\section{Introduction}
We work over complex numbers. Severi inequality is proposed by Severi in ([Sev32]), which states that:
\begin{Theorem}[Severi inequality] Let $S$ be a smooth minimal surface of general
type with $Albdim(S) = 2$. Then
$$K_S^2 \geq 4\chi(S)$$
\end{Theorem}
But Severi provided a wrong proof, and Catanese pointed it out and posed the inequality as a conjecture (cf. \cite{Ca}). The conjecture was proved by Manetti when $K_S$ is ample (cf. \cite{Ma}), what's more he showed that when the equality is attained, then $S$ is birational to a double cover of an Abelian surface. Using completely different method, Pardini finally proved the conjecture (cf. \cite{Par}). However, her method, heavily relying on Xiao's inequality, gave little geometric information. In particular, when the equality is attained, i.e., $K_S^2 = 4\chi(S)$, it is conjectured that the canonical model of $S$ is a double cover over an Abelian surface branched along a divisor having at most negligible singularities.

Before we explain our idea, let's recall Manetti's method: Let $S$ be a smooth minimal surface of general
type with $Albdim(S) = 2$. A curve $C$ on $S$ is called of
\begin{itemize}
\item
type 0: if it is contracted by the Albanese map;
\item
type 1: if at a general point of $C$ the differential of the Albanese map has rank 1;
\item
type 2: if at a general point of $C$ the differential of the Albanese map has rank 2.
\end{itemize}
Let $p: \mathbb{P}_S(T_S) \rightarrow S$ be the natural projection, $L$ the anti-tautological line bundle. Then we have
$$(p^*K_S + L)L^2 = 3(K_S^2 - 4\chi(S))$$
It suffices to prove
$$(p^*K_S + L)L^2\geq 0$$
Take general $L_1, L_2 \in |L|$, and write $L_1 \cap L_2 = V + \Gamma_0 + \Gamma_1 + \Gamma_2$ where $V$ is the vertical part, $\Gamma_i$ is composed with some curves which are mapped to curves of type $i$. The term $\Gamma_0(p^*K_S + L)$ may be $<0$. Manetti decomposed $(p^*K_S + L)L_1L_2$ into the summation of terms depending on certain points and curves. If $p(\Gamma_0)$ contains no $(-2)$-curves, by careful analysis, Manetti showed that $(p^*K_S + L)L_1L_2 \geq 0$ and the equality is attained if and only if the Albanese map is a smooth double cover over an Abelian surface. Using Manetti's idea, Mends Lopes and Pardini proved some stronger inequalities (cf. \cite{MP}).

We will apply Manetti's argument to the canonical model $\bar{S}$ of $S$, which comes from contracting all the $(-2)$-curves of $S$. $\bar{S}$ has at most quotient singularities (cf. \cite{Mat}, Chap. 4, Sec. 6), hence has an orbifold structure, which is associated with a stack $\mathcal{S}$.  Similarly we define curves of type $0,1,2$ on $\bar{S}$. We will do intersections on $\mathcal{S}$ and $\mathbb{P}_{\mathcal{S}}(T_{\mathcal{S}})$ which is also a stack induced by an orbifold.
If the curves of type 0 do not pass through the singularities of $S$, we reduced Severi inequality to an interesting and seemingly easy algebraic problem. However, due to lack of capability, the author failed in solving that problem except for a special case, so there arises Conjecture \ref{conjg}. Granted this conjecture, we proved
\begin{Theorem}\label{main}
Let $S$ be the canonical model of a smooth surface of general type and of maximal Albanese dimension, and denote by $alb_S: S \rightarrow A$ the Albanese map.
Assume that no curves of type 0 pass through the singularities of $S$. If Conjectures \ref{conjg} is true, then
$$K_S^2 \geq 4\chi(S)$$
moreover the equality is attained if and only if $alb_S: S \rightarrow A$ is a double cover over an Abelian surface
branched along a curve with at most negligible singularities.

In particular if $q(S) = 2$ and for two elements $\alpha, \beta \in H^0(S, \Omega_S^1)$ the divisor given by $\alpha \wedge \beta \in H^0(S, K_S)$ is reduced, then the assertion above is true.
\end{Theorem}

\begin{Acknowledgments} This work was done during the years in Peking University. I am grateful to Prof. Jinxing Cai, Wenfei Liu and Yifan
Chen for their valuable suggestions and many useful discussions
during the preparation of this paper.
\end{Acknowledgments}

\section{Preliminaries}
As preliminaries, let's recall the results about orbifolds, stacks, $V$-free sheaves, and the intersection theory on stacks. We will omit the detailed proof involved, for which the readers can refer to \cite{Bl} and \cite{Vis}.
\subsection{Orbifold (V-manifold) and V-free sheaves}

\begin{Definition}Let $X$ be a variety. $X$ is called an
orbifold (or V-manifold), if every point $x \in X$ is attached a pair $((Y, y), G_x)$ where $G_x$ is a finite group acting faithfully on the smooth germ $(Y,y)$ such that the germ
$(X,x) \cong (Y,y)/G_x$. Naturally, we can associate $X$ with a smooth Deligne-Mumford stack $\mathcal{X}$. In the following, we usually denote by $\pi: \mathcal{X} \rightarrow X$ the natural morphism, and if no confusion occurs, the local quotient map $(Y,y) \rightarrow (X,x)$ is also denoted by $\pi$.
\end{Definition}

\begin{Definition'}[\cite{Bl}]Let $X$ be an orbifold. For $x\in X$, assume it is attached the pair $((Y, y), G_x)$,
and denote by $\pi: (Y,y)\rightarrow (X,x)$ the quotient map. A
coherent sheaf $\mathcal{F}$ on $X$ is called V-free at $x$, if it
satisfies one of the following equivalent conditions:
\begin{enumerate}
\item[(i)]{there exists a free $G_x$-invariant sheaf $\tilde{\mathcal{F}}$ on $(Y,
y)$ such that $\mathcal{F} = \pi_*^{G_x}\tilde{\mathcal{F}}$;}
\item[(ii)]{$\mathcal{F}$ is reflexive and $\tilde{\mathcal{F}} =
(\pi^*\mathcal{F})^{**}$ is free.}
\end{enumerate}
We say $\mathcal{F}$ is V-free if it is V-free at every point of $X$.
\end{Definition'}
The natural correspondence $\mathcal{F} \mapsto (\pi^*\mathcal{F})^{**}$ gives an $1:1$ correspondence between the V-free sheaves on $X$ and the vector bundles on $\mathcal{X}$.

