## 数学代写|黎曼曲面代写Riemann surface代考|PHYS5010

2023年2月1日

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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 §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.

Suppose $\left(Y, Y^{\prime}\right)$ is a pair consisting of a complex manifold and a closed analytic subset. Suppose $g: Y \rightarrow \mathbf{C}$ is a holomorphic function, and $b$ is a holomorphic form of top degree. Suppose that for each $\zeta$ in the universal cover of $D^(s, \epsilon)$, we have a cycle in relative homology $\eta(\zeta)$, such that $\zeta$ is not contained in the support of $g . \eta(\zeta)$, and such that for $\zeta^{\prime}$ near $\zeta, \eta\left(\zeta^{\prime}\right)$ is homologous to $\eta(\zeta)$ by a homology $\kappa\left(\zeta, \zeta^{\prime}\right)$ whose support doesn’t meet $\zeta$ (in other words, $\eta\left(\zeta^{\prime}\right)-\eta(\zeta)=\partial \kappa\left(\zeta, \zeta^{\prime}\right)$ in relative homolory). Then the function $$f(\zeta)=\int_{\eta(\zeta)} \frac{b}{g-\zeta}$$ is a multivalued analytic function on $D^(s, \epsilon)$.
Proposition 10.1 Suppose that $Y^{\prime}$ is a divisor with normal crosings in $Y$, and suppose that the critical point sets of the function $g$ on $\left(Y, Y^{\prime}\right)$ are compact. Suppose there is a compact subset $K \subset Y$ such that for any $\zeta$ in the universal cover of $D^*(s, \epsilon)$, the cycle $\eta(\zeta)$ is contained in $K$. Suppose also that the homologies $\kappa\left(\zeta, \zeta^{\prime}\right)$ are contained in $K$. Then the function $f(\zeta)$ has regular singularities.

Proof: Since the critical point sets are compact, we may make a resolution of singularities $\tilde{Y} \rightarrow Y$, such that the fibers of $g$ are divisors with normal crossings and the strict transform $\tilde{Y}^{\prime}$ of $Y^{\prime}$ is a divisor with normal crossings, which crosses fibers of $g$ normally $[12]$.

## 数学代写|黎曼曲面代写Riemann surface代考|Uniformity of N

To complete the proof of condition (2.5.4), we need to obtain a uniform bound for the numbers $N$ in the expansions of $f_n(\zeta)$. Let $U^I$ be a neighborhood of some component of the critical point set in $Z_I / \Gamma_I$. Here everything looks like in the algebraic situation, because the critical point set is a compact subvariety. Let $s$ be the point to which the component maps under $g$. For any $\zeta$ close to $s$, let $U_\zeta^I=g^{-1}(\zeta) \cap U^I$. For any $J$ and $\alpha: J \rightarrow I$ let $U^{J, \alpha} \subset Z_J / \Gamma_J$ denote $\alpha^{-1} U^I$, and similarly for $U_\zeta^{J, \alpha}$. Then there is an action of the monodromy operator $T$ on the homology $H_*\left(U_\zeta^I, \cup_\alpha \alpha U_\zeta^{J, \alpha}\right)$. There is a number $N_I$ such that $\left(T^{N_I}-I\right)^K=0$ on this homology group for some $K$. We have to show that there is a uniform bound for this number $N_I$, independent of the index $I$ or the component of the critical point set.

There is a spectral sequence for relative homology, which converges to $H_\left(U_\zeta^I, U_\alpha \alpha U_\zeta^{J, \alpha}\right)$. The $E^2$ term is a direct sum of homology groups of the form $H_\left(U_\zeta^J\right)$. On each of these groups we have $\left(T^{M_J}-I\right)^K=0$ for some $K$. The number $N_I$ is the least common multiple of the $M_J$ which occur for the terms in the spectral sequence. Thus it suffices to show that there is a bound for the $M_J$.

Now recall from above that we can write $Z_J=Z^a \times Z^b$ where $g$ is constant on $Z^b$ and has isolated critical points on $Z^a$. Then $Z_J / \Gamma_J=Z^a / \Gamma^{\prime} \times X^b$. We may enlarge $U^{J, \alpha}$ until it has the form $U=U^{\prime} \times X^b$, for a relatively compact $U^{\prime} \subset Z^a / \Gamma^{\prime}$. Then $U_\zeta=U_\zeta^{\prime} \times X^b$. Further we may choose a realization of the monodromy operator which is constant in the $X^b$ direction. By the Künneth formula it suffices to bound the exponent $M$ such that $\left(T^M-I\right)=0$ on $H_*\left(U_\zeta^{\prime}\right)$. In other words, we may assume that $g$ has isolated singularities on $U$.

Suppose $u$ is a class in $H_*\left(U_\zeta\right)$. There is a retraction $R$ from $U$ to $U_s$, and we may assume that this retraction commutes with the monodromy operator. In particular, $R(T-I) u=0$. Note that $(T-I)$ is a factor in $\left(T^M-I\right)$. Therefore it suffices to bound the number $M$ such that $\left(T^M-I\right)^K u=0$ for some $K$ for all $u$ such that $R u=0$. But the singular fiber $U$, has isolated singularities, so if $R u=0$ then $u$ is a sum of classes supported on small neighborhoods of the singularities.

A singularity has the form $\left(s_1, \ldots, s_n\right)$, where $s_k$ are singularities of $g^k=$ $g_{j_{k-1} j_k}$ on $Z$. Let $U^k$ be a small neighborhood of $s_k$, and we may assume $U=U^1 \times \ldots U^n$. Choose local coordinates $z_k$ so that $s_k$ is given by $z_k=0$, and $g^k=\left(z_k\right)^{\nu_k}$ (we may change $g^k$ by constants-this moves the point $s$ to 0 ). The $\nu_k$ come from a finite set of numbers, depending on the orders of zeros of the one-forms $a_i$. Note that the $g^k$ are not identically zero because of the condition that the singularities are isolated.

# 黎曼曲面代写

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

$$f(\zeta)=\int_{\eta(\zeta)} \frac{b}{g-\zeta}$$

## 数学代写|黎曼曲面代写Riemann surface代考|Uniformity of N

$R u=0$. 但单一的纤维 $U$ ，有孤立的奇点，所以如果
$R u=0$ 然后 $u$ 是在奇点的小邻域上支持的类的总和。

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