\input{"preamble.tex"}

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\title{
\rule{\linewidth}{1pt} \\
\textbf{
    K3 Surfaces
  }
    \\ {\normalsize Lectures by Phil Engel. University of Georgia,
Spring 2023} \\
  \rule{\linewidth}{2pt}
}
\titlehead{
    \begin{center}
  \includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figures/cover.png}
  \end{center}
       \begin{minipage}{.35\linewidth}
    \begin{flushleft}
      \vspace{2em}
      {\fontsize{6pt}{2pt} \textit{Notes: These are notes live-tex'd
from a graduate course in K3 surfaces taught by Phil Engel at the
University of Georgia in Spring 2023. As such, any errors or
inaccuracies are almost certainly my own. } } \\
    \end{flushleft}
    \end{minipage}
    \hfill
    \begin{minipage}{.65\linewidth}
    \end{minipage}
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2023-11-16 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{tuesday-january-10}{%
\section{Tuesday, January 10}\label{tuesday-january-10}}

\begin{remark}

References:

\begin{itemize}
\tightlist
\item
  Beauville, ``Complex Algebraic Surfaces''
\item
  Huybrechts, ``Lectures on K3 Surfaces''
\item
  Gathmann, ``Algebraic Geometry'' (2002)
\end{itemize}

\end{remark}

\begin{remark}

K3s are amazing and used in many fields! Named after Kähler, Kodaira,
and Kummer. An accomplishment of the early 1900s Italian school of
algebraic geometry was classification of complex surfaces (4 real
dimensions, admitting holomorphic charts to \({\mathbf{C}}^2\)). These
can be studied topologically or using algebraic geometry.

A rough plan:

\begin{itemize}
\tightlist
\item
  Review algebraic varieties:

  \begin{itemize}
  \tightlist
  \item
    Riemann-Roch,
  \item
    Curves,
  \item
    Divisors,
  \item
    Line bundles,
  \item
    Picard group,
  \item
    The canonical bundle.
  \end{itemize}
\item
  Complex analytic tools:

  \begin{itemize}
  \tightlist
  \item
    The exponential exact sequence,
  \item
    Betti numbers,
  \item
    Topological Euler characteristic.
  \end{itemize}
\item
  K3s:

  \begin{itemize}
  \tightlist
  \item
    Examples of K3s,
  \item
    Enriques-Kodaira classification,
  \item
    The intersection form.
  \end{itemize}
\item
  Hodge theory:

  \begin{itemize}
  \tightlist
  \item
    Periods, etc.
  \end{itemize}
\end{itemize}

\end{remark}

\begin{remark}

Recall the definition of an affine variety,
e.g.~\(V(x_1, x_2) \subseteq {\mathbf{A}}^2_{/ {k}}\) the union of the
coordinate axes, or the cone
\(V(x_1^2-x_2^2-x_3^2) \subseteq {\mathbf{A}}^3_{/ {k}}\).
Alternatively, view them as schemes:
\(X = V(f_1,\cdots, f_m) = \operatorname{Spec}k[X]\) where
\(k[X] \coloneqq k[x_1,\cdots, x_n]/\left\langle{f_1, \cdots, f_m}\right\rangle\)
is the ring of regular functions on \(X\). The association: for any
point \((a_1,\cdots, a_n) \in k^n\) one can take the maximal ideals
\(\left\langle{x_1-a_1,\cdots, x_n-a_n}\right\rangle\), this is a
bijection by the Nullstellensatz.

\end{remark}

\begin{remark}

An example of why schemes are useful: consider \(V(y)\) and \(V(y-x^2)\)
in \({\mathbf{A}}^2_{/ {k}}\). The set-theoretic intersection is
\(X\coloneqq V(y, y-x^2) = \left\{{0}\right\} \in {\mathbf{A}}^2\), but
note that
\(k[X] = [x,y]/\left\langle{y, x^2}\right\rangle \neq k = k[x,y]/\left\langle{x,y}\right\rangle\),
so although e.g.~\(V(x) = V(x^2)\) these are distinguished as schemes by
remembering the regular functions. A scheme like \(V(x^2)\) is often
drawn as a point with a tangent direction -- the scheme remembers not
only the values of the \(x_i\), but also their various partial
derivatives.

\end{remark}

\begin{remark}

Recall that varieties carry the Zariski topology: the closed sets are of
the form \(V_X(I)\) for \(I {~\trianglelefteq~}k[X]\). From a
scheme-theoretic perspective,
\begin{align*}
V(I) \coloneqq\left\{{\text{prime ideals } p\in X {~\mathrel{\Big\vert}~}p \supseteq I}\right\}
.\end{align*}

\end{remark}

\begin{exercise}[?]