\begin{Example} \label{surf}Let $S$ be a normal surface with at most canonical
singularities. Then $S$ has at most isolated quotient
singularities of type $A_i,i=1,2,..., D_i,i\geq 4, E_6,E_7,E_8$, so for every $x \in S$, in the usually way we can assume the germ $(S, x) \cong (\mathbb{C}^2, 0)/G_x$ where $G_x \subset SL(2, \mathbb{C})$ is finite group and the action $G_x$ on $(\mathbb{C}^2, 0)$ is induced by its natural representation on $\mathbb{C}^2$. By attaching $x\in S$ the pair $((\mathbb{C}^2, 0), G_x)$, we get an orbifold structure on $S$. Let $i: S^{sm}\subset S$ be the smooth part. Then
$\Omega_S^1:=i_*\Omega_{S^{sm}}^1$ is a V-free sheaf since $(\pi^*\Omega_S^1)^{**} \cong \Omega_{\mathcal{S}}^1$.
\end{Example}

\subsection{Intersection theory and Chern classes}
Let $X$ be an orbifold of dimension $n$ associated with a smooth Deligne-Mumford stack $\mathcal{X}$. In \cite{Vis}, the author built an intersection theory on such $\mathcal{X}$. By that theory, if $D_i,i = 1,2,...,n$ are $(n-1)$-cycles on $X$, then $\pi^*D_1 \cdot \pi^*D_2 \cdot ... \cdot \pi^*D_n = D_1 \cdot D_2 \cdot ... \cdot D_n$, precisely for $x \in X$ and $\pi: (Y, y) \rightarrow (X , x)$ the local covering, if $D_1, D_2,..., D_n$ intersect properly, then $(\pi^*D_1 \cdot \pi^*D_2 \cdot ... \cdot \pi^*D_n)_y/|G_x| = (D_1 \cdot D_2 \cdot ... \cdot D_n)_x$.


Let $\mathcal{F}$ be a V-free
sheaf on $X$ of rank $r$.
First we introduce the definition of Chern classes by means of metric (cf. \cite{Bl}).
\begin{Definition} A \emph{V-metric} is a hermitian metric $h$ defined on
$\mathcal{F}|_{X^{sm}}$ such that $\pi^*h$ extends to a
hermitian metric on $(\pi^*\mathcal{F})^{**}$ locally. In turn, we can associate the \emph{V-metric} h the $i^{th}$ Chern class \emph{$c_i(X,\mathcal{F},h) \in H^i(X, \mathbb{Q})$} by means of the curvature matrix of $\pi^*h$ on $(\pi^*\mathcal{F})^{**}$.
\end{Definition}


Another definition is purely algebraic.
Let $P(X, \mathcal{F}) =Proj\oplus_i \mathcal{O}_X(S^i(\mathcal{F})^{**})$. Consider the local quotient map $\pi: (Y,y) \rightarrow (X,x)$. $G_x$ acts naturally on $Proj\oplus_i
\mathcal{O}_Y(S^i((\pi^*\mathcal{F})^{**}))$ since $(\pi^*\mathcal{F})^{**}$ is a $G_x$-invariant sheaf, and one can prove that
$P((X,x), \mathcal{F}): =Proj\oplus_i
\mathcal{O}_{(X,x)}(S^i(\mathcal{F})^{**}) \cong Proj\oplus_i
\mathcal{O}_{(Y,y)}(S^i((\pi^*\mathcal{F})^{**}))/G_x$. In this way, we attach $P(X, \mathcal{F})$ an orbifold structure with Deligne-Mumford stack $P(\mathcal{X}, (\pi^*\mathcal{F})^{**})$.
Let $L$ be the V-line bundle $\mathcal{O}_{P(X, \mathcal{F})}(1)$, and
$\rho: P(X, \mathcal{F}) \rightarrow X$ the natural map.
We can inductively define the $\hat{c_i}(X,\mathcal{F})'s$ by the following equations (cf. \cite{Ka}):
\begin{equation}\label{ch}
L^r - L^{r-1}\rho^*\hat{c_1}(X,\mathcal{F}) +\cdot\cdot\cdot +
(-1)^r\rho^*\hat{c_r}(X,\mathcal{F}) = 0, r= 1,2,...
\end{equation}

In the paper \cite{Ka}, Y. Kawamata proved that
\begin{Proposition}
$\hat{c_i}(X,\mathcal{F})$ coincides with $c_i(X,\mathcal{F},h)$.
\end{Proposition}


\begin{Example}
Fix a point $p \in \mathbb{P}^1$. Associate an orbifold structure on $\mathbb{P}^1$ by attaching a point $x \neq p$ the pair $((\mathbb{C}, 0), \{1\})$ and the point $p$ the pair $((\mathbb{C}, 0), \{1, \sigma\})$ where $\sigma$ is an involution given by $\sigma(t) = -t$. We get an orbifold curve $C$ with Deligne-Mumford stack $\mathcal{C}$. Its cotangent bundle $\Omega_\mathcal{C}^1$ gives an orbifold line bundle $L$ on $C$. Since $2\Omega_\mathcal{C}^1 \equiv \pi^*(2\Omega_{\mathbb{P}^1}^1 + p)$, so $deg(\hat{c_i}(L)) = -\frac{1}{2}$.
\end{Example}


\subsection{The curves on an orbifold surface}\label{cins}

Let $S$ be a surface with at most canonical singularities. We attach $S$ an orbifold structure with Deligne-Mumford stack $\mathcal{S}$ as in Example \ref{surf}. Let $D \subset S$ be an irreducible curve on $S$, and $\sigma: \tilde{D} \rightarrow D$ the normalization. Note that $\mathcal{O}_S(D): = i_*\mathcal{O}_{S^{sm}}(D)$ is a V-free sheaf on $S$, since $\mathcal{O}_{\mathcal{S}}(\pi^*D)$ is a line bundle on $\mathcal{S}$.

Let $\mathcal{D} \rightarrow \pi^*D$ be the normalization. Then we get a natural map $\mathcal{D} \rightarrow \tilde{D}$ which induces an orbifold structure on $\tilde{D}$ in the following way.
For $x \in D$, assume it is attached the pair $((\mathbb{C}^2, 0), G_x)$ and denote by $\pi: (\mathbb{C}^2, 0) \rightarrow (S,x)$ the quotient map. Let $\amalg_i(\mathcal{D}_i, y_i) \rightarrow (\pi^*D, 0)$ be the normalization. The action of $G_x$ on $\pi^*D$ induces an action on $\amalg_i(\mathcal{D}_i, y_i)$ naturally. Write that $\sigma^{-1}(D, x) = \amalg_j(\tilde{D}, x_j)$. For every $j$, we find a $(\mathcal{D}_i, y_i)$ and a subgroup $G_{ij} \subset G_x$ which fixes $\mathcal{D}_i$ such that $(\tilde{D}, x_j) \cong (\mathcal{D}_i, y_i)/G_{ij}$, up to an isomorphism the quotient map is independent of the choices of $i$. Therefore, we get a natural map $\mathcal{D} \rightarrow \tilde{D}$, and it induces an orbifold structure on $\tilde{D}$ with Deligne-Mumford stack $\mathcal{D}$.



