of a bundle over a Riemann surface and a logarithmic connection with four singular points. This implies that this representation cannot be realized as the monodromy of a logarithmic connection on a semistable bundle of any degree. Such examples exist over the Riemann sphere and over Riemann surfaces of positive genus alike. For the Riemann sphere this is a trivial consequence of known results, whereas for Riemann surfaces of positive genus this result is new. Consider the following four generators constructed by Bolibrukh (see [5], Example 5.2.2):
$$
\begin{equation*}
\begin{gathered} \, \begin{pmatrix} 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & -1 & -1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 2 & 2 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 & 1 & 1 & -1 & 1 & 0 \\ 0 & -1 & 1 & 1 & -1 & -1 & 0 \\ 0 & 0 & -1 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \\ G_1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad G_2 \\ \begin{pmatrix} 1 & 0 & 1 & 0 & -1 & 1 & 1 \\ 0 & -1 & -1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & -1 & 2 & 1 \\ 0 & 0 & 0 & -1 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 & -1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 & 1 & -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & -2 & 1 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}. \\ G_3 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad G_4 \end{gathered}
\end{equation*}
\notag
$$
Bolibrukh showed that this set of generators is not realizable as a set of monodromy generators of a Fuchsian system on the Riemann sphere (see [5]). We construct an example that generalizes Bolibrukh’s example to the case of a Riemann surface of arbitrary genus and then modify the original proof to obtain the following result. Theorem 1. There is no semistable pair $(F,\nabla)$ consisting of a bundle and a logarithmic connection on a Riemann surface $X$ of genus $g$ with singular points $a_1,a_2, a_3,a_4$ and monodromy representation (1.1) with generators Proof. We argue by contradiction. Assume that there exists a semistable pair $(F^{\Lambda,S},\nabla^{\Lambda,S})$ with monodromy (1.1), (2.1). Since the monodromy is upper triangular, there exists a nested sequence of invariant subbundles (stabilized by the connection $\nabla$) of rank $\operatorname{rk} F_j=j$ corresponding to invariant subspaces of all dimensions. Each subbundle $F_j$ must satisfy the inequalities We show that no appropriate admissible sets of matrices $\Lambda$ and $S$ (such that the matrices (1.8) are holomorphic at $a_k$) exist. This will lead to a contradiction.
We make significant use of the structure of this representation. Each generator $G_k$, $k=1,2,3,4$, is conjugate to a matrix with two Jordan blocks, one of which corresponds to eigenvalue $1$, and the other to eigenvalue $-1$. In this case, for each singular point there exists an associated basis in which the monodromy at this point has a Jordan form (see the definition in [1], Lecture 5, Example 5.2). Consider the valuation matrix $\Lambda_k$ (see (1.5)), and let $\lambda_k^j := (\Lambda_k)_{jj}$ denote the diagonal elements of $\Lambda_k$ (valuations). A valuation matrix $\Lambda_k$ defines a logarithmic connection at a point $z=a_k$ if and only if the matrix (1.8) is holomorphic at $z=a_k$. In our case, this means that if $\lambda_k^i$ and $\lambda_k^j$, $i<j$, correspond to the same Jordan block of the matrix $E_k$, then $\lambda_k^i \geqslant \lambda_k ^j$. The sets of diagonal elements $\rho_k^j$ of the matrices $E_k$ at singular points have the form
