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# 数学代写|黎曼曲面代写Riemann surface代考|MAT565 REGULARITY OF INDIVIDUAL TERMS

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## 数学代写|黎曼曲面代写Riemann surface代考|REGULARITY OF INDIVIDUAL TERMS

We have now shown that the individual terms in the Laplace transform add up to the Laplace transform of the monodromy. Finally we have to show that the individual terms have regular singularities. This will verify properties (2.5.4) and (2.5.5) from $\S 2$. The formal sum of the power series for the individual terms will then give the power series for the singularities of the Laplace transform of the monodromy, due to the estimates given in the previous section.
Condition (2.5.4)
First we must prove that the terms $f_n(\zeta)$ have locally finite regular singularities. To do this we use the following proposition, a technical extension of the wellknown regularity of the Gauss-Manin connection.

## 数学代写|黎曼曲面代写Riemann surface代考|We now apply this to our situation

We now apply this to our situation. We would like to express the integrals $f_I(\zeta)$ as integrals in a space where the critical point sets of $g$ are compact. This will take some work, making use of the assumption that our original Riemann surface $S$ was compact.

For each index $I$, let $\Gamma_I$ denote the subgroup of translations of $Z_I$ by elements of $\pi_1(S)^n$ which preserves the function $g$. Then $b_I$ and $g$ descend to the quotient $Z_I / \Gamma_I$. Furthermore, if $\alpha: I^{\prime} \rightarrow I$ is an elementary arrow then the map $\alpha: Z_{I^{\prime}} \rightarrow Z_I$ descends to a map $Z_{I^{\prime}} / \Gamma_{I^{\prime}} \rightarrow Z_I / \Gamma_I$. To see this, suppose $\gamma^{\prime}=\left(\gamma_1^{\prime}, \ldots, \gamma_m^{\prime}\right) \in \Gamma_{I^{\prime}}$. We will find $\gamma \in \Gamma_I$ such that $\alpha \circ \gamma^{\prime}=\gamma \circ \alpha$. Suppose $\alpha$ is determined by a number $l$ as in $\S 4$. Set $\gamma_k=\gamma_k^{\prime}$ for $k \leq l$ and $\gamma_{k+1}=\gamma_k^{\prime}$ for $k \geq l$. Then $\alpha\left(\gamma^{\prime} z\right)k=\gamma_k^{\prime} z_k=\gamma_k \alpha(z)_k$ if $k \leq l$ and $\alpha\left(\gamma^{\prime} z\right){k+1}=\gamma_k^{\prime} z_k=\gamma_{k+1} \alpha(z)_{k+1}$ if $k \geq l$, so $\alpha \circ \gamma^{\prime}=\gamma \circ \alpha$. Therefore the integrals can be defined as integrals over relative homology classes on the spaces $Z_I / \Gamma_I$. We have to show that the connected components of the critical point sets are compact.

## MATLAB代写

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