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arXiv 1810.06752math.AG
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关于抛物型向量丛与抛物联络的直象

On the direct images of parabolic vector bundles and parabolic connections

Indranil Biswas, Francois-Xavier Machu

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摘要：

设$φ: Y \rightarrow X$是光滑复射影曲线之间的有限满射，其中$X$是不可约的，但$Y$不必是不可约的.设$V_*$是$Y$上的抛物向量丛.我们在$X$上的直接像$φ_* V$上构造了一个抛物线结构，其中$V$是$V_*$的向量丛。用这种方法得到的$X$上的抛物向量丛$φ_* V_*$有一个分歧环面子丛;它是$φ_* V_*$在抛物点处满足一定条件的抛物因子之外的$\text{Ad}(φ_* V)$的环面丛。反之，给定$X$上的抛物向量丛$E_*$，以及它的分歧环面子丛$\mathcal T$，我们构造$X$的分歧覆盖$Z$和$Z$上的抛物向量丛$W_*$，使得抛物丛$E_*$是$W_*$的直像. $V_*$上的连接会产生$φ_* V_*$上的连接。$φ_* V_*$的分歧环面子丛由$φ_* V_*$上的对数连接所诱导的$\text{End}(φ_* V)$上的对数连接所保持。如果$X$上的抛物向量丛$E_*$具有一个联络$D$，使得由它导出的自同态丛上的联络保持分歧环面子丛$\mathcal T$，那么我们证明了$Z$上的抛物向量丛$W_*$具有一个联络，它产生直接象$E_*$上的联络$D$.

英文摘要：

Let $φ: Y \rightarrow X$ be a finite surjective morphism between smooth complex projective curves, where $X$ is irreducible but $Y$ need not be so. Let $V_*$ be a parabolic vector bundle on $Y$. We construct a parabolic structure on the direct image $φ_* V$ on $X$, where $V$ is the vector bundle underlying $V_*$. The parabolic vector bundle $φ_* V_*$ on $X$ obtained this way has a ramified torus sub-bundle; it is a torus bundle of $\text{Ad}(φ_* V)$ outside the parabolic divisor for $φ_* V_*$ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle $E_*$ on $X$, and a ramified torus sub-bundle $\mathcal T$ for it, we construct a ramified covering $Z$ of $X$ and a parabolic vector bundle $W_*$ on $Z$, such that the parabolic bundle $E_*$ is the direct image of $W_*$. A connection on $V_*$ produces a connection on $φ_* V_*$. The ramified torus sub-bundle for $φ_* V_*$ is preserved by the logarithmic connection on $\text{End}(φ_* V)$ induced by this connection on $φ_* V_*$. If the parabolic vector bundle $E_*$ on $X$ is equipped with a connection $D$ such that the connection on the endomorphism bundle induced by it preserves the ramified torus sub-bundle $\mathcal T$, then we prove that the corresponding parabolic vector bundle $W_*$ on $Z$ has a connection that produces the connection $D$ on the direct image $E_*$.

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