\section{Characterization of the double cover over an Abelian surface}
The following theorem gives a characterization of the double cover over an Abelian surface by the ramification divisor. The idea is from \cite{De}, in particular if $R$ is irreducible, our theorem follows from applying Prop. 3.1 in \cite{De} directly.
\begin{Theorem}\label{char}
Let $X$ be a normal surface, $A$ an Abelian surface, $f: X \rightarrow A$ a finite surjective morphism, $R$ the ramification divisor of $f$. Assume that
\begin{itemize}
\item
$R$ is reduced and ample; and
\item
for every $r \in R$, the restriction map to the germ $f|_{(R, r)}$ is an isomorphism to its image.
\end{itemize}
Then there exists an Abelian surface $T$ such that $f$ factors through a double cover $g:X \rightarrow T$ which is ramified along $R$.
\end{Theorem}
\begin{proof}
Consider the map $p: X \times R \rightarrow A$ given by $(x, r) \mapsto f(x) - f(r)$. $p$ has a factorization
$$p = \pi \circ q: X \times R \rightarrow T \rightarrow A$$
where $T = spec_{\mathcal{O}_A}(p_*\mathcal{O}_{X \times R})$, hence $q$ is a connected morphism and $\pi$ is finite.
Our theorem follows from the following lemma.
\begin{Lemma}
The map $\pi: T \rightarrow A$ is an $\acute{e}$tale cover, hence $T$ is an Abelian surface. And there exists a morphism $g:X \rightarrow T$ such that $f = \pi \circ g + a_0$ for some $a_0 \in A$ and $q(x,r)= g(x) - g(r)$, in particular the fiber $q^{-1}(0) \cong R \times_T R$ is connected.
\end{Lemma}
Now we begin to prove this lemma.

Step 1:
For every $a \in A$, we can find an analytic open neighborhood $a \in U \subset A$ such that:
\begin{itemize}
\item
$f^{-1}(U) = V_1 \cup V_2 \cup ... \cup V_k$ where $V_i \cap V_j = \emptyset, i \neq j$;
\item
$f|_{V_i \cap R}$ is an isomorphism if $V_i \cap R \neq \emptyset$;
\item
$f|_{V_i}: V_i \rightarrow U$ is a double cover ramified along $V_i \cap R$ if $V_i \cap R \neq \emptyset$ and is an isomorphism otherwise.
\end{itemize}
We say an open set $V \subset X$ is \emph{good} if $f|_V$ is either an isomorphism or a standard double cover ramified along $V \cap R$. So for every $x \in X$, there exists a good neighborhood containing $x$.

Let $B = f(R)$. For $a' \in A$, denote by $\Delta_{a'} \subset (B+ a') \times B$ the set of the points like $(b+ a', b)$. Then $p^{-1}(a') = F^{-1}(\Delta_{a'})$ where $F = f \times f|_R: X \times R \rightarrow A \times B$. For a good open set $V \subset X$ and an open subset $R_s$ of $R$ such that $f|_{R_s}$ is an isomorphism, putting $U = f(V), R_V = V \cap R, B_V = f(R_V), B_s = f(R_s)$, we have the following two facts:
\begin{enumerate}
\item[(i)]{$p^{-1}(a') \cap (V \times R_s) = F|_{(V \times R_s)}^{-1}(\Delta_{a'})\cong f|_V^{-1}(B_s + a')$;}
\item[(ii)]{$p^{-1}(a') \cap (V \times R_s)$ is connected if and only if $(B_s + a') \cap U$ is connected and in addition $(B_s + a') \cap B_V \neq \emptyset$ if $f|_{V}$ is a double cover;}
\end{enumerate}
where $(i)$ follows from the fact that $R_s \rightarrow B_s$ is an isomorphism, and $(ii)$ is from $(i)$ and the fact that $V$ is good.

Step 2: $\pi$ is unramified if seen as a map between two analytically topological spaces.

We argue by contrary, so assume that $\pi$ is ramified at $t \in T$ and let $a = \pi(t)$. Then we can find a neighborhood $U_a$ of $a$ and a connected component $D_a$ of $p^{-1}(a)$, such that for every $(x, r) \in D_a$, there exists a neighborhood $V_x \times R_r$ of $(x,r)$ such that
\begin{itemize}
\item[($c_1$)]{$V_x$ is a good open neighborhood of $x$, and $f|_{R_r}$ is an isomorphism;}
\item[($c_2$)]{$p^{-1}(a) \cap (V_x \times R_r) = D_a \cap (V_x \times R_r)$ is connected and for general $a' \in U_a$ $p^{-1}(a') \cap (V_x \times R_r)$ is not connected.}
\end{itemize}
Put $$U_x = f(V_x),~B_r = f(R_r),~R_x = V_x \cap R,~ B_x = f(R_x)$$

Since $(B_r + a) \cap U_x$ is connected, shrinking $V_x, U_x$ and $U_a$ if necessary, we can assume
\begin{itemize}
\item[($c_3$)]{$(B_r + a') \cap U_x$ is connected for $a' \in U_a$.}
\end{itemize}
Then $(ii)$ in Step 1 implies that $f|_{V_x}$ is a double cover, $(B_r + a) \cap B_x \neq\emptyset$ and $(B_r + a') \cap B_x  = \emptyset$ for general $a' \in A$. However, we will show it is impossible below.

First remark that
\begin{itemize}
\item[(a)]{if $B_r + a$ and $B_x$ intersect properly or one of them is reducible, then shrinking $U_a$ if necessary, for every $a' \in U_a$, $(B_r + a') \cap B_x \neq \emptyset$, so we conclude that both $B_r + a$ and $B_x$ are irreducible and $B_r + a = B_x$;}
\item[(b)]{since $R$ is ample it is connected by Hodge Index Theorem, so from $(a)$ and Step 1 $(i)$, we conclude that $D_a \cap (X \times R)$ is mapped onto $R$ via the natural projection $X \times R \rightarrow X$.}
\end{itemize}
Now take finite $(x_i, r_i)$ and finite open sets $\{V_{x_i} \times R_{r_i}\}_i$ satisfying $(c_{1-3})$ and covering $D_a$. Shrinking $U_a$ if necessary, for $a' \in U_a$, the union of $\{V_{x_i} \times R_{r_i}\}_i$ covers a component $D_{a'}$ of $p^{-1}(a')$. Let $I_{a'}$ denote the image of $D_{a'}$ via the natural projection $X \times R \rightarrow X$. Since $R$ is ample, $I_{a'} \cap R \neq \emptyset$. By $(b)$ the $V_{x_i}$'s cover $R$, so we can find a $V_{x_i}$ such that $I_{a'} \cap R_{x_i}\neq \emptyset$, hence $f(I_{a'} \cap R_{x_i}) \subset (B_{r_i} + a') \cap B_{x_i}\neq \emptyset$, and a contradiction follows.