$$
\begin{equation*}
\begin{aligned} \, a_1&\colon\ \ \biggl\{ 0 , \frac{1}{2}, 0 , 0 , 0 , \frac{1}{2} , 0 \biggr\}; \qquad a_2\colon\ \ \biggl\{ 0, \frac{1}{2}, \frac{1}{2} , 0 , \frac{1}{2}, \frac{1}{2} , 0 \biggr\}; \\ a_3&\colon\ \ \biggl\{ 0 , \frac{1}{2} , 0 , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , 0 \biggr\}; \qquad a_4\colon\ \ \biggl\{ 0 , \frac{1}{2} , \frac{1}{2} , \frac{1}{2} , 0 , \frac{1}{2} , 0 \biggr\}; \end{aligned}
\end{equation*}
\notag
$$
see formula (1.6). The sets of exponents at the singular points are given by
$$
\begin{equation*}
\begin{aligned} \, a_1&\colon\ \ \lambda_{1}^{1},\,\underline{\lambda_{1}^{2}}+\frac{1}{2},\,\lambda_{1}^{3},\, \lambda_{1}^{4},\, \lambda_{1}^{5},\, \underline{\lambda_{1}^{6}}+\frac{1}{2},\, \lambda_{1}^{7}; \\ a_2&\colon\ \ \lambda_{2}^{1},\, \underline{\lambda_{2}^{2}}+\frac{1}{2},\,\underline{\lambda_{2}^{3}}+\frac{1}{2},\, \lambda_{2}^{4},\, \underline{\lambda_{2}^{5}}+\frac{1}{2},\, \underline{\lambda_{2}^{6}}+\frac{1}{2},\, \lambda_{2}^{7}; \\ a_3&\colon\ \ \lambda_{3}^{1},\, \underline{\lambda_{3}^{2}}+\frac{1}{2},\,\lambda_{3}^{3},\, \underline{\lambda_{1}^{4}}+\frac{1}{2},\, \underline{\lambda_{3}^{5}}+\frac{1}{2},\, \underline{\lambda_{3}^{6}}+\frac{1}{2},\, \lambda_{3}^{7}; \\ a_4&\colon\ \ \lambda_{4}^{1},\, \underline{\lambda_{4}^{2}}+\frac{1}{2},\, \underline{\lambda_{4}^{3}}+\frac{1}{2},\, \underline{\lambda_{4}^{4}}+\frac{1}{2},\, \lambda_{4}^{5},\, \underline{\lambda_{4}^{6}}+\frac{1}{2},\, \lambda_{4}^{7}. \end{aligned}
\end{equation*}
\notag
$$
Furthermore, the valuations $\lambda_k^j$ without underscore, as well as the valuations $\underline{\lambda_{k}^{j}}$ with underscore, form nonstrictly decreasing sequences for each singular point. This follows from the condition that the matrix (1.8) is holomorphic at $z=a_k$.
Now we prove that the corresponding set of valuation matrices $\Lambda$ does not exist. Note that each sequence $\rho_k^j$ at $a_k$ begins and ends with zero: and therefore Let denote the degree of the subbundle of rank $j$ generated by the first $j$ solutions of the corresponding associated bases. All subbundles $F_j$ are stabilized by the connection $\nabla$.We can assume without loss of generality that $\lambda_k^7=0$ for $k=1,2,3,4$. Indeed, subtracting the scalar matrices $I \cdot \lambda_k^7$ from all the $\Lambda_k$ we obtain a new bundle with logarithmic connection with the same monodromy, which also form a semistable pair. Now, by assumption we have
$$
\begin{equation*}
\boldsymbol{s}_7=\boldsymbol{s}_6+\lambda_1^7+\lambda_2^7+\lambda_3^7 +\lambda_4^7=\boldsymbol{s}_6,
\end{equation*}
\notag
$$
and it follows from the semistability of the pair that
$$
\begin{equation*}
\mu(F_6)=\frac{1}{6}\boldsymbol{s}_6\leqslant\frac{1}{7}\boldsymbol{s}_7=\mu(F_7),
\end{equation*}
\notag
$$
that is, $\boldsymbol{s}_7\leqslant 0$. Again, from the semistability of $(F,\nabla)$ we obtain
$$
\begin{equation*}
\mu(F_1)=\boldsymbol{s}_1\leqslant \frac{1}{7}\boldsymbol{s}_7=\mu(F_7)\leqslant 0,
\end{equation*}
\notag
$$
and from the decreasing sequence of valuations we obtain the inequality
$$
\begin{equation*}
\boldsymbol{s}_1=\lambda_1^1+\lambda_2^1+\lambda_3^1+\lambda_4^1\geqslant \lambda_1^7+\lambda_2^7+\lambda_3^7+\lambda_4^7=0.
