# Endomorphism of involution invariant vector bundles

Let $C$ be an hyperelliptic curve with an involution $\sigma$, and let $E$ be a rank $2$ stable involution invariant vector bundle on $C$, that is $\sigma^{*}E \cong E$. Let $(End(E) \otimes K_C)_{+}$ be the vector bundle on $C$ whose local sections are $2 \times 2$ matrices $$\left(\begin{array}{cc}\omega_1(t) & \omega_2(t)\\ \omega_3(t) & \omega_4(t) \end{array}\right),$$

where $\omega_i(t)$ are differential forms such that $\omega_1(p) = \omega_2(p) = 0$ if $p \in C$ is a fixed point of $\sigma$.

More explicitly, we define a sheaf $(End(E) \otimes K_C)_{+}$ as the functor associating to any open subset $U \subset C$, $\mathcal{O}_{C}(U)$-module $End(E) \otimes K_C)_{+}(U) \subseteq End(E) \otimes K_C)(U)$ whose elements are the $\phi(U) \in End(E) \otimes K_C)(U)$ such that the following diagram is commutative $\require{AMScd}$ \begin{CD} E(U) @>\phi(U)>> E(U)\otimes K_C(U)\\ @V \varphi(U) V V @VV \varphi(U)\otimes\rho(U) V\\ \sigma^{*}E(U) @>> \sigma^{*}\phi(U)> \sigma^{*}E(U)\otimes \sigma^{*}K_C(U) \end{CD} where $\varphi: E \to \sigma^{*}E$ is a linearisation of $E$ and $\rho: K_C \to \sigma^{*}K_{C}$ is a linearisation of $K_{C}$ and $\varphi \otimes \rho:E \otimes K_{C} \to \sigma^{*}E \otimes \sigma^{*}K_{C}$.

So $(End(E) \otimes K_C)_{+}$ is a subsheaf of $(End(E) \otimes K_C)$, and since $(End(E) \otimes K_C)$ is torsion free, then $(End(E) \otimes K_C)_{+}$ is torsion-free as well and since $C$ is a smooth curve, it is also a vector bundle.

Suppose $E'$ is another stable involution invariant vector bundle such that

$(End(E) \otimes K_C)_{+} \simeq (End(E') \otimes K_C)_{+}$, then can we conclude

$End(E) \otimes K_C \cong End(E') \otimes K_C$?

• I don’t understand your condition in the definition of the positive part $(End(E) K)_+$. Do you mean that there is a fixed line $L_p$ in $E_p$ at every Weierstrass point such that the image of the endomorphism valued 1-form is contained in $L_pK_p$ whence evaluated at $p$ for any section in the + bundle? – Sebastian May 14 '18 at 11:13
• And what is $V_P$? – Sasha May 14 '18 at 11:19
• @user45397: I think then your definition of $(...)_+$ is not compatible with the exact sequence. For instance, what is the second map in it? – Sasha May 14 '18 at 17:32
• @Sebastian: I have defined $(End(E)\otimes K_{C})_{+}$ more explicitly. – user45397 May 15 '18 at 16:42