Step 3: $T$ is smooth, hence $\pi: T \rightarrow A$ is an $\acute{e}$tale cover, and we can assume $\pi: T \rightarrow A$ is an isogeny between two Abelian surfaces.

Considering the normalization $\tilde{T}$ of $T$, by Step 2 it is topologically isomorphic to $T$ and the composed map $\tilde{T} \rightarrow A$ is $\acute{e}$tale, hence $\tilde{T}$ is smooth. For a point $t \in T$, denote by $\tilde{t} \in \tilde{T}$ the point over $t$ and $a \in A$ the image of $t$ via $\pi$. Then the composed map of germs $(\tilde{T}, \tilde{t}) \rightarrow (T, t) \rightarrow (A, a)$ is an isomorphism, so $(T, t)$ must be a smooth germ, and our assertion follows.

Step 4: $q^{-1}(0) \cong R \times_T R$ is connected.

Take $r_0 \in R$, and define $g: X \rightarrow T$ by $x \mapsto q(x, r_0)$. Then $g$ is ramified along $R$ since $\pi$ is $\acute{e}$tale. Consider the map $q': X \times R \rightarrow T$ defined by $(x, r) \mapsto g(x) - g(r)$. Since $\pi$ is an isogeny, check that
\begin{equation}
\begin{split}
\pi\circ q'(x,r) &= \pi (g'(x) - g'(r)) \\
&= \pi(q(x,r_0) - q(r, r_0)) \\
&= \pi \circ q(x,r_0) - \pi \circ q(r,r_0) \\
&= f(x) - f(r_0) - (f(r) - f(r_0)) = \pi \circ q(x,r)
\end{split}
\end{equation}
i.e., $\pi\circ q = \pi \circ q'$. We conclude that $q'$ is connected, so there exists isomorphism $\sigma: T\rightarrow T$ such that $q = \sigma \circ q'$. Replacing $\pi$ by $\pi \circ \sigma$, we can assume $q = q'$. It follows that that $q(x,r) = g(x) - g(r)$ and $f(x) = \pi\circ g(x) + f(r_0)$. By Step 1 $(i)$, $R \times_T R$ is composed with some connected components of $q^{-1}(0)$, hence $q^{-1}(0) = R \times_T R$ is connected, and we are done.

\end{proof}





\section{Proof of Theorem \ref{main}}

\subsection{Reduce to the intersections of cycles}

Let $S$ be assumed as in Theorem \ref{main}, and $\sigma:
\tilde{S}\rightarrow S$ the minimal resolution. As in Example \ref{surf}, we give $S$ an orbifold structure with Deligne-Mumford stack $\mathcal{S}$. For a point $x \in S$, we assume it is attached the pair $((\mathbb{C}^2, 0), G_x)$, and denote by $\pi: (\mathbb{C}^2, 0) \rightarrow (S,x)$ the quotient
map and by $(u,v)$ a coordinate of $(\mathbb{C}^2, 0)$. Let $\Omega_{S}
= i_*\Omega_{S^{sm}}$. Note that $\Omega_{S} \cong \sigma_*\Omega_{\tilde{S}}$,
and then $\sigma_*: H^0(\tilde{S}, \Omega_{\tilde{S}})
\simeq H^0(S, \Omega_{S})$.
For
$\alpha \in H^0(\tilde{S},\Omega_{\tilde{S}})$,
$\pi^*\sigma_*\alpha |_{(\mathbb{C}^2, 0)}$ is a closed holomorphic 1-form, hence
exact, i.e., $\exists f \in \mathbb{C}[[u, v]]^{G_x}$ such that
$\pi^*\sigma_*\alpha = df$ locally.

By Definition 2.13 and the theorem in \cite{Bl} P.25, we have
$$c_2(\Omega_S) = e_{orb}(S) = e(S) - \sum_{x \in
Sing(S)}(1-1/|G_x|)$$ and
$$c_1(\Omega_S) = K_S$$

Let $X = P(S, \Omega_S^1)$ and $ L = \mathcal{O}_X(1)$. Using Eq. \ref{ch}, we have
$$(K_S+L)L^2 = 2c_1(\Omega_S)^2 -
c_2(\Omega_S) = 2K_S^2 - e(S) + \sum_{x \in Sing(S)}(1-1/|G_x|)$$
By $e(\tilde{S}) = e(S) + \sum_{x \in Sing(S)}(\#\{conjugencies~
of~ G_x\} - 1)$, it follows that
$$(K_S+L)L^2 = 2K_{\tilde{S}}^2 - e(\tilde{S}) + \sum_{x \in
Sing(S)}(\#\{conjugencies~ of ~G_x\} - 1/|G_x|)$$
By Neother's formula, we have $K_{\tilde{S}}^2 - 4\chi(\tilde{S})= \frac{1}{3}(2K_{\tilde{S}}^2 - e(\tilde{S}))$, so we reduce Severi inequality to
$$(K_S+L)L^2 \geq \sum_{x \in
Sing(S)}(\#\{conjugencies~ of ~G_x\} - 1/|G_x|)$$

If the Albanese image is of dimension 2, then there exist two
elements $\alpha,\beta \in H^0(S, \Omega_S^1)$ such that
$\alpha\wedge \beta \neq 0$ as a nonzero element in $H^0(S,K_S)$. Note that $\alpha$ and $\beta$ define two hyperplanes $L_\alpha$ and $L_\beta$ in $X$. Therefore, the Severi inequality is
equivalent to $$(\rho^*K + L)L_\alpha L_\beta \geq \sum_{x \in
Sing(S)}(\#\{conjugencies ~of ~G_x\} - 1/|G_x|)$$
We will divide the left hand side into the summation of some terms in local form.

\subsection{Reduce to a local inequality}\label{gl}
Assume $L_\alpha \cap L_\beta = \sum_i n_iC_i + \sum_{q \in S} m_{q}F_q + \sum_jl_jE_j$ where every $C_i$ is a reduced and irreducible horizontal curve, $F_k$ a vertical curve and $E_j$ the pull-back $\rho^*e_j$ of a curve $e_j \subset S$. Put $D_i = \rho_* C_i$. Then it follows that
$(\alpha \wedge \beta) = \sum_i n_iD_i + \sum_jl_je_j$.