\end{equation*}
\notag
$$
It follows that and all valuations without underscores are also equal to zero. Indeed, if the first and last elements of a nondecreasing sequence are equal to zero, then all elements in-between also vanish.
The semistability of the pair $(F,\nabla)$ and the relation $0 = \boldsymbol{s}_1 \leqslant \frac{1}{7}\boldsymbol{s}_7 \leqslant 0$ we have established imply that Now consider the valuations with underscores. We introduce the following notation:
$$
\begin{equation*}
A=\sum_{k=0}^4 \lambda_k^2\quad\text{and} \quad \overline{A}=\sum_{k=0}^4 \lambda_k^6.
\end{equation*}
\notag
$$
The semistability of $(F,\nabla)$ implies that Since the sequence of underscored valuations decreases, it follows that We have $0=\boldsymbol{s}_6=\boldsymbol{s}_5+\overline{A}$, hence $\boldsymbol{s}_5=-\overline{A}\geqslant 0$. From the semistability assumption on the pair $(F,\nabla)$ we obtain
$$
\begin{equation*}
\frac{1}{5}\boldsymbol{s}_5\leqslant\frac{1}{7}\boldsymbol{s}_7=0,
\end{equation*}
\notag
$$
hence This implies that the underscored valuations are equal to each other at each point. Therefore, the degree of $F$ is
$$
\begin{equation*}
\deg F=7+2 \lambda_1^2 + 4 \lambda_2^2 + 4 \lambda_3^2 + 4 \lambda_4^2=0,
\end{equation*}
\notag
$$
where $7$ is the sum of all eigenvalues $\rho_k^j$. This leads to a contradiction, since an even integer cannot be equal to an odd one. We have shown that there is no admissible set of valuation matrices $\Lambda$ and therefore no semistable pair $(F,\nabla)$. Theorem 1 is proved. Theorem 1 immediately implies the following. Remark 1 (corollary). The representation in Theorem 1 cannot be the monodromy representation of a semistable bundle of degree zero with logarithmic connection which has four singular points. Indeed, the existence of such a bundle with connection would contradict Theorem 1, since a semistable bundle with a connection is always a semistable pair. § 3. Estimates for slopes with bounded connection exponentsIn this section we prove the inequality for the slopes of the associated quotients $\mu(E_j/E_{j-1})$ of the filtration (1.9). The results in this section generalize the results in [6] in a certain sense. To prove Lemma 3 we use the following facts from [7] (see Proposition 4.4 there). Lemma 1. Let there exist a nonzero homomorphism $f\colon V \to W$ between the bundles $V$ and $W$, where $V$ is semistable, and let $W_1=f(V)$. Then the inequality $\mu(V)>\mu(W)$ implies that $\mu(W_1)>\mu(W)$. Lemma 2 (corollary to Lemma 1). If there exists a nontrivial homomorphism $f\colon V \to W$ between semistable bundles, then Lemma 3. Let $E$ be a bundle with logarithmic connection on a compact Riemann surface $X$ of genus $g$ with divisor $D=\{ a_1,\dots,a_n\}$. If for some $l$ the slopes of filtration (1.9) satisfy the inequality then the subbundle $E_l$ is stabilized by the connection $\nabla$. Proof. We argue by contradiction. Suppose that inequality (3.1) holds and the subbundle $E_l$ is not stabilized by the connection $\nabla$. Let $k$ be the least index $k>l$ such that the subbundle $E_k$ in (1.9) is stabilized by $\nabla$. Such $k$ exists since $E_m=E$ is stabilized by $\nabla$ and $m>l$. This means that $\nabla$ takes sections of $E_k$ to sections of $E_k\otimes \mathcal{O}^{1,0} (\log D)$, where $\mathcal{O}^{1, 0} (\log D)$ denotes the sheaf of meromorphic $1$-forms with simple poles at the points of the divisor $D$.
