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arXiv 2102.09723math.AGmath.CVmath.SG
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模空间上Poisson结构的比较

Comparison of Poisson structures on moduli spaces

Indranil Biswas, Francesco Bottacin, Tomás L. Gómez

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摘要：

设$X$是复不可约光滑射影曲线，${\mathbb L}$是$X$上具有非零截面$σ_0$的代数线丛.令$\mathcal{M}$表示稳定Hitchin对$(E,\, θ)$的模空间，其中$E$是$X$上的固定秩$r$和度$δ$的代数向量丛，并且$θ\, \in\, H^0(X,\, End(E)\otimes K_X\otimes{\mathbb L})$。结合每个稳定Hitchin对的谱数据，得到了$\mathcal{M}$与全空间$K_X\otimes{\mathbb L}$上纯一维稳定层的模空间$\mathcal{P}$的同构.模空间$\mathcal{P}$和$\mathcal{M}$都具有代数Poisson结构，它们是用$σ_0$构造的.这里我们证明了$\mathcal{P}$和$\mathcal{M}$之间的上述同构保持泊松结构。

英文摘要：

Let $X$ be a complex irreducible smooth projective curve, and let ${\mathbb L}$ be an algebraic line bundle on $X$ with a nonzero section $σ_0$. Let $\mathcal{M}$ denote the moduli space of stable Hitchin pairs $(E,\, θ)$, where $E$ is an algebraic vector bundle on $X$ of fixed rank $r$ and degree $δ$, and $θ\, \in\, H^0(X,\, End(E)\otimes K_X\otimes{\mathbb L})$. Associating to every stable Hitchin pair its spectral data, an isomorphism of $\mathcal{M}$ with a moduli space $\mathcal{P}$ of stable sheaves of pure dimension one on the total space of $K_X\otimes{\mathbb L}$ is obtained. Both the moduli spaces $\mathcal{P}$ and $\mathcal{M}$ are equipped with algebraic Poisson structures, which are constructed using $σ_0$. Here we prove that the above isomorphism between $\mathcal{P}$ and $\mathcal{M}$ preserves the Poisson structures.

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