For a $D_i$ not of type 0, we calculate $LC_i$. We will consider the intersections of some cycles on the stack $\mathcal{X} = P(\mathcal{S}, \Omega_\mathcal{S}^1)$ and $\mathcal{S}$. We also denote by $\pi: \mathcal{X} \rightarrow X$ the natural map and by $\mathcal{L}$ the pull-back $\pi^*L$.

Let $\tilde{D_i}$ be the normalization of $D_i$ and denote by $j$ the composed map $\tilde{D_i} \rightarrow D_i \hookrightarrow S$. As in Section \ref{cins}, attach $\tilde{D_i}$ an orbifold structure with Deligne-Mumford stack $\pi: \mathcal{D}_i \rightarrow \tilde{D_i}$. The natural morphism $\mathcal{D}_i \rightarrow \mathcal{S}$ is also denoted by $j$ if no confusion occurs. Then there exists a divisor $R_i$ on $\mathcal{D}_i$ fitting into the following exact sequence
$$\mu: j^*\Omega_{\mathcal{S}}^1 \rightarrow \Omega_{\mathcal{D}_i}^1(-R_i) \rightarrow   0 $$
Set $\mathcal{Y} = P(\mathcal{D}_i,j^*\Omega_{\mathcal{S}}^1)$ which is a $\mathbb{P}^1$ bundle over $\mathcal{D}_i$. The kernel of $\mu$ defines a section $\mathcal{B}_i \subset \mathcal{Y}$. By abuse of notations, we also denote by $\mathcal{L}$ the line bundle $\mathcal{O}_\mathcal{Y}(1)$ and by $j$ the natural map from $\mathcal{Y}$ to $\mathcal{X}$ if no confusion occurs.

In the following we assume $\alpha, \beta$ are general, so we can assume
\begin{itemize}
\item[]
{$e_j$ is of type 0;}
\item[$\diamondsuit$]
{$\alpha$ is mapped to a nonzero element in $\Omega_{\mathcal{D}_i}^1(-R_i)$ via $\mu$, hence it defines the section $\mathcal{C}_i = j^*\pi^*C_i\subset \mathcal{Y}$ which is different from $\mathcal{B}_i$.}
\end{itemize}
Then we have
$$deg(\Omega_{\mathcal{D}_i}^1(-R_i)) = (\mathcal{C}_i + \sum_j\beta_j\mathcal{F}_j)\mathcal{B}_i$$
thus
\begin{equation}\label{key1}
\begin{split}
LC_i &= j^*\mathcal{L}(j^*\mathcal{L}  - \sum_j\beta_j\mathcal{F}_j) \\
     &= j^*\mathcal{L}^2 - deg(\Omega_{\mathcal{D}_i}^1(-R_i)) + \sum_i\mathcal{C}_i\mathcal{B}_i\\
     &= K_\mathcal{S}\pi^*D_i - (K_\mathcal{S }+ \pi^*D_i)\pi^*D_i + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1) + deg(R_i) + \mathcal{C}_i\mathcal{B}_i\\
     &=  -\pi^*D_i^2 + ((K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1) + \mathcal{C}_i\mathcal{B}_i)  + deg(R_i)
\end{split}
\end{equation}

Following the calculation of \cite{Ma}, we have
\begin{equation}
\begin{split}
&(\rho^*K_S + L)L_\alpha L_\beta = (\rho^*K_S + L)(\sum n_iC_i + \sum_p m_{p}F_p + \sum_jl_jE_j) \\
& = (\rho^*K_S + L)(\Sigma_{D_i~is ~not~of~ type ~0} n_iC_i) + \Sigma_{p~is~not~on~a~curve~of~type~0}m_{p}F_p) +  I_0' \\
& = I_1' + I_0'
\end{split}
\end{equation}
Now we focus on $I_1'$, so assume that none of the $D_i$'s appearing in the following equations are of type 0 and the point $p$ is not on a curve of type 0. Writing $K_\mathcal{S} \equiv \sum_jn_j\pi^*D_i + \pi^*D'$, then by \ref{key1} we have
\begin{equation}\label{key2}
\begin{split}
I_1'= &\sum_in_i(K_\mathcal{S}\pi^*D_i - \pi^*D_i^2 + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1) + \mathcal{C}_i\mathcal{B}_i+ deg(R_i)) + \sum_pm_{p} \\
=&\sum_i(n_i - 1)K_\mathcal{S}\pi^*D_i  \\
& + \sum_i((K_\mathcal{S} - n_i\pi^*D_i)\pi^*D_i + n_i((K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + deg(R_i)) + \sum_pm_{p}\\
=&\sum_i(n_i - 1)K_\mathcal{S}\pi^*D_i  + \sum_iD'D_i\\
& + \sum_in_i(\sum_{j \neq i}\pi^*D_j\pi^*D_i + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + deg(R_i)) + \sum_pm_{p}
\end{split}
\end{equation}
Let $I_0 = I_0' + \sum_iD'D_i$, and $I_1 = I'_1 - \sum_iD'D_i = \sum_i(n_i - 1)K_\mathcal{S}\pi^*D_i + \sum_in_i(\sum_{j \neq i}\pi^*D_j\pi^*D_i + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + deg(R_i)) + \sum_pm_{p}$. By assumption, curves of type 0 do not pass through singularities of $S$, completely the same argument as in \cite{Ma} shows that
\begin{itemize}
\item[$\spadesuit$]{$I_0 \geq 0$, and $I_0 = 0$ holds only if the divisor $(\alpha \wedge \beta)$ contains no curves of type 0, i.e., $D' = 0$.}
\end{itemize}
We will try to show the following more precise inequality
\begin{equation}\label{key3}
\begin{split}
&I_1 - \sum_i(n_i - 1)K_\mathcal{S}\pi^*D_i \\
&=\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + deg(R_i)) + \sum_pm_{p} \\
&\geq  \sum_{x \in
Sing(S)}(\#\{conjugencies ~of ~G_x\} - 1/|G_x|)
\end{split}
\end{equation}
If Inequality \ref{key3} were proved to be true and the equality in Severi inequality were attained, since $K_S$ is ample we could conclude that
\begin{itemize}
\item[$\heartsuit$]{$K_S \equiv (\alpha \wedge \beta) = \sum_i D_i$ where $D_i$ are distinct reduced and irreducible curves of type $1$ or $2$.}
\end{itemize}
We remark that $(K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)$ only depends on the singularities of $\pi^*D_i$, i.e., the singularities of $S$ and $D_i$, so for every $x \in S$, we can calculated its contributions to $(K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)$ (cf. Lemma \ref{ga}). And since $\pi^*D_i$ and $\pi^*D_j~ i \neq j$, $ \mathcal{C}_i$ and $\mathcal{B}_i$ intersect properly, the divisor $R_i$ is effective, for a point $x \in S$, we can naively define the contributions of $x$ to $\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + deg(R_i)) + \sum_pm_{p}$, which we denote by
$$[\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + deg(R_i)) + \sum_pm_{p}]_x$$
We will calculate the contributions of the singularities on $S$ in next section. To show Inequality \ref{key3}, it suffices to show that for every $x \in Sing(S)$
\begin{equation}\label{key3'}
\begin{split}
&[\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i) + \sum_pm_{p}]_x \\
&\geq \#\{conjugencies ~of ~G_x\} - 1/|G_x|
\end{split}
\end{equation}