Let $\mathrm{pr}_s\colon E_s\to E_s/E_{s-1}$ be the fibrewise projection onto the quotient bundle $E_s/E_{s-1}$ and $i_1$ be the smallest integer for which the image of the map
$$
\begin{equation}
\mathrm{pr}_k \circ\nabla \colon E_{i_1}\to E_k/E_{k-1}\otimes \mathcal{O}^{1,0} (\log D)
\end{equation}
\tag{3.2}
$$
is nontrivial. Such $i_1$ exists because $E_{k-1}$ is not stabilized by the connection $\nabla$ (this follows from the choice of $k$). Note that if for some $i_1$ the image of (3.2) is nontrivial, then the $\mathrm{pr}_k\circ\nabla$-images of the subbundles $E_l$ such that $i_1<l\leqslant k $ are nontrivial too. The two composite maps
$$
\begin{equation}
\begin{gathered} \, E_{i_1} \xrightarrow{\nabla} E \otimes \mathcal{O}^{1,0} (\log D) \xrightarrow{\mathrm{pr}_k} E_k/E_{k-1} \otimes \mathcal{O}^{1,0} (\log D), \\ E_{i_1} \xrightarrow{\mathrm{pr}_{i_1}} E_{i_1}/E_{i_1-1} \xrightarrow{\nabla} E_k/E_{k-1} \otimes \mathcal{O}^{1,0} (\log D), \end{gathered}
\end{equation}
\tag{3.3}
$$
coincide, that is, $\mathrm{pr}_k\circ\nabla=\nabla\circ \mathrm{pr}_{i_1}$. We set $\tau_1 :=\nabla \circ \mathrm{pr}_{i_1}$; then
$$
\begin{equation}
\mathrm{pr}_{k} \circ \tau_1 := \mathrm{pr}_{k} \circ \nabla \circ \mathrm{pr}_{i_1} =\mathrm{pr}_k \circ \nabla.
\end{equation}
\tag{3.4}
$$
Indeed, the map $\nabla \circ \mathrm{pr}_k$ vanishes on $E_{i_1-1}$, so we can take the quotient $E_{i_1}/E_{i_1-1}$ first and then apply $\mathrm{pr}_k \circ \nabla$.
Note also that the map $\tau_1$ is linear over the sheaf of holomorphic functions. Let $f$ be a function and $\mathbf s$ be a section of $E_{i_1}/E_{i_1 - 1}$. Consider the image of $\mathbf s$ under the natural map of $E_{i_1}/E_{i_1 - 1}$ to $E_k$. Since
$$
\begin{equation}
\tau_1(f\mathbf s)=\mathrm{pr}_k (f \nabla(\mathbf s)) + \mathrm{pr}_k (df \wedge \mathbf s )=f\tau_1(\mathbf s) \in \Gamma(E \otimes \mathcal{O}^{1,0}( \log D)),
\end{equation}
\tag{3.5}
$$
the term $\mathrm{pr}_k(df\wedge s)$ is zero. Since the map is nonzero, it is a linear homomorphism and the image $\tau_{1}(E_{i_1} / E_{i_1 -1 })$ is a holomorphic subbundle.
