# Lefschetz Theorem
Using Real Morse Theory

Nima Rose Manjila  NM: Department of Mathematics, Indian Institute of Science
Education and Research, Dr. Homi Bhabha Road, Pune 411008, India.
<EMAIL_ADDRESS>and  A J Parameswaran  AJ: School of
Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai 400005,
India<EMAIL_ADDRESS>
###### Abstract.

Taking [2] as the primary reference, we prove some of the important theorems
solely using Real Morse theory and exact sequences. The Proof of Lefschetz
hyperplane section theorem is a deviation from the conventional one which uses
vanishing cycles, thimbles and monodromy. Instead, we prove using simpler
machinery. By making the observation that we can compose the Lefschetz Pencil
with a Real Morse function to get a map from the variety to $\mathbb{R}$ which
is ”close” to being a Real Morse function, we prove the genus formula for
plane curves and then we prove the Riemann Hurwitz formula for ramified maps
between curves by employing techniques from deformation theory. Lastly, we
prove the Lefschetz Hyperplane Section Theorem.

###### Key words and phrases:

Complex Projective variety, Riemann-Hurwitz Theorem, Homology group, Relative
homology, Pencil, Ehressmann’s fibration theorem, Lefschetz hyperplane section
theorem

###### 2010 Mathematics Subject Classification:

14Q05, 14C99, 37B30, 53A20, 55R10, 55U25, 57N65

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Genus Formula for Plane Curves
  4. 4 Riemann-Hurwitz Theorem
  5. 5 Lefschetz Hyperplane Section Theorem

##  1\. Introduction

A $C^{\infty}$ function $f:M\to\mathbb{R}$ is said to be a Morse function if
every critical point is non-degenerate and for every $u$ in $\mathbb{R}$,
$f^{-1}[-\infty,u]$ is compact. A critical point is non-degenerate iff the
Hessian matrix of $f$ at $c$ denoted as $H^{f}_{c}$ is non-singular. Let
$Crit\ f$ denote the set of critical points of $f$. The $index$ of a non-
degenerate critical point $c\in Crit\ f$ is the number of negative eigenvalues
of $H^{f}_{c}$. Let $Crt_{\lambda}f$ denote the set of critical points of $f$
of index $\lambda$. For a critical point $c$, in an arbitrary small
neighbourhood of $c$, it is well known that, for appropriate coordinates
$(u_{1},...,u_{n})$, $f$ is of the form
$f(u_{1},...,u_{n})=f(c)-\sum_{j=1}^{\lambda}u_{j}^{2}+\sum_{j=\lambda+1}^{n}u_{j}^{2}$.

For an arbitrary $r\in\mathbb{R}$, let

$M^{r}_{f}:=f^{-1}(-\infty,r]=\\{x\in M\mid f(x)\leq r\\}$

We heavily use the fact that the topology of
$M^{r}_{f}=f^{-1}(-\infty,r],-\infty<r<\infty$ changes only at critical
points. Where, at each critical point of index $\lambda$, we attach a cell of
dimension $\lambda$. The relation between the topology of $M$ and the Morse
function is the topic of [3]. We discussed the main results of Morse theory in
section 2.

Let $X\subset\mathbb{P}_{N}$ be a smooth algebraic subvariety in the complex
projective space $\mathbb{P_{N}}$. Let $A\subset\mathbb{P}_{N}$ be a
codimension two linear subspace ($\cong\mathbb{P}_{N-2}$) that intersect $X$
transversally. Then the pencil of hyperplane sections of $X$ with axis
$X^{\prime}$ is the map $P:Y\to\mathbb{P}_{1}$ where $Y$ is the blow up of $X$
along $X^{\prime}$ and $\mathbb{P}_{1}$ is the space of hyperplanes containing
$A$. A pencil is called Lefschetz pencil if every fibre is either smooth or
has at most one quadratic singularity.

Lamotke [2] proves Lefschetz hyperplane section theorem by using monodromy,
thimbles and vanishing cycles of Lefschetz pencil, whereas we do it by
composing the pencil with a Morse function on the complex projective line
$\mathbb{P}_{1}$.

The composite function will be Morse in the case of plane curves. However, we
prove that it is enough to study

$h:\mathbb{A}^{2}\to\mathbb{A}^{2}$ $(x,z)\to(x,F(x,z))$

restricted to $C$, where $\mathbb{A}^{2}$ is the affine plane. We study the
Jacobian of $h$, denoted as $J(h)$ and use Bézout’s theorem (that states: two
planes curves without common components will intersect in the product of
degrees number points) to prove Theorem 1.1 in Section 3.

######  Theorem 1.1 (Genus Formula).

Let $g(C)$ denote the topological genus of a curve $C$ of degree $d$. Then

$g(C)=\frac{1}{2}(d-1)(d-2)$

Now, Let $C$ and $B$ are both compact Riemann surfaces, and $f:C\to B$ be a
non-constant holomorphic map. Then $f$ is called a _ramified covering map_ of
$B$ and $C$ is the _ramified cover_. There exist a finite set of points of
$B$, outside of which we do find an honest covering. These exceptional points
on $C$ are called the _ramification points_ and its image under $f$ are called
_discriminant points_. An _unramified_ covering or simply, covering is then
the occurrence of an empty ramification locus. And $f$ is locally(around a
point $p$) given by $z\mapsto z^{n_{p}},n_{p}\geq 1$. If $n_{p}>1$, the point
is called a _ramified point_ with _ramification index_ $n_{p}$. We prove the
following theorem in Section 4.

######  Theorem 1.2 (Riemann Hurwitz Theorem).

Let $C$ and $B$ be two compact connected Riemann surfaces $f:C\to B$ be a non-
constant holomorphic map of degree $d$. The genus of $C$, denoted as $g(C)$ is
given by

$g(C)-1=d(g(B)-1)+\frac{1}{2}\sum_{p\in C}(n_{p}-1)$

where $g(B)$ is the genus of $B$ and $n_{p}$ is the ramification index of a
point $p$.

However, for a higher dimensional manifold, the composite function will have
very bad singularities at the preimages of critical points $c^{0}$ of index
$0$ and $c^{2}$ of index $2$ of $\mathbb{P}_{1}$. Studying the topology of
complex varieties involve three steps. In the first step, we find the index of
critical points of $Y$ coming from the Pencil. Followed by understanding the
topology of the blow-up variety $Y$. The final step is comparing the topology
of $Y$ with that of $X$.

Let $D_{c^{0}}$ be a closed real-dimension 2 disc with boundary around $c^{0}$
that does not contain any critical values of the Pencil and $D_{c^{2}}$ be a
closed real-dimension 2 disc with boundary around $c^{0}$ that does not
contain any critical values of the Pencil. Let $B_{0}$, $B_{2}$ be the
preimages of $D_{c^{0}}$, $D_{c^{2}}$ respectively. We use Morse theory to its
full strength to find the relative homology $H_{\lambda}(Y\char
92\relax\mathring{B_{2}},B_{0})$, where $\mathring{B_{2}}$ denotes the
interior of $B_{2}$. Further considering appropriate homology long exact
sequences, we prove the following theorem in Section 5.

######  Theorem 1.3 (Lefschetz Hyperplane Section Theorem).

The inclusion $X_{c^{0}}\hookrightarrow X$ induces isomorphisms of the
homology groups $H_{\lambda}(X_{c^{0}})\to H_{\lambda}(X)$ for every
$\lambda<n-1$ and a surjection for $\lambda=n-1$.

