Co-Higgs Bundles and Poisson Geometry

Abstract

The relationship between co-Higgs bundles and holomorphic Poisson structures was first studied by Polishchuk, who established a one-to-one correspondence between rank 2 holomorphic co-Higgs fields $\Phi$ and Poisson structures on the total space of $\mathbb{P}(V)$. Matviichuk later extended this work by showing how to obtain co-Higgs fields from coisotropic Poisson structures $\sigma_{\Phi}$ on $\text{Tot}(V)$ and $\pi_{\Phi}$ on $\text{Tot}(\mathbb{P}(V))$ for rank greater than or equal to $3$. For diagonalizable holomorphic co-Higgs fields, Matviichuk introduced the notion of strong integrability as a necessary condition for the existence of these corresponding Poisson structures, focusing his analysis on open sets $\mathcal{U} \subset \mathbb{C}$ and $\mathbb{P}^1$.

We extend these results in several directions. First, we prove that a Poisson structure on $\mathbb{P}(V)$ lifts to a quadratic Poisson structure on $V$ if and only if $p^* \omega_X \otimes p^*\det V^*$ admits a Poisson module structure, extending Matviichuk's result to the non-Calabi-Yau case. For co-Higgs fields over $\mathcal{U} \subset \mathbb{C}^n$, we classify all cases where $\sigma_\Phi$ is integrable, proving that $\Phi$ must be either a function multiple of a constant matrix or have only one nonzero column. We also show that for curves of genus $g \geq 1$, all co-Higgs fields are strongly integrable and characterize their explicit forms based on genus.

Finally, we investigate the relationship between stability and Poisson geometry. We establish that $\Phi$-invariant subbundles correspond to Poisson subvarieties of $\mathbb{P}(V)$ and prove that the eigenvariety $E$ coincides with $\text{Zeroes}(\pi_\Phi)$. For rank 2 co-Higgs bundles over $\mathbb{P}^1$, we show the zero locus of $\pi$ decomposes into two components: a 2:1 cover corresponding to the spectral curve $S_\Phi$ and fibers over the zeros of $\Phi$. This decomposition provides geometric criteria for understanding stability. In particular, we prove that the spectral curve is irreducible and the zero locus consists only of the 2:1 cover component if and only if $(V,\Phi)$ is stable.