Next we show that It is known that the degree of the sheaf of $1$-forms with $n$ simple poles on a one-dimensional manifold (a Riemann surface of genus $g$) is equal to The slope of the image bundle in (3.3) is
$$
\begin{equation}
\begin{aligned} \, \nonumber \mu(E_k/E_{k-1} \otimes \mathcal{O}^{1,0} (\log D) ) &=\frac{\deg(E_k/E_{k-1}) + rk (E_k / E_{k-1})(2g-2+n) }{rk(E_k/E_{k-1})} \\ &= \mu_k + 2g - 2 + n. \end{aligned}
\end{equation}
\tag{3.7}
$$
Note that $\tau_1$ defines a linear map that takes $E_{i_1}/E_{i_1-1}$ to a subbundle of $E_k/E_{k-1}$. The bundle $E_k/E_{k-1}$ is semistable as a quotient of successive terms in the Harder–Narasimhan filtration, so the slope of any subbundle is not greater than the slope of the whole bundle. Applying Lemma 2 to the map
$$
\begin{equation*}
\tau_1\colon E_{i_1}/E_{i_1-1} \to E_k/E_{k-1} \otimes \mathcal{O}^{1,0} (\log D)
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation}
\mu_{i_1}=\mu (E_{i_1}/E_{i_1-1}) \leqslant \mu(E_k/E_{k-1} \otimes \mathcal{O}^{1,0} (\log D) ) =\mu_k + 2g - 2 + n.
\end{equation}
\tag{3.8}
$$
Condition (4) in Definition 10 implies that Hence which contradicts the assumption. The lemma is proved. Theorem 2. Let $E$ be a holomorphic vector bundle on a compact Riemann surface $X$ of genus $g$ with logarithmic connection $\nabla$ and singular divisor $D=\{a_1,\dots,a_n\}$. Assume that the following inequalities hold for the exponents of the connection at all $n$ singular points: Proof. Let the bundle $E$ have rank $p$, and let the ranks of the subbundles $E_1,\dots, E_m$ in the Harder–Narasimhan filtration (1.9) be given by The slopes $\mu_i$ of the quotients $E_i/E_{i-1}$ ($E_0=\{ 0\},$ $d_0=0$) are given by the formula
We consider two cases. Case 1. Let
$$
\begin{equation}
\mu_i-\mu_{i+1}\leqslant n+2g-2, \qquad i=1,\dots,m-1.
\end{equation}
\tag{3.11}
$$
The degree of $E$ satisfies the inequalities
$$
\begin{equation}
0\leqslant\deg E=\sum_{i=1}^n\sum_{j=1}^p\beta_i^j<Mnp,
\end{equation}
\tag{3.12}
$$
and we have Inequality (3.11) implies that $\mu_1-\mu_m\leqslant (m-1)(n+2g-2)$. Note that if $\mu_1$ and $\mu_m$ are of different signs, then (3.11) implies that $|\mu_i|\leqslant (m-1)(n+2g-2)$. If $\mu_1$ and $\mu_m$ have the same sign, then they are positive, for otherwise this contradicts the first inequality in (3.12). In this case all $\mu_1,\dots,\mu_m$ are positive, and it follows from (3.13) and the second inequality in (3.12) that $\mu_m p<Mnp$. Therefore, $ \mu_m<Mn$, which, together with (3.11), implies that $\mu_i<Mn+(m-1)(n+2g-2)$.
Case 2. Let $1\leqslant k_1<\dots<k_l < m$ be integers such that if and only if $j=k_s$ for some $s=1,\dots,l$. It follows from the assumptions that $l\geqslant 1$. In this case Lemma 3 implies that the subbundles are stabilized by the connection. This means, in particular, that
$$
\begin{equation*}
\deg E_{k_s}=\sum_{i=1}^n\sum_{j=1}^{d_{k_s}}\beta_i^j, \qquad s=1,\dots,l,
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation}
\deg (E_{k_s}/E_{k_{s-1}})=\sum_{i=1}^n\sum_{j=d_{k_{s-1}+1}}^{d_{k_s}}\beta_i^j, \qquad s=2,\dots,l.
