By Lerner M. E., Repin O. A.

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Zn in which the sheaf E has basis e: set ( ) n P = (Pij )1 i,j d and Ω = (ωij )1 i,j d with ωij = =1 ϕij dz . 7 Example. The connection matrix of d on the trivial bundle OM in the canonical basis. b Operations on bundles with connection Let (E, ∇) and (E , ∇ ) be two holomorphic bundles equipped with holomorphic connections. One can equip in a natural way the bundles E ⊕E , L(E, E ) and E ⊗ E with holomorphic connections that we denote by ∇ . If s and s are two holomorphic local sections of E and E respectively, we set for any vector ﬁeld ξ, • • • ∇ξ (s ⊕ s ) := ∇ξ (s) ⊕ ∇ξ (s ), in other words ∇ = ∇ ⊕ ∇ ; ∇ξ (s ⊗ s ) := (∇ξ (s) ⊗ s ) + (s ⊗ ∇ξ (s )), in other words ∇ = (∇ ⊗ Id) + (Id ⊗∇ ); if ϕ : E → E is a homomorphism, ∇ξ (ϕ) is the local section of L(E, E ) deﬁned by ∇ξ (ϕ)(s) = ∇ξ (ϕ(s)) − ϕ (∇ξ (s)) .

44 0 The language of ﬁbre bundles Compatibility with a “metric”. c, a symmetric nondegenerate bilinear form g on T M , that we also call a “metric”, we will say that the Higgs ﬁeld Φ, or the product , is compatible with the metric g if, for any triple (ξ1 , ξ2 , ξ3 ) of vector ﬁelds on M , g(ξ1 ξ2 , ξ3 ) = g(ξ1 , ξ2 ξ3 ). Unlike the case of connections, a “metric” does not naturally deﬁne by itself a symmetric Higgs ﬁeld compatible with it. If a unit ﬁeld exists, one can consider the diﬀerential 1-form e∗ deﬁned by adjunction.

28 0 The language of ﬁbre bundles Proof. If one applies the Leibniz rule to ∇ − ∇ , the terms which are independent of the connection cancel, hence the OM -linearity. 3 Example (Connections on the trivial bundle). The trivial bundle 1 of rank one is equipped with the connection d : OM → ΩM . In an analogous way, the trivial bundle of rank δ is equipped with the direct sum connection δ 1 δ → ΩM . Any other connection ∇ on the trivial bundle of rank δ can d : OM 1 ) of be written as ∇ = d + Ω where Ω is a global section of the sheaf Mδ (ΩM 1 δ × δ matrices with entries in ΩM .