### Acknowledgement:

This work came out of first author’s Masters Thesis and is supported partly by
Infosys SHE scholarship. The authors would like to thank Amit Hogadi, Krishna
Kaipa and Steven Spallone for the encouragement and helpful comments.

##  2\. Preliminaries

In this section, we state results regarding Morse Theory and Lefschetz Pencil
necessary for proving Theorems 1.1, 1.2 and 1.3.

Stated below is the main theorem of Morse Theory that describes the role of
critical points in general. The proof of it follows from homotopy theory and
results about topological surgery. See [3] and [1].

######  Theorem 2.1.

Let $M^{r}_{f}:=f^{-1}(-\infty,r]=\\{x\in M\mid f(x)\leq r\\}$. Suppose there
exists $s$ in $Img\ f$ such that $s>r$ and $f^{-1}[r,s]$ is compact,

  1. (1)

if $f^{-1}[r,s]$ does not contain any critical point, then $M^{s}_{f}$ is
homotopic to $M^{r}_{f}$. Moreover, $M^{s}_{f}$ is a deformation retract of
$M^{r}_{f}$.

  2. (2)

If $f^{-1}[r,s]$ contains a critical point $c^{\lambda}_{i}$ and let
$f(c^{\lambda}_{i})=r^{\lambda}_{i}$ be the corresponding critical value then
$M^{s}_{f}$ is homotopic to $M^{r}_{f}$ attached with a $\lambda$\- cell.

  3. (3)

If $M^{r^{\lambda}_{i}}_{f}$ is compact $\forall c^{\lambda}_{i}\in Crit\ f$
then M has the homotopy type of a C-W complex with one cell of dimension
$\lambda$ attached at every critical point $c^{\lambda}_{i}$.

From Theorem 2.1, we know that each critical point of index $\lambda$
corresponds to a $\lambda$ cell in the C-W decomposition of $M$. Therefore we
make the natural definition,

$C_{\lambda}(M):=\langle Crit_{\lambda}f\rangle$

where $R$ is an arbitrary ring and $\langle Crit_{\lambda}f\rangle$ is the
free abelian group generated by the set $Crit_{\lambda}f$. We define
$\partial_{\lambda}$ with respect to the moduli space between the critical
points.

For all $c^{\lambda}_{i}\in Crit_{\lambda}f$ we have

$\partial_{\lambda}:C_{\lambda}\to C_{\lambda-1}$
$\partial_{\lambda}(c^{\lambda}_{i})=\sum_{c^{\lambda-1}_{j}\in\
Crit_{\lambda-1}f}\\#M_{f}(c^{\lambda}_{i},c^{\lambda-1}_{j})c^{\lambda-1}_{j}$

where $\\#M_{f}(c^{\lambda}_{i},c^{\lambda-1}_{j})$ is the signed number of
gradient flow lines between $c^{\lambda}_{i}$ and $c^{\lambda-1}_{j}$. The
Morse Theory complex is defined as

(2.1)
${0}$${C_{n}(f,R)}$${\cdots}$${C_{\lambda}(f,R)}$${\cdots}$${C_{0}(f,R)}$${0}$$\scriptstyle{0}$$\scriptstyle{\partial_{n}}$$\scriptstyle{\partial_{\lambda}}$$\scriptstyle{\partial_{0}}$

This complex is equivalent to the homology complex associated with the C-W
Complex because each index $\lambda$ critical point corresponds to a $\lambda$
cell.

######  Lemma 2.2.

On an $n$ dimensional manifold $M$, the Euler characteristic of $M$ denoted as
$e(M)$ is the alternating sum of the number of index $\lambda$\- cells for
each $0\leq\lambda\leq n$. [3]

######  Lemma 2.3.

Given a Morse function $f:M\to\mathbb{R}$, there exist a C-W Complex homotopic
to $M$, and the $\lambda$ cells are critical points of index $\lambda$.

For a proof see [3].

######  Proposition 2.4.

Let $M$ be a compact, connected and oriented manifold and $f:M\to\mathbb{R}$
be a Morse function. Let $p:X\to M$ be a finite connected topological cover,
then $f^{\prime}:X\to\mathbb{R}$ given by $f^{\prime}=f\circ p$ is a Morse
function and $e(X)=de(M).$

###### Proof.

A finite connected cover of a compact connected manifold of dimension $n$ is a
compact connected topological manifold of dimension $n$. Therfore, $X$ admits
a Morse decomposition. Since $p$ is a local isomorphism, critical points of
$f^{\prime}$ of index $\lambda$ are precisely $p^{-1}(Crit\ f)$. We
appropriately call $p^{-1}(c_{i}^{\lambda})$ as
$a^{\lambda}_{i_{1}},a^{\lambda}_{i_{2}},\cdots a^{\lambda}_{i_{d}}$. Since
there are $d$ copies of each cell, by Lemma 2.2, $e(X)=de(M)$. ∎

Let $\mathbb{P}_{N}$ denote $N$-dimensional complex projective space.

######  Definition 2.5.

Let $X\subset\mathbb{P}_{N}$ be a smooth irreducible variety. A _pencil_ in
$\mathbb{P}_{N}$ consists of all hyperplanes which contain a fixed
$(N-2)$-dimensional projective subspace $A$, which is called the _axis_ of the
pencil. This axis is uniquely determined by intersecting any two hyperplanes
of the Pencil.

######  Definition 2.6.

Let $X\subset\mathbb{P}_{N}$ be a closed irreducible variety of dimension $n$.
The points of tangential intersection of the pencil $P$ with $X$ are called
singular points of the pencil. Such a pencil that only admits finitely many
tangential intersections with $X$ and the singular points are quadratic is
called a _Lefschetz Pencil_.

Define

$X_{t}:=X\cap H_{t},\ t\in\mathbb{P}_{1}$

such that

$X=\cup_{t\in\mathbb{P}_{1}}X_{t}.$

Let

$X^{\prime}=X\cap A$

and

$Y=\\{(x,t)\in X\times\mathbb{P}_{1}\mid x\in H_{t}\\}.$

The map $p:Y\to X$ is called the blowup map and $Y$ is called the _blowup
variety_ or _modification_ of $X$ along $X^{\prime}$. Let

(2.2) $Y^{\prime}:=p^{-1}(X^{\prime})=X^{\prime}\times\mathbb{P}_{1}$

and $Y_{t}:=P^{-1}(t)$. Then $P\mid_{Y_{t}}:Y_{t}\to X_{t}$ is a biregular
map. Therefore $Y_{t}$ is isomorphic to $X_{t}$.

######  Definition 2.7.

The space of all hyperplanes of $\mathbb{P}_{N}$ which are tangent to $X$
denoted as $\check{X}\subsetneq\check{\mathbb{P}}_{N}$ is called the _dual
variety_ of $X$ and is a proper closed subvariety of $\check{P}^{N}$.

##  3\. Genus Formula for Plane Curves

Equipped with Morse theory, we study the topology of projective varieties, the
simplest being curves. In this section, we prove Theorem 1.1

Let $f$ be a degree $d$ irreducible polynomial in homogenous coordinates
$(x:y:z)$. We define $C:=V(f)$, the curve defined by $f$ and assume that $C$
is smooth.