\end{equation}
\tag{3.15}
$$
This allows us to apply Case 1 to the bundles $E_{k_1}$ and $E_{k_s}/E_{k_{s-1}}$, $s=2,\dots,l$, with the induced connections, since they satisfy fully the assumptions of that case. Indeed, the sequence of associated quotients $E_i/E_{i-1}$, $i=1,\dots,m$, is split into the subsequences $E_i/E_{i-1}$, where $i=k_s+ 1,\dots,k_{s+1}$ and $s=0,\dots,l-1$ (we assume that $k_0=0$). We prove the inequalities for the slopes of the associated quotients of each subsequence separately, using the fact that the subsequence $E_i/E_{i-1}$, $i=k_s+1,\dots,k_{s+1}$, is the sequence of associated quotients of the bundle $ E_{k_{s+1}}/E_{k_s}$. By the assumptions of Case 2 and Lemma 3 the connection $\nabla$ induces an invariant connection on the quotient $E_{k_{s+1}}/E_{k_s}$. The exponents of this connection coincide with some exponents of $\nabla$. Consequently, the inequality for exponents (3.9) is satisfied for this connection. We can apply the argument of Case 1 to the quotient $E_{k_{s+1}}/E_{k_s}$ and obtain the even stronger inequality
$$
\begin{equation}
|\mu_i|<Mn+(k_{s+1}-k_s-1)(n+2g-2), \qquad i=k_s+1,\dots,k_{s+1}.
\end{equation}
\tag{3.16}
$$
This holds for each $s=1,\dots,l$, that is, for all slopes. Thus, the proof of Theorem 2 is complete. § 4. Riemann–Hilbert problem with completely reducible monodromyIn this section we generalize the results obtained in [8] and [9] to the case of a Riemann surface of arbitrary genus. Let $X$ be a Riemann surface of genus $g$, and let $D=\{a_1,\dots,a_n\}\subset X$ be a divisor of $n$ points. In this part we consider completely reducible representations that are decomposed into a direct sum of two representations:
$$
\begin{equation*}
\chi=\chi'\oplus\chi'', \qquad \chi', \chi''\colon \pi_1(z_0,X\setminus D )\to \operatorname{GL}(p^l,\mathbb{C}).
\end{equation*}
\notag
$$
Let
$$
\begin{equation}
G_1^l,\dots,G_n^l\quad\text{and} \quad H_1^l,\dots,H_{2g}^l, \qquad l=\,','',
\end{equation}
\tag{4.1}
$$
be the generators of the representations $\chi'$ and $\chi''$ of the fundamental group $\pi_1(X\setminus D)$, where the $G_k^l$, $k=1,\ldots,n$, correspond to loops around singular points $a_1,\dots,a_n$, and the $H_k$, $k=1,\ldots,2g$, correspond to $A$- and $B$-cycles on the surface $X$. The generators (4.1) satisfy the relation
$$
\begin{equation*}
G_1^l\dotsb G_n^l H_1^l H_2^l {H_1^l}^{-1}{H_2^l}^{-1}\dotsb {H_{2g-1}^l}{H_{2g}^l}(H_{2g-1}^l)^{-1}(H_{2g}^l)^{-1}=I, \qquad l=\,',''.
\end{equation*}
\notag
$$
Theorem 3. Let $\chi=\chi'\oplus \chi''$ be the monodromy of a logarithmic connection $\nabla$ on a semistable bundle $E$ of degree zero on $X$. Assume that at each of the singular points $a_1,\dots,a_n$ the generators of the representations $\chi'$ and $\chi''$ have disjoint spectra: Then $\chi'$ and $\chi''$ can be realized as the monodromy representations of logarithmic connections $\nabla'$ and $\nabla''$ on semistable bundles of degree zero over $X$. Proof. Let $E'$ and $E''$ be the subbundles of $E$ corresponding to the invariant subspaces of the subrepresentations $\chi'$ and $\chi''$. The degrees of $E'$ and $E''$ are nonpositive since $E$ is semistable, that is, $\deg E'\leqslant 0$ and $\deg E''\leqslant 0$.