In a special case, we proved this inequality.
\begin{Lemma}\label{key4}
Suppose that the divisor given by $(\alpha \wedge \beta)$ is reduced near a singularity $x \in S$. Then Inequality \ref{key3'} holds, and the equality is attained only when the local Albanese map $a_S: (S, x) \rightarrow (\mathbb{C}^2, 0)$ given by path integral via $\alpha ~and~ \beta$, is a standard double cover branched along a curve with a negligible singularity at $0$.
\end{Lemma}
We postpone the proof to Sec. \ref{prooflm}.

\begin{Remark}
The proof of Lemma \ref{key4} also applies for a smooth point not lying on a curve of type 0, so we conclude that
\begin{itemize}
\item[$\clubsuit$]{for a smooth point $x \in S$ not lying on a curve of type 0, if $(\alpha \wedge \beta)$ is reduced near $x$, then Inequality \ref{key3'} holds, and the equality is attained if and only if $(\alpha \wedge \beta)$ is smooth at $x$, and the local Albanese map defined by $\alpha ~and~ \beta$ is a standard double cover branched along a smooth curve.}
\end{itemize}
\end{Remark}

\subsection{Proof of Theorem \ref{main}}
By assumption Conjecture \ref{conjg} holds, which implies Inequality \ref{key3} by the argument in Sec. \ref{rc}, hence Severi inequality is true.

Now we assume the equality in Severi inequality is attained. Then
\begin{itemize}
\item
$\spadesuit$ tells that there exist no curves of type 0 on $S$;
\item
$\heartsuit$ tells that $(\alpha \wedge \beta)$ is reduced;
\item
$q = 2$, because otherwise we can find $\alpha, \beta$ satisfying assumptions $\diamondsuit$ and that the divisor $(\alpha \wedge \beta)$ is singular at a smooth point, which contradicts $\clubsuit$.
\end{itemize}
So the Albanese map $a_S: S \rightarrow A$ is either an isomorphism or a standard double cover locally, finally our theorem is an easy consequence of Theorem \ref{char}.

The converse is from direct calculation.

If $q = 2$ and $(\alpha \wedge \beta)$ is reduced, our assertion follows from applying Lemma \ref{key4} and Theorem \ref{char}.



\subsection{Local calculation}\label{rc}

\begin{Notation}\label{not}
Let $(S,x) = (\mathbb{C}^2,0)/G_x$. Assume $\pi^*\alpha, \pi^*\beta$ are defined by $df_x, dg_x$ where $f_x, g_x\in k[[u,v]]^{G_x}$ whose constant term is assumed to be 0, and assume $\pi^*D_i$ is defined by the equation $h_{xi} \in k[[u,v]]$, let $h_x = h_{x1}\cdot\cdot\cdot h_{xl}$. By assumption of the theorem, none of the $h_{xi}$'s divides $f_x$. Denote by $L_{dh_{xi}}$ the hyperplane determined by $dh_{xi}$ in $P((\mathbb{C}^2,0), \Omega_{(\mathbb{C}^2,0)}^1)$, and write that
$$j^*L_{dh_{xi}} = B_i + \sum_j\gamma_{xij} F_{x_j},~j^*L_{df_{x}} = C_i + \sum_j\beta_{xij} F_{x_j}~and~L_{df_{x}}\cap L_{dg_{x}} = \sum_in_iC_i + m_0F_0$$
where $j:\mathcal{D}_i \rightarrow \pi^*D_i \hookrightarrow (\mathbb{C}^2,0)$ the normalization of $\pi^*D_i$ and the $x_j$'s are the points on $\mathcal{D}_i$ over $0$. For $f \in k[[u,v]]$, $(f)$ denotes the divisor given by $f = 0$; and for a 2-form $f du \wedge dv$, $(f du \wedge dv)$ denotes the divisor $(f)$. In this section, for $f,g \in k[[u,v]]$, $(f)(g)$ means the intersection number of the divisors $(f)$ and $(g)$ at $0$.
\end{Notation}

Now the key point is the following lemma.
\begin{Lemma}\label{ga}
$$[(K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1) + \mathcal{C}_i\mathcal{B}_i +\sum_pm_{p}]_x =  \frac{(df_x\wedge dh_{xi})(h_{xi})-\sum_j\beta_{xij}+m_0}{|G_x|}$$
\end{Lemma}
\begin{proof}
Note that the pull-back of residue of $\frac{du \wedge dv}{dh_{xi}}$ is a local generator of the line bundle $j^*(K_\mathcal{S} + \pi^*D_i)$ on $\mathcal{D}_i$, and $dt$ is a local generator of $\Omega_{\mathcal{D}_i}^1$ where $t$ is a parameter of a component of $\mathcal{D}_i$. Comparing the two generators, we get
$$[(K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)]_x =\sum_j \gamma_{xij}/|G_x|$$

Then the lemma follows from the calculation $(df_x\wedge dh_{xi})(h_{xi}) = j^*L_{dh_{xi}}\cdot j^*L_{df_{x}} = C_iB_i + \sum_j(\gamma_{xij} + \beta_{xij})$ and the facts that $[\mathcal{C}_i\mathcal{B}_i]_x = \frac{C_iB_i}{|G_x|}$ and $[\sum_pm_{p}]_x = \frac{m_0}{|G_x|}$.
\end{proof}

\begin{Claim}\label{cl}With the notations as in \ref{not}, we have
\begin{enumerate}
\item[(i)]{$df_x\wedge dg_x = c\cdot h_{x1}^{n_1}\cdot\cdot\cdot h_{xl}^{n_l}du\wedge dv$ with $c \in k[[u,v]]$ such that $c(y) \neq 0$;}
\item[(ii)]{$(df_x \wedge dh_x)(df_x \wedge dg_x) = \sum_in_i((df_x\wedge dh_{xi})(h_{xi}) + \sum_{i \neq j}(h_{xj})(h_{xi}))$;}
\item[(iii)]{$L_{dH_{x}}L_{df_{x}}L_{dg_{x}} = (df_x \wedge dH_x)(df_x \wedge dg_x) - \sum_i\sum_jn_i\beta_{xij} + m_0$ for $H_x \in k[[u,v]]$ such that $H_x$ has no common divisor with $f_x$.}
\end{enumerate}
\end{Claim}
\begin{proof}
$i)$ is trivial.

