# Geometry ag.algebraica – Analogue of the main package for the Hodge package

Leave $$X$$ Being a complex projective variety connected without problems.

A holomorphic Higgs package is a pair. $$(E, theta)$$ consists of a pack of holomorphic vectors $$E$$ in $$X$$ together with a Higgs field $$theta in H ^ 0 (X, End (E) otimes Omega_X ^ 1)$$ with $$theta wedge theta = 0$$ in $$H ^ 0 (X, End (E) otimes Omega ^ 2_X)$$.

A holomorphic Higgs package. $$(E, theta)$$ It is said to have a structure of a Hodge package system if $$E$$ It has a holomorphic decomposition of direct sum. $$E = bigoplus limits_ {i = 0} ^ n E_i$$ such that $$theta (E_i) subseteq E_ {i-1} otimes Omega_X ^ 1$$, for all $$1 leq i leq n$$.

Now let $$G$$ be an algebraic cognate reducer group connected through $$mathbb {C}$$.

A holomorphic director. $$G$$-Higgs package in $$X$$ it's a pair $$(E_G, theta)$$, where $$E_G$$ he is a holomorphic director $$G$$-Bundle in $$X$$ Y $$theta in H ^ 0 (X, text {ad} (E_G) otimes Omega_X ^ 1)$$ such that $$theta wedge theta = 0$$ in $$H ^ 0 (X, text {ad} (E_G) otimes Omega_X ^ 2)$$; here $$text {ad} (E_G): = E_G times ^ G mathfrak {g}$$ is the attached vector pack associated with the attached representation $$ad: G longrightarrow End ( mathfrak {g})$$Y $$mathfrak {g}$$ is the lie algebra of $$G$$.

For example when $$G = GL_n$$, $$text {ad} (E_G) = End (E_ {GL_n})$$, where $$E_ {GL_n}$$ is the vector pack corresponding to the $$GL_n$$-make $$E_G$$.

Question:
Is there any analog of Hodge's package system for the principal case? $$G$$-Higgs packages?