######  Lemma 3.1.

The Axis of a Pencil in $\mathbb{P}_{2}$ is a point. We choose a point $p$ as
the axis such that $p\notin C$. Let $P:C\to\mathbb{P}_{1}$ be the Lefschetz
pencil whose axis $A$ is $p\notin C$. We choose coordinates $(x:y:z)$ such
that $p=(0:0:1)$. Let the projection from the Axis be given by

$\displaystyle\pi_{p}:\mathbb{P}_{2}\char 92\relax\\{p\\}$
$\displaystyle\to\mathbb{P}_{1}$ $\displaystyle(x:y:z)$
$\displaystyle\mapsto(x:y).$

  1. (1)

$\pi_{p}$ is well defined

  2. (2)

the fiber of $\pi_{p}$ is the same as the fiber of $P$. Therefore Lefschetz
Pencil on the Curve $C$ is exactly the projection map $\pi_{p}|_{C}$

###### Proof.

For every $(x:y:z)\neq(0:0:1)$, $\pi_{p}(x:y:z)=(x:y)\neq(0,0)$. Therefore
$\pi_{p}(x:y:z)=(x:y)\in\mathbb{P}_{1}$.

For a point $(x_{0}:y_{0})\in\mathbb{P}_{1}$ the fiber
$\pi_{p}^{-1}(x_{0}:y_{0})=(x_{0}:y_{0}:z)=(x_{0}:y_{0}:0)+z(0:0:1)$. ie, any
point on the subspace $(x_{0}:y_{0}:z)$ is given by
$\lambda_{1}(x_{0}:y_{0}:0)+\lambda_{2}z(0:0:1)$ where
$\lambda_{1},\lambda_{2}\in\mathbb{C}$. Therefore $(x_{0}:y_{0}:z)$ is the
line passing through $(0:0:1)$. Therefore $\pi_{p}$ corresponds to the
Lefschetz pencil whose axis is $(0:0:1)$. ∎

For any two points $c^{0}$ and $c^{2}$ on $\mathbb{P}_{1}$, by Lemma 2.3 and
Reeb’s Theorem [3][]Theorem 4.1], there exists a Morse function such that
$c^{0}$ is its index $0$ critical point, $c^{1}$ is its index $2$ critical
point and there are no index $1$ critical points.

######  Proposition 3.2.

Consider the map

$C\stackrel{{\scriptstyle\mathrm{{\pi_{p}|_{C}}}}}{{\longrightarrow}}{\mathbb{P}_{1}}\stackrel{{\scriptstyle\mathrm{\emph{g}}}}{{\longrightarrow}}{\mathbb{R}}$

where $g$ is a morse function on $\mathbb{P}_{1}$ with exactly $2$ critical
points ,one of index $0$ and other of index $2$ such that the index 0 and
index 2 critical points of $g$ are regular values of $\pi_{p}$. The composite
map $(g\circ\pi_{p})|_{C}$ is Morse.

###### Proof.

The curve $C$ is smooth, therefore locally isomorphic to 1-dimensional Disc.
Consider such a disc $D$ around a critical point and let $z=x+iy\in D$.
$\pi_{p}|_{D}:D\to D$ is given by

(3.1) $z\longrightarrow z^{2}$ (3.2) $x+iy\longrightarrow x^{2}-y^{2}+i2xy$

Since a Morse function $g$ on $\mathbb{P}_{1}$ with exactly 2 critical points
say $c^{0}$ and $c^{2}$, and the Jacobian has maximal rank on
$\mathbb{P}_{1}\char 92\relax\\{c^{0}\cup c^{2}\\}$ and therefore is of the
form

(3.3) $g(X,Y)=aX+bY+\ \text{higher order terms},\ a\neq 0\ or\ b\neq 0$

Since for a $z\in C$, $\partial(g\circ\pi_{p})(z)=\partial
g(\pi_{p}(z))\partial\pi_{p}(z)$,

(3.4) $Crit\ g\circ\pi_{p}=\pi_{p}^{-1}(Crit\ g)\cup Crit\ \pi_{p}$

Let us call the index $0$ and index $2$ critical points of $g$ as $c^{0}$ and
$c^{2}$ respectively. Since $\pi_{p}$ is a local isomorphism in the
neighborhood of a critical point of $g$, $\pi_{p}^{-1}(c^{0})$ are non-
degenerate, index $0$ critical points of $(g\circ\pi_{p})|_{C}$. Similarly,
$\pi_{p}^{-1}(c^{2})$ are non-degenerate, index $2$ critical points of
$(g\circ\pi_{p})|_{C}$. Therefore it is enough to prove that critical point of
$\pi_{p}$ are non-degenerate index 1 critical points of $(g\circ\pi_{p})$,
i.e, for every $z=x+iy\in Crit\ \pi_{p}$ the Hessian matrix of $\pi_{p}$ is
non-singular.

(3.5) $(g\circ\pi_{p})(z)=a(x^{2}-y^{2})+2bxy+\ higher\ order\ terms$
$H^{(g\circ\pi_{p})}(z)(0)=\begin{pmatrix}2a&2b\\\ 2b&-2a\\\ \end{pmatrix}$
$det(H^{(g\circ\pi_{p})}(x+iy)(0))=det\begin{pmatrix}2a&2b\\\ 2b&-2a\\\
\end{pmatrix}=-4a^{2}-4b^{2}$ $-4(a^{2}+b^{2})=0\Leftrightarrow a=0\ and\ b=0$

Since either $a\neq 0$ or $b\neq 0$,we have that the Hessian matrix of
$\pi_{p}$ is non-singular and critical points of $g\circ\pi_{p}$ are non-
degenerate. ∎

Consider $\mathbb{P}^{2}\char 92\relax\\{(0:0:1)\\}$. This space is defined by
$(x:y:z)$ such that $x\neq 0$ or $y\neq 0$. Take $y=1$ and

$(x:1:z)\cong(x:z)$

######  Remark.

Let $\pi_{x}:\mathbb{A}^{2}\to\mathbb{A}^{1}$ be the projection $(x:z)\to x$.
This is exactly $\pi_{p}|_{y=1}$. Therefore one can easily see that our
familiar coordinate projection is a special case of $\pi_{p}$.

We prove that critical points of $\pi_{p}$ are exactly index $1$ critical
points. This can be easily seen since $\pi_{p}$ is a local isomorphism and the
preimage of local maxima/minima of $g$ would be local maxima/minima of
$\pi_{p}\circ g$. Another proof is by explicitly calculating the eigenvalues.
The characteristic polynomial of $H^{g\circ\pi_{p}}(x+iy)$ is given by
$(X-2a)(X+2a)-(-2b)^{2}$.

$(X-2a)(X+2a)-(-2b)^{2}=X^{2}-4a^{2}-4b^{2}$ (3.6) $X=\pm\sqrt{4a^{2}+4b^{2}}$

Therefore the number negative eigenvalues of $H^{g\circ\pi_{p}}$ is $1$.

Define $F(x:z):=f(x:1:z)$. It is seen that degree of $F$ is equal to $d$.
Otherwise $y$ would be a factor of $f$ in which case $f$ is not irreducible.
Consider the map

$h:\mathbb{A}^{2}\to\mathbb{A}^{2}$ $(x,z)\to(x,F(x,z))$

######  Proposition 3.3.

The critical points of $\pi_{p}|_{C\cap{\mathbb{A}^{2}}}$ are exactly those of
$h$ when restricted to $F(x,z)=0$.

###### Proof.

$h|_{C}:(x,z)\longrightarrow(x,0)$ $(x_{0},z)\longleftarrow x_{0}$
$\pi_{p}|_{C\cap{\mathbb{A}^{2}}}:(x:1:z)|_{\mathbb{A}^{2}}\longrightarrow(x:1)$
$(x_{0},z)\longleftarrow x_{0}$

The fibers of both the maps are the same. ∎

######  Lemma 3.4.

Lefschetz pencil $P$ has exactly $d(d-1)$ critical points.