Note that a weak Levelt basis exists in a neighbourhood of each singular point $z=a_k$. A weak Levelt basis is the union of Levelt bases of all eigensubspaces of the operator $G_k=G_k'\oplus G_k''$ (see [1] for the details). A weak Levelt basis is associated with a filtration, which means that the degrees of bundles can be calculated as the sum of the asymptotics of weak Levelt bases over all singular points. The invariant subspaces corresponding to $\chi'$ and $\chi''$ are spanned by different parts of a weak Levelt basis (see [1]). Namely, the first $\operatorname{rk} \chi'$ basis vectors span the first invariant subspace, and the remaining $\operatorname{rk} \chi''$ basis vectors span the second invariant subspace. By assumption the operators $G_k'$ and $G_k''$ have different eigenvalues, which means that a vector in a weak Levelt basis belongs to exactly one of the two invariant subspaces. Hence the degree of the subbundle $E'$ is equal to the sum of the connection exponents corresponding to the subrepresentation $\chi'$, that is, and the degree of $E''$ is equal to It follows that This implies, in particular, that $\deg E'=0$ and $\deg E''=0$. Hence $E'$ and $E''$ are subbundles of degree zero of a semistable bundle $E$, which also has degree zero. Thus, $E'$ and $E''$ are also semistable, and the proof is complete.The converse statement holds in the case when the spectra of the matrices that form direct summands have a nonempty intersection at at least one singular point. The proof of the following theorem essentially repeats the proof of Theorem 2 in [8]. The main difference is not related to additional generators arising in the monodromy representation, but to the fact that we do not construct a trivial bundle from the representation, but rather a semistable bundle of degree zero. Theorem 4. For each representation $\chi'$, Proof. To prove the theorem we construct the required representation $\chi''$ from the representation $\chi'$. Let $\chi'$ be defined by generators $G_1',\dots,G_n'$ at the points $a_1,\dots,a_n$ and generators $H_1',\dots,H_{2g}'$ corresponding to cycles on $X$. We can assume without loss of generality that the matrix $G_1'$ is in Jordan normal form, since the monodromy representation is defined up to global conjugation. The representation $\chi''$ is defined by matrices
$$
\begin{equation}
G_1'',\dots,G_n''\quad\text{and} \quad H_1'',\dots,H_{2g}'',
\end{equation}
\tag{4.3}
$$
which we must specify. Consider a set (4.3) satisfying the following conditions:
Now we must prove that the set of matrices (4.3) defining a representation $\chi''$ and satisfying conditions (a), (b) and (c) exists for any representation $\chi'$. Then we show that a representation $\chi=\chi'\oplus\chi''$ satisfying these conditions can be realized as the monodromy representation of a logarithmic connection on a semistable bundle of degree zero. The existence of such a set (4.3) is proved in the same way as in [8] (see the proof of Theorem 3 there), in the case of the Riemann sphere. We follow the proof of Theorem 3 in [8]. Each pair $(E,\nabla)=(E^{\Lambda,S},\nabla^{\Lambda,S})$ can be defined by its coordinate description as explained in § 1.3. Recall that the coordinate description of a pair $(E^{\Lambda,S},\nabla^{\Lambda,S})$ with prescribed monodromy determines fully the set of admissible valuation matrices $\Lambda=\{\Lambda_1,\dots,\Lambda_n\}$ and the set $S= \{S_1,\dots,S_n\}$ of conjugation matrices (see [1], Lecture 8). Up to holomorphic gauge equivalence, there is a unique fundamental system of horizontal sections on $X\setminus D$ with prescribed monodromy $\chi$. Consider the analytic continuation of the basis as a basis of horizontal sections of the connection in a neighbourhood of the point $z=a_k$, along fixed paths from neighbourhoods $U_k$ of singular points $a_k$ to the distinguished nonsingular point $z_0$. We have a formula for the basis of horizontal sections at $z_0$:
$$
\begin{equation*}
g_{\alpha k}Y_k=S_k(z-a_k)^{-E_k}(z-a_k)^{-\Lambda_k}(z-a_k)^{\Lambda_k}(z-a_k)^{E_k}=S_k,
\end{equation*}
\notag
$$
where the index $\alpha$ refers to the open set $U_{\alpha}$ containing the point $a_k$ on its boundary. Note that the basis does not change upon analytic continuation from $U_{\alpha}$ to $z_0$ along the fixed path. This follows from the construction of bundles using the coordinate description.