$ii)$ Notice that $dh_x = h_{x2}\cdot\cdot\cdot h_{xl}dh_{x1} + ... + h_{x1}\cdot\cdot\cdot h_{x(l-1)}dh_{xl}$. Then we have
\begin{equation}
\begin{split}
&(df_x \wedge dh_x)(df_x \wedge dg_x) \\
&=\sum_i n_i(h_{x2}\cdot\cdot\cdot h_{xl}df_x \wedge dh_{x1} + ... + h_{x1}\cdot\cdot\cdot h_{x(l-1)}df_x \wedge dh_{xl})(h_{xi})\\
&= n_1(h_{x2}\cdot\cdot\cdot h_{xl}df_x \wedge dh_{x1} + h_{x1}(...))(h_{x1}) + ... + n_l(h_{x1}\cdot\cdot\cdot h_{xl-1}df_x \wedge dh_{xl} + h_{xl}(...))(h_{xl})\\
&= n_1(h_{x2}\cdot\cdot\cdot h_{xl}df_x \wedge dh_{x1})(h_{x1}) + ... +n_l(h_{x1}\cdot\cdot\cdot h_{xl-1}df_x \wedge dh_{xl})(h_{xl})\\
&= \sum_in_i((df_x\wedge dh_{xi})(h_{xi}) + \sum_{i \neq j}(h_{xj})(h_{xi}))
\end{split}
\end{equation}

$iii)$ Similarly by $j^*L_{df_{x}} = C_i + \sum_j\beta_{xij}F_{x_j}$, we have $L_{dH_{x}} C_i = (dH_x \wedge df_x)(h_{xi}) - \sum_j\beta_{xij}$. Then $iii)$ follows easily.
\end{proof}

With the help of Lemma \ref{ga} and Claim \ref{cl} above, we obtain
\begin{equation}\label{key3e}
\begin{split}
&[\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)+ \mathcal{C}_i\mathcal{B}_i + \sum_pm_{p}]_x \\
&=\frac{L_{dh_{x}}L_{df_{x}}L_{dg_{x}}}{|G_x|} \\
\end{split}
\end{equation}
So we reduce Inequality \ref{key3'} to
\begin{equation}\label{key5}
\begin{split}
&L_{dh_{x}}L_{df_{x}}L_{dg_{x}}\\
&\geq \#\{conjugencies~
of~ G_x\}|G_x| -1
\end{split}
\end{equation}

For Inequality \ref{key5}, we proposed a pure algebraic inequality as a conjecture.
\begin{Conjecture}\label{conjg}
Let $G$ be a finite subgroup of $SL(2,\mathbb{C})$ acting naturally on $(\mathbb{C}^2, 0) = spec(\mathbb{C}[[u,v]])$, and let $f,g \in \mathbb{C}[[u,v]]$ be two $G$-invariant formal polynomials whose constant term is $0$. Write $df \wedge dg = h_1^{n_1} \cdot\cdot\cdot h_l^{n_l}du \wedge dv$ where the $h_i$'s are reduced, irreducible and distinct to each other, and put $h = h_1 \cdot\cdot\cdot h_l$. Assume that none of $h_i$ divides $f$ (which means that the curve defined by $h_i = 0$ is not of type 0). Denote $X = P((\mathbb{C}^2, 0), \Omega_{(\mathbb{C}^2, 0)}^1)$. For a polynomial $H \in \mathbb{C}[[u,v]]$, $L_{dH}$ denotes the hyperplane defined by $dH$ in $X$. Then
\begin{equation}\label{conj}
L_{dh}L_{df}L_{dg} \geq \#\{conjugencies~of~ G\}|G| - 1
\end{equation}
\end{Conjecture}



\begin{Remark}\label{br}
During the proof of Lemma \ref{key4}, when $(df_x \wedge dg_x)$ is reduced near $0$, we in fact proved a more precise inequality
\begin{equation}
\begin{split}
&[\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)) + \sum_pm_p]_x \\
&\geq \#\{conjugencies ~of ~G_x\} - 1/|G_x|
\end{split}
\end{equation}
Therefore, by Eq. \ref{key3e} Conjecture \ref{conjg} holds when $(df \wedge dg)$ is reduced.
\end{Remark}

\subsection{Proof of Lemma \ref{key4}}\label{prooflm}

For a singularity $x \in S$, we now calculate part of its contributions $[\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)) + \sum_pm_p]_x$. Let the notations be as in \ref{not}.

For simplicity, we consider the case when $x$ is an $A_{n-1}$-singularity. In this case, $G_x \cong Z_n \cong \{1, \epsilon, \epsilon^2, ..., \epsilon^{n-1}\}$ where $\epsilon = e^{\frac{2\pi i}{n}}$, its action on $\mathbb{C}[[u, v]]$ is given by $\epsilon: u\mapsto \epsilon u, v \mapsto \bar{\epsilon}v$, hence $\mathbb{C}[[u,v]]^{G_x} = \mathbb{C}[[u^n, v^n, uv]]$. Then observe that $$mult_0(df_x \wedge dg_x) \geq n$$

Let $\sigma: V \rightarrow (\mathbb{C}^2,0)$ be the blowing up map at $0$, denote the exceptional $(-1)$-curve by $E$, and write $\sigma^*\pi^*D_i = \tilde{D}_i + a_iE$ where $\tilde{D}_i$ denotes the strict transform of $\pi^*D_i$. Then $[(K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)]_x$ is no less than $(a_ i - 1)a_i/n$, and the equality is attained if and only if $\tilde{D}_i$ is smooth. Therefore, we have
\begin{equation}
\begin{split}
&[\sum_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)]_x \\
&\geq (\sum_i(\sum_{j \neq i}a_ia_j + (a_ i - 1)a_i))/n \\
&= (\sum_ia_i(\sum_ia_i - 1))/n = (mult_0(df_x \wedge dg_x)(mult_0(df_x \wedge dg_x) - 1))/n \\
& \geq n - 1
\end{split}
\end{equation}
If the equality in the first inequality is attained, then the $\tilde{D}_i$'s are smooth, and any two of them are disjoint.