###### Proof.

By Proposition 3.3, it is enough to calculate the number of critical points of
$h|_{F(x:z)=0}$. The Jacobian of $h$ is give by

$J(h)=\begin{pmatrix}\partial x/\partial x&\partial F/\partial x\\\ \partial
x/\partial z&\partial F/\partial z\\\ \end{pmatrix}=\begin{pmatrix}1&\partial
F/\partial x\\\ 0&\partial F/\partial z\\\ \end{pmatrix}$
$det(J(h))=det\begin{pmatrix}1&\partial F/\partial x\\\ 0&\partial F/\partial
z\\\ \end{pmatrix}=0\Leftrightarrow\partial F/\partial z=0$

Therefore, the critical points of $h|_{C}$ is given by the intersection of the
two curves $f=0$ and $\partial F/\partial z=0$. Now by Bézout’s theorem, the
intersection has $d(d-1)$ points. ∎

######  Corollary 3.5.

The degree of the dual curve is the intersection number of the dual curve with
a Projective line. This translates to the number of lines in the Lefschetz
pencil that is tangent to $X$. That is, $Class\ C=degree\ of\ \check{C}$ is
equal to $d(d-1)$.

###### Proof of Theorem 1.1.

It is easily seen that the number of index 0 critical pints is the cardinality
of $\pi_{p}^{-1}(0)$ is equal to the cardinality of $\mathbb{P}_{1}\cap C$ is
equal to $d$. Similarly, for index $2$ critical points. Index one critical
points come from $\pi_{p}$ and by the previous proposition, there is exactly
$d(d-1)$ critical points of index $1$.

Therefore, the Homology complex is given by

${0}$${C_{2}}$${C_{1}}$${C_{0}}$${0}$${\mathbb{Z}^{d}}$${\mathbb{Z}^{d(d-1)}}$${\mathbb{Z}^{d}}$$\scriptstyle{0}$$\scriptstyle{\partial_{2}}$

$\scriptstyle\cong$

$\scriptstyle{\partial_{1}}$

$\scriptstyle\cong$

$\scriptstyle{\partial_{0}}$

$\scriptstyle\cong$

$\scriptstyle{\partial_{n}}$$\scriptstyle{\partial_{n}}$$\scriptstyle{\partial_{0}}$

Taking as a matter of fact, that for a connected, compact orientable surface,
both $H_{0}$ and $H_{2}$ are isomorphic to $\mathbb{Z}$ the following
calculation,

$H_{0}\cong\frac{Ker\ \partial_{0}}{Im\
\partial_{1}}\cong\mathbb{Z}\implies\frac{\mathbb{Z}^{d}}{Im\
\partial_{1}}\cong\mathbb{Z}\implies Im\
\partial_{1}\cong\mathbb{Z}^{d-1}\implies Ker\
\partial_{1}\cong\mathbb{Z}^{d(d-1)-(d-1)}$

$H_{2}\cong Ker\ \partial_{2}\cong\mathbb{Z}\implies Im\
\partial_{2}\cong\mathbb{Z}^{d-1}$ $H_{1}\cong\frac{Ker\ \partial_{1}}{Im\
\partial_{2}}\cong{\mathbb{Z}^{d(d-1)-2(d-1)}}\cong{\mathbb{Z}^{(d-2)(d-1)}}$

yields the result , genus of $C$ is given by $g(C)=\frac{1}{2}(d-1)(d-2)$.

∎

##  4\. Riemann-Hurwitz Theorem

In this section, we prove Theorem 1.2.

The Riemann-Hurwitz Theorem, named after Bernhard Riemann and Adolf Hurwitz,
describes the relationship of the genera of two compact Riemann surfaces when
one is a ramified covering of the other. We use deformation Theory explicitly
and construct a deformation function with our required properties to fit into
the setup of Morse theory.

######  Definition 4.1.

For a map $f$, where $p$ is any point in $C$, one can choose an isomorphism
$\phi_{C}$ from the open unit disc $D$ to the neighborhood of $p$ and another
isomophism $\phi_{B}$ from $D$ to a neighborhood of $f(p)$ and by open mapping
theorem $\phi_{B}^{-1}\circ f\circ\phi_{C}\mid_{D}:D\to D$ is given by
$z\mapsto z^{n_{p}},n_{p}\geq 1$, where $n_{p}$ is the order of vanishing of
$f$ at $p$ for $z$ in a neighborhood of $p$. If $n_{p}=1$, the point is called
an _unramified point_ of $f$. If $n_{p}>1$, the point is called a _ramified
point_ with _ramification index_ $n_{p}$. The _degree_ of $f$ is the
cardinality of any regular fibre.

With abuse of notation, we write $f:D\to D$ such that the local coordinates
are chosen to satisfy $p=f(p)=0$.

An equivalent way of thinking about ramified point is that $p$ is said to be
ramified with respect to $f$ if there exists a small neighborhood $U$ of $p$
such that $f(p)$ has exactly one preimage in $U$, but the image of any other
point in $U$ has exactly $n_{p}$ preimages in $U$.

###### Proof of 1.2.

We first prove case where $f$ has only non-degenerate critical points, i.e,
$f:D\to D$ is given by $z\mapsto z^{n_{p}}$ where $n_{p}\leq 2$ for all $p\in
C$.

Let $g$ be a Morse function on $B$ such that critical points of $g$ are
regular values of $f$. Consider the composite map

$C\stackrel{{\scriptstyle\mathrm{{f}}}}{{\longrightarrow}}{B}\stackrel{{\scriptstyle\mathrm{\emph{g}}}}{{\longrightarrow}}{\mathbb{R}}.$

Let $l_{\lambda}^{B}$, $l_{\lambda}^{C}$ denote the number of index
$\lambda$-critical points in $B$ and $C$ respectively, and $\mu$ the number of
critical points of $f$ then from (3.4) and (3.6), we have that

(4.1) $\begin{split}l_{0}^{C}&=d\times l_{0}^{B}\\\ l_{2}^{C}&=d\times
l_{2}^{B}\\\ l_{1}^{C}&=d\times l_{1}^{B}+\mu\end{split}$

Substitituting (4.1) in $e(C)=l_{0}^{C}-l_{1}^{C}+l_{2}^{C}$,

$\begin{split}e(C)&=(d\times l_{0}^{B})-(d\times l_{1}^{B}+\mu)+(d\times
l_{2}^{B})\\\ &=d(l_{0}^{B}-l_{1}^{B}+l_{2}^{B})-\mu\end{split}$

That is,

$e(C)=de(B)-\mu$

Let $g$ denote the genus of the curve. Substituting $e=2-2g$, we get

(4.2) $g(C)-1=d(g(B)-1)+\frac{1}{2}\mu$

We can quickly verify our genus formula for smooth plane curves of degree $d$
as in Section 3 by replacing $B$ in (4.2) with $\mathbb{P}_{1}$. It follows
from Lemma 3.4 that $\mu=d(d-1)$, where $d$ is the degree of the curve.
Therefore,

(4.3) $\begin{split}g(C)-1&=d(g(B)-1)+\frac{1}{2}d(d-1)\\\
g(C)&=\frac{1}{2}(d^{2}-3d+2)\\\ &=\frac{1}{2}(d-1)(d-2)\end{split}$

as desired.

Now, we look at the case where $C$ has degenerate critical points, i.e,
$f:D\to D$ is given by $z\mapsto z^{n_{p}}$, where $n_{p}>2$. Inorder to make
the critical points non-degenerate, we deform the function locally by a small
perturbation in the unit disc $D$ centered at $p$.

