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\begin{document}
\title{MAT 545 Fall 2025\\ Homework 3}
\maketitle
\begin{itemize}

\item[{\bf 1.}] (Leray's covering) Let $\underline{U}=\{U_{\al}\}_{\al\in A}$ be an open cover of a complex manifold $X$ such that
$H^{0,1}_{\delb}(U_{\al}) =0$ for every $\al\in A$, and let $\mathcal{O}$ be the structure sheaf of $X$.
\begin{itemize}
\item[(a)] Let $\rho_{\al}$ be a partition of unity subordinated to $\underline{U}$ and let $\sigma=\{\sigma_{\al\beta}\}\in Z^{1}(\underline{U},\mathcal{O})$ be a $1$-cocycle. Show that
$$\vp_{\al}=\sum_{\beta}\rho_{\beta}\sigma_{\al\beta}\in C^{\infty}(U_{\al})$$
satisfies $\delta\vp=\sigma$ and that $\theta_{\al}=\delb\vp_{\al}$ are restrictions to $U_{\al}$ of a $\delb$-closed $(0,1)$-form $\theta$ on $X$.
 \item[(b)] Prove that the mapping $\sigma\mapsto\al(\sigma)=\theta$ defines a linear map $\al:H^{1}(\underline{U},\mathcal{O})\to H_{\delb}^{0,1}(X)$ from
\v{C}ech cohomology to Dolbeault cohomology.
\item[c)] Prove that the map $\alpha$ establishes the isomorphism 
$$H^{1}(\underline{U},\mathcal{O})\simeq H_{\delb}^{0,1}(X).$$
\end{itemize}

\vspace{2mm}


\item[{\bf 2.}] Let $X$ be a complex manifold  and let $L\to X$ be a holomorphic line bundle. Prove that $L$ is a trivial bundle if and only if it has a nowhere vanishing global holomorphic section.
\vspace{2mm}


\item[\bf{3.}]  Let $X$ be a complex manifold, $\dim X=n$, and let $\mathcal{T}_{X}$ be the holomorphic tangent bundle over $X$. The \emph{first Chern class} $c_{1}(X)$ of the complex manifold $X$ is the first Chern class of line bundle $\Lambda^{n}\mathcal{T}_{X}$. Let $\Omega_{X}$ be the holomorphic cotangent bundle and $\mathcal{K}_{X}= \Lambda^{n}\Omega_{X}$ be the so-called canonical line bundle over $X$. 
Prove that $c_{1}(X)=-c_{1}(\mathcal{K}_{X})$.

\vspace{2mm}

\item[{\bf 4.}] Let $X$ be a complex manifold and let $V$ be a smooth divisor on $X$ (a complex submanifold of codimension $1$). The \emph{normal bundle} $\mathcal{N}_{V}$ to $V$ is the holomorphic line bundle
$$\mathcal{N}_{V}=\left.\mathcal{T}_{X}\right|_{V}/\mathcal{T}_{V},$$
the quotient of $\left.\mathcal{T}_{X}\right|_{V}$, restriction to $V$ of the holomorphic tangent bundle $\mathcal{T}_{X}$ of $X$, over the the holomorphic tangent bundle $\mathcal{T}_{V}$ of $V$. Prove the fllowing adjunction formulas
\begin{itemize}
\item[(a)] $N_{V}\simeq \left.[-V]\right|_{V}$
\item[(b)]
$\mathcal{K}_{V}\simeq \left.(\mathcal{K}_{X}\otimes[V])\right|_{V}.$
\end{itemize}
\vspace{2mm}

\item[\bf{5.}] Prove that the following complex submanifolds in $\mathbb{P}^{n}$ have vanishing first Chern class by showing that they admit a nowhere vanishing top degree holomorphic differential form (so they are Calabi-Yau manifolds).
\begin{itemize}
\item[(a)] $X$ is a Fermat cubic in $\mathbb{P}^{2}$, defined by the equation 
$$F(z_{0},z_{1},z_{2})=z^{3}_{0}+z_{1}^{3}+z^{3}_{2}=0$$
(actually, $X$ is an elliptic curve). Let $\omega$ be a $(1,0)$-form on $X$, which in the coordinate chart $X\cap U_{0}$ is defined by
$$\omega_{0}=\frac{du_{1}}{u_{2}^{2}},\quad\text{where}\quad u_{1}=\frac{z_{1}}{z_{0}},\;u_{2}=\frac{z_{2}}{z_{0}}.$$
Show that $\omega$ is a nowhere vanishing holomorphic $(1,0)$-form on $X$.

\vspace{1mm}
\emph{Hint}: Prove that on $X\cap U_{0}$ one has $du_{1}/(\del f/\del u_{2})=-du_{2}/(\del f/\del u_{1})$.

\vspace{1mm}
\item[(b)] $X$ is Fermat quartic in $\mathbb{P}^{3}$, defined by the equation 
$$F(z_{0},z_{1},z_{2}, z_{3})=z^{4}_{0}+z_{1}^{4}+z^{4}_{2}+ z^{4}_{3}=0$$
(actually, $X$ is a K3 surface). Let $\omega$ be a $(2,0)$-form on $X$, which in the coordinate chart $X\cap U_{0}$ is defined by
$$\omega_{0}=\frac{du_{1}\wedge du_{2}}{u_{3}^{3}},\quad\text{where}\quad u_{1}=\frac{z_{1}}{z_{0}},\;u_{2}=\frac{z_{2}}{z_{0}},\;  u_{3}=\frac{z_{3}}{z_{0}}.$$
Show that $\omega$ is a nowhere vanishing holomorphic $(2,0)$-form on $X$.

 \item[(c)] $X$ is a  Fermat quintic in $\mathbb{P}^{4}$, defined by the equation 
$$F(z_{0},z_{1},z_{2}, z_{3},z_{4})=z^{5}_{0}+z_{1}^{5}+z^{5}_{2}+ z^{5}_{3}+z_{4}^{5}=0$$
(actually, $X$ is a Calabi-Yau threefold). Let $\omega$ be a $(3,0)$-form on $X$, which in the coordinate chart $X\cap U_{0}$ is defined by
$$\omega_{0}=\frac{du_{1}\wedge du_{2}\wedge du_{3}}{u_{4}^{4}},\quad\text{where}\quad u_{1}=\frac{z_{1}}{z_{0}},\;u_{2}=\frac{z_{2}}{z_{0}},\;  u_{3}=\frac{z_{3}}{z_{0}},\; u_{4}=\frac{z_{4}}{z_{0}}.$$
Show that $\omega$ is a nowhere vanishing holomorphic $(3,0)$-form on $X$.
\end{itemize}
\end{itemize}
\end{document}  