Calculate that
$$L_{d(u^n)}\cap L_{d(v^n)} \geq (n-1)^2F_{0}, L_{d(u^n)}\cap L_{d(uv)}\geq (n-1)F_{0}, L_{d(v^n)}\cap L_{d(uv)}\geq (n-1)F_{0}$$
Since $df_x, dg_x$ are generated by $d(u^n), d(v^n)$ and $d(uv)$, we conclude that $m_0 \geq n-1$, and since $m_x = m_0 /n$,
\begin{equation}
\begin{split}
&[\sum_in_i(\sum_{j \neq i}\pi^*D_i\pi^*D_j + (K_\mathcal{S} + \pi^*D_i)\pi^*D_i - deg(\Omega_{\mathcal{D}_i}^1)) + \sum_pm_p]_x \\
&\geq n - \frac{1}{n} = \#\{conjugencies ~of ~G_x\} - 1/|G_x|
\end{split}
\end{equation}

The equality above is attained if and only if
\begin{itemize}
\item
$mult_0(df_x \wedge dg_x) = n$;
\item
blowing up $(\mathbb{C}^2,0)$ at $0$, the singularity $0 \in \sum_i\pi^*D_i$ is resolved, hence $0$ is a simple singularity of $\sum_i\pi^*D_i$.
\end{itemize}
Write $df_x \wedge dg_x = hdu \wedge dv$, and $h_x(u, v) = h_n(u,v) + h_{\geq n+1}$ (for $F \in \mathcal{C}[[u,v]]$ $F_{\geq m}$ means the summation of the terms of degree $\geq m$ appearing in $F$). Then the divisor $(h_n)$ has a simple singularity at $0$, precisely the germ $((h_n), 0) \cong (\sum_i\pi^*D_i, 0)$.
Easy calculation gives that
$$du^n \wedge duv = nu^n du\wedge dv, dv^n \wedge duv = nv^n dv\wedge du, du^n \wedge dv^n = n^2(uv)^{n-1} du\wedge dv$$
Up to a coordinate transform (this is needed only when $n = 2$) and multiplying by a constant, we can write write $f_x = uv + f_{\geq 3}$, and $g_x = u^n + cv^n + uvg' + g_{\geq 2n}$ where $c \neq 0$. Then $d(uv) \wedge d(u^n + cv^n) = h_n du \wedge dv$, hence $h_n = n(u^n - cv^n)$. Replacing $f_x$ by $f_x - \mu g_x$ for some constant $\mu$, we can write $f_x = uv + c_1u^n + uvf' + f_{\geq 2n}$, $g_x = u^n + cv^n + uvg'' + g_{\geq 2n}$. Replacing $v$ by $v- c_1u^{n-1}$, we can write that
$$f_x = uv + uvf'' + f_{\geq 2n}, g = u^n + cv^n + uvg''' + g_{\geq 2n}$$
The local Albanese map $a: (\mathbb{C}^2,0) \rightarrow (\mathbb{C}^2, 0)$ defined by $df_x, dg_x$ coincides with the map given by $z \mapsto (f_x(z), g_x(z)) \in \mathbb{C}^2$. We conclude that
\begin{itemize}
\item
$a$ is of degree $2n$ by the formula above;
\item
it factors through the quotient map $(\mathbb{C}^2,0) \rightarrow (S, x)$;
\item
$a$ is ramified along $\sum_iD_i$.
\end{itemize}
Then it is easy to see that the local Albanese map $a_S$ given by path integral via $\alpha, \beta$ is a standard double cover branched along a curve with a negligible singulary at $0$.



\section{Some Remarks}


\subsection{About the assumption}

In Theorem \ref{main}, we assumed that no curves of type 0 pass through the singularities of $S$. To eliminate this assumption, we need to form a local inequality analogous to Inequality \ref{key3'}, the left hand side of which should involve a term contributed by the curves of type 0 passing through $x$. Due to lack of capability, the author failed in even proposing this inequality. This paper can be seen as one step to attain our goal.


\subsection{About Conjecture \ref{conjg}}
Remark \ref{br} says that Conjecture \ref{conjg} is true when the divisor $(df \wedge dg)$ is reduced. When proving Lemma \ref{key4}, we calculated part of left hand side of Inequality \ref{key3'} instead, hence got a stronger inequality in this special case. The advantage of Conjecture \ref{conjg} is that the left hand side of Inequality \ref{conj} is the intersection of 3 cycles; $h$ is determined by $f$ and $g$, and if $(df \wedge dg)$ is reduced, $h$ is also $G$-invariant; for explicit $f,g$, it is easier to check Inequality \ref{conj} than \ref{key3'}. On the other hand, the definition of $h$ is very unnatural. However, we checked a lot of examples which provide evidences for Inequality \ref{conj}, here we pose an easy example with non-reduced $(df \wedge dg)$.

\begin{Example}
Let all the notations be as in Conjecture \ref{conjg}. We assume that $G$ is a cyclic group of order $n$ with an action on $(\mathbb{C}^2, 0)$ such that $(\mathbb{C}^2, 0)/G$ has an $A_{n-1}$ singularity. Let
$$f = (u+v^{n-1})^n, g = uv$$
It follows that
$$df = n(u+ v^{n-1})^{n-1}d(u+ v^{n-1}) = n(u+ v^{n-1})^{n-1}(du + (n-1)v^{n-2}dv), dg = vdu + udv$$
and
$$df \wedge dg = n (u+ v^{n-1})^{n-1}(u - (n-1)v^{n-1})du \wedge dv$$
hence
$$h = (u+ v^{n-1})(u - (n-1)v^{n-1}), ~dh = (u+ v^{n-1})d(u - (n-1)v^{n-1}) + (u - (n-1)v^{n-1})d(u+ v^{n-1})$$
Note that $L_{df} = (n-1)(u+ v^{n-1}) + L_{d(u+ v^{n-1})}$ where $(u+ v^{n-1})$ denotes a divisor. We calculate that
\begin{equation}
\begin{split}
&L_{dh}L_{df}L_{dg} \\
= &(n-1)[(u+v^{n-1})\cdot L_{udv+vdu} \cdot (u - (n-1)v^{n-1}) + (u+v^{n-1})\cdot L_{udv+vdu} \cdot L_{d(u+ v^{n-1})}]\\
&+ L_{d(u+ v^{n-1})}\cdot L_{udv+vdu} \cdot (u+v^{n-1}) + L_{d(u+ v^{n-1})}\cdot L_{udv+vdu}\cdot L_{d(u-(n-1) v^{n-1})}\\
= &(n-1)[(n-1) + (1+(n-2))] + (1 + (n-2)) + (n-2) \\
= &n^2 - 1 + n(n-2)
\end{split}
\end{equation}
It is easy to check that our conjecture is true for this $f$ and $g$.
\end{Example}