$F:D_{\frac{1}{2}}\times D\to D\times D$

given by

$F(z,t):=(f_{t}(z),t)$

where $D_{\frac{1}{2}}$ is a disc of radius $\frac{1}{2}$ contained in $D$ and
$f_{t}$ is locally given by

$f_{t}:D_{\frac{1}{2}}\to D$ (4.4) $f_{t}(z):=z^{n_{p}}-tz.$
$J^{F}(z,t)=\begin{pmatrix}\partial f_{t}/\partial z&\partial t/\partial z\\\
\partial f_{t}/\partial t&\partial t/\partial t\\\
\end{pmatrix}=\begin{pmatrix}n_{p}z^{n_{p}-1}-t&0\\\ -z&1\\\ \end{pmatrix}$
$det(J^{F_{p}})=n_{p}z^{n_{p}-1}-t=0\Rightarrow
z=\sqrt[{n_{p}-1}]{\tfrac{t}{n_{p}}}$

Therefore, for a fixed $\epsilon<<\frac{1}{2}$ we choose a $t$ such that $\mid
t\mid<n_{p}\epsilon^{n_{p}-1}$. Then $f_{t}(z):D_{\frac{1}{2}}\to D$ has
$n_{p}-1$ distinct critical points contained in $D_{\epsilon}$. And the
Hessian of $f_{t}$ given by

$\begin{split}H^{f_{t}}(z)&=\frac{\partial^{2}f_{t}}{\partial z^{2}}\\\
&=n_{p}(n_{p}-1)z^{n_{p}-2}\\\ &=0\Leftrightarrow z=0\end{split}$

But $z=0$ is not a critical point of $f_{t}$. Therefore, all the $n_{p}-1$
critical points of $f_{p}$ are non-degenerate and clearly contained in
$D_{\epsilon}$.

For the triple $D$, $D_{\epsilon}$, $C\char
92\relax\overline{D_{\frac{1}{2}}}$, by partitions of unity theorem, there
exists a $\mathscr{C}^{\infty}$ function $\phi:C\to[0,1]$ such that $\phi$ is
identically $1$ on $D_{\epsilon}$, $0$ on $C\char
92\relax\overline{D_{\frac{1}{2}}}$ and $0\leq\phi\leq 1$ on
$\overline{D_{\frac{1}{2}}\char 92\relax D_{\epsilon}}$, the closure of
$D_{\frac{1}{2}}\char 92\relax D_{\epsilon}$. Here the closure of
$D_{\frac{1}{2}}$, $\overline{D_{\frac{1}{2}}}$ is the compact support of
$\phi$. Recall that $f-f_{t}=tz$ from (4.4).

We define

(4.5) $\begin{split}f_{1}&=(1-\phi)f+\phi f_{t}\\\ &=f-\phi(f-f_{t})\\\
&=f-tz\phi\end{split}$

Since $f_{1}=f$ on $C\char 92\relax\overline{D_{\frac{1}{2}}}$. Therefore the
critical points of $f_{1}$ are the same as the critical points of $f$ on
$C\char 92\relax\overline{D_{\frac{1}{2}}}$. Again, $f_{1}=f_{t}$ on
$D_{\epsilon}$. Therefore, the critical points of $f_{1}$ are the same as the
critical points of $f_{t}$ on $D_{\epsilon}$.

By applying the following lemma to $\psi=z\phi$, gives us that $f_{1}$ does
not have any critical point in the region $\overline{D_{\frac{1}{2}}\char
92\relax D_{\epsilon}}$.

######  Lemma 4.2.

Suppose $f:D\to\mathbb{C}$ be differentiable map that is a local isomorphism
(i.e, $f$ has maximal rank everywhere) on $\overline{D_{\frac{1}{2}}\char
92\relax D_{\epsilon}}$. Let $\psi$ be a continuously differentiable function
on $D$, then $f+t\psi$ is also a local isomorphism on
$\overline{D_{\frac{1}{2}}\char 92\relax D_{\epsilon}}$ for sufficiently small
$t>0$.

###### Proof.

Consider the map

$\Phi:\overline{D_{\frac{1}{2}}\char 92\relax D_{\epsilon}}\times[0,1]\to
M_{2}(\mathbb{R})$ $(z,t)\mapsto J(f+t\psi).$

Then $\Phi$ is continuous. As $GL_{2}(\mathbb{R})$ is open in
$M_{2}(\mathbb{R})$, $\Phi^{-1}(GL_{2}(\mathbb{R}))$ is open in
$\overline{D_{\frac{1}{2}}\char 92\relax D_{\epsilon}}\times[0,1]$ in the
product topology.

Since $J(f)$ has maximal rank, $\overline{D_{\frac{1}{2}}\char 92\relax
D_{\epsilon}}\times\\{0\\}\in\Phi^{-1}(GL_{2}(\mathbb{R}))$. Now, as
$\overline{D_{\frac{1}{2}}\char 92\relax D_{\epsilon}}$ is compact, there
exists an open set $\overline{D_{\frac{1}{2}}\char 92\relax
D_{\epsilon}}\times[0,t_{0})\supset\overline{D_{\frac{1}{2}}\char 92\relax
D_{\epsilon}}\times\\{0\\}$ in $\Phi^{-1}(GL_{2}(\mathbb{R}))$. Therefore, for
every $t<t_{0}$, $J(f+t\psi)$ has maximal rank.∎

This leads us to the following theorem.

######  Theorem 4.3.

Let $C$ and $B$ be compact connected Riemann surfaces. Let $f:C\to B$ be a
$\mathscr{C}^{\infty}$ function that is analytic at every critical point.
Assume that $f$ has $n$ degenerate critical points and $n_{1}$ non-degenerate
critical points. Let $p$ be a non degenerate critical point of $f$. Then,
there exists a $\mathscr{C}^{\infty}$ function $f_{1}:C\to B$ which has
$n_{p}-1$ analytic non-degenerate critical points in a small neighborhood of
$p$ and coincides with $f$ outside the neighborhood. Furthermore, there exists
a $\mathscr{C}^{\infty}$ function $f_{n}:C\to B$ which has $0$ degenerate
critical points and $n_{1}+\sum_{p\in C}n_{p}-1$ analytic non-degenerate
critical points.

###### Proof.

The first part of the proof follows from our explicit construction of the
$\mathscr{C}^{\infty}$ function $f_{1}$ in (4.5) and subsequent verifications.
Now we apply this to $f_{1}$. Suppose $q$ is a degenerate critical point of
$f_{1}$ then, there exists a $\mathscr{C}^{\infty}$ function $f_{2}$ which has
$n-2$ degenerate critical points and $n_{1}+(n_{p}-1)+(n_{q}-1)$ non
degenerate critical points. Inductively construct $f_{3},\cdots,f_{n}$. Note
that $f_{n}$ has 0 degenerate critical points and

$n_{1}+\sum_{p\ \text{degenerate}}(n_{p}-1)$

non degenerate index $1$ critical points. Since $n_{p}=2$ for a non degenerate
critical point,

(4.6) $\sum_{p\ \text{non-degenerate}}(n_{p}-1)=\sum_{p\ \text{non-
degenerate}}1=n_{1}.$

Also note that for a regular point $n_{p}=1$ and $\sum_{p\
\text{regular}}(n_{p}-1)=0$.Therefore, $f_{n}$ has