The pair $(E,\nabla)$ with monodromy representation $\chi=\chi_1\oplus\chi_2$ is defined by sets of matrices $\Lambda=\{\Lambda_1,\dots,\Lambda_n\}$ and $S=\{S_1,\dots,S_n\}$. We define the conjugation matrix $S_1$ by where $I$ denotes the identity matrix of size $p$. We take as $S_2,\dots,S_n$ some admissible matrices, for example, matrices conjugating $G_2,\dots,G_n$ to a Jordan normal form. The valuation matrices are defined by
$$
\begin{equation*}
\Lambda_1=\begin{pmatrix} d\cdot I & 0 \\ 0 & 0 \end{pmatrix}\quad\text{and} \qquad \Lambda_2=\dots=\Lambda_n=0,
\end{equation*}
\notag
$$
where $d$ is a sufficiently large positive integer, which we specify at the end of the proof. These data define the required bundle with connection $(E^{\Lambda,S},\nabla^{\Lambda,S})$.
We need to prove that the pair $(E^{\Lambda,S},\nabla^{\Lambda,S})$ is stable. The theorem will follow from this stability since, by Theorem 2, the existence of a stable pair with prescribed monodromy implies the existence of a semistable bundle of degree zero with logarithmic connection and the same monodromy. First, we describe all invariant subspaces of the representation $\chi$. It is easy to see that the subrepresentations of $\chi$ are: all the subrepresentations $\chi_{\alpha}'$ of $\chi'$, the direct sums $\chi_{\alpha}'\oplus \chi''$, and also the representations $\chi'$ and $\chi''$ themselves. Indeed, an invariant subspace of the generator $G=G_2'\oplus G_2''$ of the monodromy representation $\chi$ at the point $a_2$ can uniquely be decomposed into a direct sum of invariant subspaces of the operators $G_2^1$ and $G_2 ^2$, which have intersection zero, because all eigenvalues of the matrices $G_2^1$ and $G_2^2$ are different by construction. This exhausts all subrepresentations since $\chi''$ is irreducible. Next we prove that the stability condition for the pair $(E,\nabla)$ is satisfied. That is, we need to prove that the conditions on the slopes are satisfied for any proper subbundle $E_{\mathrm{sub}}$ of each of the four subbundles $E_{\chi'}$, $E_{\chi_{\alpha}'}$, $E_{\chi_{\alpha}'\oplus\chi''}$ and $E_{\chi'}$. Only these subbundles are stabilized by the connection $\nabla$. Let us show that for sufficiently large $d$ A bundle with connection has already been constructed, and we can find the degrees of all subbundles stabilized by this connection. As is known, these degrees are equal to the sums of the corresponding asymptotics, but we do not know a priori the eigenvalues of monodromy through which the fractional parts of the asymptotics are expressed. We have
$$
\begin{equation*}
\deg(E)\in [dp,dp + 2np)\quad\text{and} \quad \mu(E)\in \biggl[\frac d2,\frac d2+n\biggr).
\end{equation*}
\notag
$$
Set $\dim\chi_{\alpha}'=q$, where $q<p$. Then the degrees and slopes satisfy the following conditions:
We see that the slope of the bundle $E$ is at least $d/2$, and the slopes of its subbundles are at most $d(p-1)/(2p-1)+n$. This value of the slope is attained for $\dim\chi_{\alpha}=p-1$. Since $d > 4pn$, the conditions for the stability of the pair are satisfied. As proved in [2], the existence of a stable pair implies the existence of a semistable bundle of degree zero with a connection that has the same monodromy. Theorem 4 is proved. 
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\Bibitem{VyuDud24} |
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