$n_{1}+\sum_{p\ \text{degenerate}}(n_{p}-1)=\sum_{p\in C}(n_{p}-1)$

non-degenerate critical points. ∎

__Proof of Proposition ( 1.2) continued. Let $D$ be the neighborhood of a
critical point of $f_{n}.$ The function $f_{n}\mid_{D}:D\to D$ is given by
$z\mapsto z^{2}$. Choose a Morse function $g:B\to\mathbb{R}$ such that
critical points of $g$ are regular values of $f_{n}$. The composite map
$g\circ f_{n}\mid_{D}:D\to\mathbb{R}$ is holomorphic and has non degenerate
critical points. From (3.6) the non degenerate critical points of $f_{n}$ have
index 1. If $l_{\lambda}^{B}$, $l_{\lambda}^{C}$ denote the number of index
$\lambda$-critical points in $B$ and $C$ of $f$ and $g\circ f$ respectively,
then from (3.4),

(4.7) $\begin{split}l_{0}^{C}&=d\times l_{0}^{B}\\\ l_{2}^{C}&=d\times
l_{2}^{B}\\\ l_{1}^{C}&=d\times l_{1}^{B}+\sum_{p}(n_{p}-1)\end{split}$

Substituting (4.7) to $e(C)=l_{0}^{C}-l_{1}^{C}+l_{2}^{C}$,

$\begin{split}e(C)&=(d\times l_{0}^{B})-(d\times
l_{1}^{B}+\sum_{p}(n_{p}-1))+(d\times l_{2}^{B})\\\
&=d(l_{0}^{B}-l_{1}^{B}+l_{2}^{B})-\sum_{p}(n_{p}-1)\end{split}$

That is,

(4.8) $e(C)=de(B)-\sum_{p}(n_{p}-1)$

Let $g$ denote the genus of the curve. Substituting $e=2-2g$ in (4.2), we get

(4.9) $g(C)-1=d(g(B)-1)+\frac{1}{2}\sum_{p\in C}(n_{p}-1)$

∎

##  5\. Lefschetz Hyperplane Section Theorem

In the previous two sections, we completed the study of pencils in dimension 1
and in fact, generalized to any map between curves. Here we study the topology
of complex projective varieties, give an illustration of the maps and prove
Theorem 1.3

###  5.1. Non Degeneracy of Critical Points of the Pencil

Let $X$ be a smooth variety of complex dimension $n>1$ and $Y$ it’s blow up
along the smooth axis $X^{\prime}$. Let $g$ be a morse function on
$\mathbb{P}_{1}$ with exactly 1 index-$0$ and 1 index-$2$ critical points
which are regular values of the Lefschetz pencil $P$ from $Y$ to
$\mathbb{P}_{1}$. From (3.3), $g$ is given by $g(x,y)=ax+by+\ \text{higher
order terms}$, where $a\neq 0\ or\ b\neq 0$. For a sufficiently small
neighborhood $D$ of a critical point $c$,

$P\mid_{D}:D\to D$

is given by

$(z_{1},z_{2},...z_{n})\mapsto\sum_{i=1}^{n}z_{i}^{2}$
$(x_{1},iy_{1},\cdots,x_{n},iy_{n})\mapsto\sum_{i=1}^{n}x_{i}^{2}-y_{i}^{2}+i2x_{i}y_{i}=(\sum_{i=1}^{n}x_{i}^{2}-y_{i}^{2},\sum_{i=1}^{n}2x_{i}y_{i})$

Consider the composite map

$Y\stackrel{{\scriptstyle\mathrm{{P}}}}{{\longrightarrow}}{\mathbb{P}_{1}}\stackrel{{\scriptstyle\mathrm{\emph{g}}}}{{\longrightarrow}}{\mathbb{R}}$

given by

$(g\circ
P)\mid_{D}(x_{1}+iy_{1},\cdots,x_{n}+iy_{n})=g(\sum_{i=1}^{n}x_{i}^{2}-y_{i}^{2},\sum_{i=1}^{n}2x_{i}y_{i})=a\sum_{i=1}^{n}(x_{1}^{2}-y_{1}^{2})+2b\sum_{i=1}^{n}(x_{1}y_{1})+O(x,y).$

Now, the Hessian matrix of $g\circ P\mid_{D}$ at a critical point is given by

$\displaystyle\begin{split}H^{(g\circ
P)\mid_{D}}(0)&=\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}\frac{\partial^{2}(g\circ
P)}{\partial x_{1}^{2}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial
x_{1}x_{n}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial^{2}(g\circ P)}{\partial
x_{n}x_{1}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial
x_{n}^{2}}\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}\frac{\partial^{2}(g\circ
P)}{\partial x_{1}\partial y_{1}}&\cdots&\frac{\partial^{2}(g\circ
P)}{\partial x_{1}\partial y_{n}}\\\ \vdots&\ddots&\vdots\\\
\frac{\partial^{2}(g\circ P)}{\partial
x_{n}y_{1}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial x_{n}\partial
y_{n}}\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}\frac{\partial^{2}(g\circ
P)}{\partial x_{1}\partial y_{1}}&\cdots&\frac{\partial^{2}(g\circ
P)}{\partial x_{1}\partial y_{n}}\\\ \vdots&\ddots&\vdots\\\
\frac{\partial^{2}(g\circ P)}{\partial
x_{n}y_{1}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial x_{n}\partial
y_{n}}\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}\frac{\partial^{2}(g\circ
P)}{\partial y_{1}^{2}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial
y_{1}x_{n}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial^{2}(g\circ P)}{\partial
y_{n}x_{1}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial y_{n}\partial
x_{n}}\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}\frac{\partial^{2}(g\circ
P)}{\partial y_{1}^{2}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial
y_{1}y_{n}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial^{2}(g\circ P)}{\partial
y_{n}y_{1}}&\cdots&\frac{\partial^{2}(g\circ P)}{\partial
y_{n}^{2}}\end{matrix}$}\end{array}\right]\\\
&=\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}+2a&&0\\\
&\ddots&\\\
0&&+2a\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+2b&&0\\\
&\ddots&\\\ 0&&+2b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+2b&&0\\\
&\ddots&\\\
0&&+2b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}+2b&&0\\\
&\ddots&\\\
0&&+2b\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}-2a&&0\\\
&\ddots&\\\ 0&&-2a\end{matrix}$}\end{array}\right]\end{split}$

where each block is an $n\times n$ diagonal matrix with 0 in every $ij$-th
entry for $i\neq j$.

(5.1) $H^{(g\circ
P)\mid_{D}}(0)=2\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}+a&&0\\\
&\ddots&\\\
0&&+a\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\ 0&&+b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}-a&&0\\\
&\ddots&\\\ 0&&-a\end{matrix}$}\end{array}\right]$

Let $a\neq 0$. We reduce the matrix to row echelon form using the the row
transformation

$R_{n+1}\mapsto R_{n+1}-\frac{b}{a}R_{1}.$
$\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}+a&&0\\\
&\ddots&\\\
0&&+a\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\ 0&&+b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}-a&&0\\\
&\ddots&\\\
0&&-a\end{matrix}$}\end{array}\right]\mapsto\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}+a&&0\\\
&\ddots&\\\
0&&+a\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\ 0&&+b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}0&&0\\\
&\ddots&\\\
0&&0\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}\frac{-(a^{2}+b^{2})}{a}&&0\\\
&\ddots&\\\ 0&&\frac{-(a^{2}+b^{2})}{a}\end{matrix}$}\end{array}\right]$

and the determinant of $H^{(g\circ P)\mid_{D}}$ at a critical point, given by

(5.2)
$\displaystyle\begin{split}Det\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}+a&&0\\\
&\ddots&\\\
0&&+a\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\ 0&&+b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}-a&&0\\\
&\ddots&\\\
0&&-a\end{matrix}$}\end{array}\right]&=Det\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}+a&&0\\\
&\ddots&\\\
0&&+a\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\ 0&&+b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}0&&0\\\
&\ddots&\\\
0&&0\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}\frac{-(a^{2}+b^{2})}{a}&&0\\\
&\ddots&\\\ 0&&\frac{-(a^{2}+b^{2})}{a}\end{matrix}$}\end{array}\right]\\\
&=(-1)^{n}\frac{(a^{2}+b^{2})^{n}}{a^{n}}a^{n}\\\
&=(-1)^{n}(a^{2}+b^{2})^{n}\\\ &\neq 0\end{split}$

Therefore, the critical points of $g\circ P$, which are the critical points of
the pencil $P$ are nondegenerate.

The characteristic polynomial of $H^{(g\circ P)\mid_{D}}$ at a critical point
is given by

(5.3)
$\displaystyle\begin{split}Det\left[\begin{array}[]{c|c}\leavevmode\hbox{\set@color$\begin{matrix}a-x&&0\\\
&\ddots&\\\
0&&a-x\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\ 0&&+b\end{matrix}$}\\\
\hline\cr\vphantom{\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}}\leavevmode\hbox{\set@color$\begin{matrix}+b&&0\\\
&\ddots&\\\
0&&+b\end{matrix}$}&\leavevmode\hbox{\set@color$\begin{matrix}-a-x&&0\\\
&\ddots&\\\
0&&-a-x\end{matrix}$}\end{array}\right]&=(x^{2}-(a^{2}+b^{2}))^{n}\end{split}$

Therefore the eigenvalues of $H^{(g\circ P)\mid_{D}(0)}$ are
$\sqrt{a^{2}+b^{2}}$ and $-\sqrt{a^{2}+b^{2}}$, with algebraic multiplicity
$n$ each. That is, all the critical points of $g\circ P$ arising from the
critical points of $P$ have index $n$.

However, the preimages of the index 0 and index $2$ critical points $c^{0}$
and $c^{2}$ of $g$, are hyperplane sections each. Hence the function $g\circ
P$ has non-isolated critical points on $Y$.

###  5.2. Lefschetz Hyperplane Section Theorem

Let $X$ be an n-dimensional smooth variety and $Y$ it’s blowup along
$X^{\prime}$. Let $P:Y\to\mathbb{P}^{1}$ be the Lefschetz Pencil. Let
$D_{c^{0}}$ be a closed real-dimension 2 disc with boundary around $c^{0}$
that does not contain any critical values of the Pencil, explicitly, if
$y_{1},y_{2}\cdots y_{\mu}$ are all the critical points of $P$, then choose an
$\epsilon<(g\circ P)(y_{i})$, for all $i\in\\{1,2,\cdots,\mu\\}$ and
$D_{c^{0}}$ is defined as $g^{-1}(-\infty,\epsilon]$. For a point
$p\in\mathbb{P}^{1}$, the fiber of $P$ at $p$ is defined as $Y_{p}=P^{-1}(p)$.
Let $B_{0}$ be the preimage of $D_{c^{0}}$. Since $D_{c^{0}}$ is closed and
does not contain any critical values of the Pencil, we have that preimage
$P^{-1}(D_{c^{0}})$ is closed and does not contain any critical points of the
Pencil.

Similarly there exists a closed real-dimension 2 disc,
$D_{c^{2}}=g^{-1}[1-\epsilon,\infty)$ containing $c^{2}$ which does not
contain any critical points of the pencil $P$ and let
$B_{2}=P^{-1}({D}_{c^{2}})$.

FIGURE 1. is an illustration of the map
$Y\stackrel{{\scriptstyle\mathrm{{P}}}}{{\longrightarrow}}{\mathbb{P}_{1}}\stackrel{{\scriptstyle\mathrm{\emph{g}}}}{{\longrightarrow}}{\mathbb{R}}$

Figure 1.

######  Lemma 5.1.

Let $B_{0}$, $B_{2}$ be the preimages of $D_{c^{0}}$, $D_{c^{2}}$
respectively. Let $\partial B_{0}$, $\partial B_{2}$ denote the boundary of
$B_{0}$ and $B_{2}$, then

$B_{0}\cong P^{-1}(c^{0})\times D_{c^{0}}\cong P^{-1}(c^{2})\times
D_{c^{2}}\cong B_{2}$

and

$\partial B_{0}\cong P^{-1}(c^{0})\times S^{1}\cong P^{-1}(c^{2})\times
S^{1}\cong\partial B_{2},$

###### Proof.

Since $D_{c^{0}}$ does not contain any critical points of the Pencil, by
Ehresmann’s fibration Theorem, $P$ fibers $B_{0}$ locally trivially.
Therefore,

$B_{0}\cong P^{-1}(D_{c^{0}})\cong P^{-1}({c^{0}})\times D_{c^{0}}$

Let $b_{1}\in\partial D_{c^{0}}$. By Ehresmann’s fibration Theorem,

$P^{-1}(\partial D_{c^{0}})\cong P^{-1}(b_{1})\times S^{1}.$

Therefore,

$\partial(B_{0})\cong\partial(P^{-1}(D_{c^{0}}))\cong(P^{-1}(\partial
D_{c^{0}}))\cong P^{-1}(b_{1})\times S^{1}.$

Since there does not exist any critical points of the Pencil in $D_{c^{0}}$,
$P^{-1}(b_{1})\cong P^{-1}({c^{0}})$ and

$\partial(B_{0})\cong P^{-1}({c^{0}})\times S^{1}.$

Similarly for $B_{2}$ and $\partial B_{2}$. ∎

Since $B_{0}$ does not contain any critical points of $P$, $B_{0}$ is
homotopic to $Y_{c^{0}}$, similarly $Y_{c^{2}}$ is homotopic to $B_{2}$.

######  Theorem 5.2.

Let $H_{\lambda}(Y\char 92\relax\mathring{B_{2}},B_{0})$ denote the homology
of $Y\char 92\relax\mathring{B_{2}}$ relative to $B_{0}$ and
$H_{\lambda}(Y\char 92\relax\mathring{B_{0}},B_{2})$ denote the homology of
$Y\char 92\relax\mathring{B_{0}}$ relative to $B_{2}$, then

$H_{\lambda}(Y\char 92\relax\mathring{B_{2}},B_{0})\cong H_{\lambda}(Y\char
92\relax\mathring{B_{0}},B_{2})=\begin{cases}\mathbb{Z}^{\mu}&\quad\lambda=n\\\
0&\quad\text{otherwise}\\\ \end{cases}$

where $\mu$ is the number of critical points of the Pencil.

###### Proof.

Since $Y\char 92\relax\mathring{B_{2}}$ is the manifold $Y_{g\circ
P}^{1-\epsilon}=(g\circ P)^{-1}(-\infty,1-\epsilon]$ and $B_{0}$ is the
manifold $Y_{g\circ P}^{\epsilon}=(g\circ P)^{-1}(-\infty,\epsilon]$, from
2.1, $Y\char 92\relax\mathring{B_{2}}$ is homotopic to $B_{0}$ with $\mu$
number of n-cells attached. Therefore,

$H_{\lambda}(Y\char
92\relax\mathring{B_{2}},B_{0})=\begin{cases}\mathbb{Z}^{\mu}&\quad\lambda=n\\\
0&\quad\text{otherwise}\\\ \end{cases}$

Similarly, $Y\char 92\relax\mathring{B_{0}}$ is the manifold $Y_{-g\circ
P}^{1-\epsilon}=(-g\circ P)^{-1}(-\infty,1-\epsilon]$ and $B_{2}$ is the
manifold $Y_{-g\circ P}^{\epsilon}=(-g\circ P)^{-1}(-\infty,\epsilon]$ and
from Morse Theory [3], $Y\char 92\relax\mathring{B_{0}}$ is homotopic to
$B_{2}$ with $\mu$ number of n-cells attached. Therefore,

$H_{\lambda}(Y\char
92\relax\mathring{B_{0}},B_{2})=\begin{cases}\mathbb{Z}^{\mu}&\quad\lambda=n\\\
0&\quad\text{otherwise}\\\ \end{cases}$

∎

######  Theorem 5.3.

Recall that $Y$ is the blow-up variety of $X$ of dimension $n$ and
$B_{0}=P^{-1}(D_{c^{0}})$. Then, $H_{\lambda+1}(Y,B_{0})\cong
H_{\lambda-1}(B_{0})$, for $\lambda\leq n$.

###### Proof.

Consider the homology long exact sequence of the triple given by $B_{0}\subset
Y\char 92\relax\mathring{B}\subset Y$,

(5.4) ${\cdots}$${{H_{\lambda}(Y\char
92\relax\mathring{B_{2}},B_{0})}}$${{H_{\lambda}(Y,B_{0})}}$${{H_{\lambda}(Y,Y\char
92\relax\mathring{B_{2}})}}$${{H_{\lambda-1}(Y\char
92\relax\mathring{B_{2}},B_{0})}}$${\cdots}$

We have that

(5.5) $\begin{split}H_{\lambda}(Y,Y\char 92\relax\mathring{B_{2}})&\cong
H_{\lambda}(B_{0},\partial B_{0})\cong H_{\lambda}(X_{c^{0}}\times
D_{1},X_{c^{0}}\times S^{1})\\\ &\cong
H_{\lambda}(X_{c^{0}}\times(D_{1},S^{1}))\\\
&\cong\bigoplus\limits_{i}H_{i}(X_{c^{0}})\otimes
H_{\lambda-i}(D_{1},S^{1})\\\ &\cong H_{\lambda-2}(X_{c^{0}})\\\ &\cong
H_{\lambda-2}(B_{0}).\end{split}$

The first isomorphism is through excision property. Since $B_{0}$ do not have
any critical points of the pencil, Ehresmann’s fibration theorem gives us the
second isomorphism An application of the Künneth formula gives the fourth
isomorphism and the final isomorphism follows from the following fact that

(5.6) $H_{q}(D_{1},S^{1})=\begin{cases}R&\quad q=2\\\
0&\quad\text{otherwise}\\\ \end{cases}$

Substituting $(\ref{eq:7})$ and Theorem 5.2 in (5.4), we have for $\lambda<n$

${{0}}$${{H_{\lambda}(Y,B_{0})}}$${{H_{\lambda-2}(B_{0})}}$${{0}}$

Therefore,

(5.7) $H_{\lambda}(Y,B_{0})\cong H_{\lambda-2}(B_{0})$

∎

######  Theorem 5.4.

Let $\nu$ be a continuous mapping between pairs of compact Euclidean
neighborhood retracts, such that $\nu:X\char 92\relax X_{1}\to Y\char 92\relax
Y_{1}$ is a homeomorphism, then $\nu$ induces an isomorphism
$\nu_{\lambda}:H_{\lambda}(X,X_{1})\to H_{\lambda}(Y,Y_{1})$.

###### Proof.

This can be proved using the Neighborhood theorem and excision property.
Homology commutes with direct limit. ∎

Since we deal with manifolds, ENR property follows. From what we have about
the homology of the modification, we are good to start finding the homology of
the variety $X$ itself.

###### Proof of 1.3.

From the long exact sequence of the pair $X_{c^{0}},X$ given by

${\cdots}$${{H_{\lambda+1}(X,X_{c^{0}})}}$${{H_{\lambda}(X_{c^{0}})}}$${{H_{\lambda}(X)}}$${{H_{\lambda}(X,X_{c^{0}})}}$${\cdots}$

we get that the statement of the theorem is equivalent to saying that
$H_{\lambda}(X,X_{c^{0}})=0$ for $\lambda\leq n-1.$

Consider the homology long exact sequence of the triple $B_{0}\subset
Y^{\prime}\cup B_{0}\subset Y$,

(5.8) ${\cdots}$${{H_{\lambda}(Y^{\prime}\cup
B_{0},B_{0})}}$${{H_{\lambda}(Y,B_{0})}}$${{H_{\lambda}(Y,Y^{\prime}\cup
B_{0})}}$${{H_{\lambda-1}(Y^{\prime}\cup B_{0},B_{0})}}$${\cdots}$

Consider our third term of the sequence,

(5.9) $\begin{split}H_{\lambda}(Y,Y^{\prime}\cup B_{0})&\cong
H_{\lambda}(Y,Y^{\prime}\cup Y_{c^{0}})\end{split}$

Since $Y\char 92\relax(Y^{\prime}\cup Y_{c^{0}})\cong X\char 92\relax
X_{c^{0}}$, by applying Theorem 5.9 in (5.9),

(5.10) $\begin{split}H_{\lambda}(Y,Y^{\prime}\cup B_{0})&\cong
H_{\lambda}(X,X_{c^{0}})\end{split}$

We aim to prove this term is zero. We already know from (5.7) that the second
term $H_{\lambda}(Y,B_{0})\cong H_{\lambda-2}(B_{0})$. And for the first term

$\begin{split}H_{\lambda}(Y^{\prime}\cup B_{0},B_{0})&\cong
H_{\lambda}(Y^{\prime},X^{\prime}),\quad\text{by theorem \ref{E}}\\\ &\cong
H_{\lambda}(X^{\prime}\times\mathbb{P}^{1},X^{\prime})\\\ \end{split}$

By Künneth Formula,

(5.11)
$\begin{split}H_{\lambda}(X^{\prime}\times\mathbb{P}^{1},X^{\prime})&\cong
H_{\lambda}(X^{\prime}\times(\mathbb{P}^{1},1))\\\
&\cong\bigoplus_{i}H_{i}(X^{\prime})\otimes H_{\lambda-i}(\mathbb{P}^{1})\\\
&\cong H_{\lambda-2}(X^{\prime})\end{split}$

Substituting (5.7), (5.10), (5.11) in (5.8), we get

(5.12)
${0}$${{H_{\lambda-2}(X^{\prime})}}$${{H_{\lambda-2}(X_{c^{0}})}}$${{H_{\lambda}(X,X_{c^{0}})}}$${0}$$\scriptstyle{K}$

Since $X^{\prime}=X_{c^{0}}\cap B_{0}$ is a smooth hyperplane section of
$X_{c^{0}}$, $K$ is an isomorphism by induction and therefore, by the
exactness of the sequence (5.12)

(5.13) $H_{\lambda}(X,X_{c^{0}})=0,\quad\text{for $\lambda\leq n-1$}$

∎

## References

  * [1] M. Audin, M. Damian, and R. Erné, Morse theory and Floer homology, Springer, 2014.
  * [2] K. Lamotke, The topology of complex projective varieties after s. lefschetz, Topology, 20 (1981), pp. 15–51.
  * [3] J. Milnor, Morse Theory.(AM-51), Volume 51, vol. 51, Princeton university press, 2016.