Consider \(X\coloneqq V(xy) \subseteq {\mathbf{A}}^2\), one has
\(k[X] = k[x,y]/\left\langle{xy}\right\rangle\) and
\(I=\left\langle{y-x-1}\right\rangle\) corresponding to the line
\(y=x+1\). What are the closed sets?

\end{exercise}

\begin{remark}

Note that \({\mathbf{A}}^1_{/ {{\mathbf{C}}}}\) with the Zariski
topology differs from \({\mathbf{A}}^1_{/ {{\mathbf{C}}}}\) with the
analytic topology. The closed sets are of the form \(V(I)\), and since
\({\mathbf{C}}[x]\) has GCDs every ideal is principal and
\(I = \left\langle{f}\right\rangle \subseteq k[X]\) for some \(f\). So
closed sets are finite or the entire space, i.e.~the cofinite topology.
By Serre's GAGA, miraculously many results and computations are the same
in either topology for compact (proper) varieties over \({\mathbf{C}}\).

\end{remark}

\begin{remark}

Affine varieties/schemes form a category and there is an equivalence
\({ {{\mathsf{Aff}}{\mathsf{Sch}}}^{\operatorname{op}}} { \, \xrightarrow{\sim}\, }\mathsf{CRing}\).

\end{remark}

\begin{example}[?]

An example of a morphism:
\begin{align*}
\phi: {\mathbf{A}}^1 &\to {\mathbf{A}}^2 \\
t &\mapsto (t^2, t^3)
.\end{align*}
This induces a map on regular functions
\(\phi^*: {\mathbf{C}}[{\mathbf{A}}^2]\to {\mathbf{C}}[{\mathbf{A}}^1]\)
which is of the form
\begin{align*}
\phi^*: {\mathbf{C}}[x,y] &\to {\mathbf{C}}[t] \\
x &\mapsto t^2 \\
y&\mapsto t^3
.\end{align*}
One could similarly define \(\phi\) with codomain \(V(y^2-x^3)\).

\end{example}

\begin{remark}

What are the regular functions on \emph{open} sets? Let
\(U \subseteq X\) in the Zariski topology, then regular functions on
\(U\) are ratios \(f/g\) of polynomials.

\end{remark}

\begin{example}[?]

Let
\(U \coloneqq{\mathbf{A}}^1\setminus\left\{{ 0, 1 }\right\} \subseteq {\mathbf{A}}^1\),
then regular functions include \({1\over x}\) and \({1\over x-1}\).

\end{example}

\begin{remark}

Recall the definition of a sheaf; we'll write \({\mathcal{O}}_X\) for
the structure sheaf and regard \({\mathcal{O}}_X(U)\) as the
\(k{\hbox{-}}\)algebra of functions on \(U\), satisfying the sheaf
axioms of existence and uniqueness of gluing.

\end{remark}

\begin{remark}

Write \({\mathcal{O}}_{\mathbf{C}}\) for the sheaf of \textbf{regular}
functions, then
e.g.~\({\mathcal{O}}_{\mathbf{C}}(\left\{{a_1,\cdots, a_n}\right\}^c) = {\mathbf{C}}[x] { \left[ \scriptstyle {{1\over x-a_1}, \cdots, {1\over x-a_n}} \right] }\),
and more generally
\({\mathcal{O}}_X(V(f)^c) = {\mathbf{C}}[x] { \left[ \scriptstyle {1\over f} \right] }\).
We'll sometimes distinguish \({\mathcal{O}}_{\mathbf{C}}^{\text{hol}}\)
which is defined on \(X^{\mathrm{an}}\) instead (in the Euclidean
topology), which is a priori different as a ringed space. Later we'll
use this in the exponential exact sequence
\begin{align*}
2\pi i \underline{{\mathbf{Z}}} \to {\mathcal{O}}&\xrightarrow{\exp} {\mathcal{O}}^{\times}\\
f&\mapsto e^f
.\end{align*}

\end{remark}

\begin{example}[?]

Schemes are useful in number theory: consider
\(X \coloneqq\operatorname{Spec}{\mathbf{Z}}\), then
\({\mathcal{O}}_X(X) = {\mathbf{Z}}\). There is a point \(p\) for every
prime, and a generic point \(0\). Note that
e.g.~\(20\in {\mathcal{O}}_X(X)\) can be regarded as a function on
\(\operatorname{Spec}{\mathbf{Z}}\), and
\(V(20) \coloneqq\left\{{p {~\mathrel{\Big\vert}~}p\supseteq\left\langle{20}\right\rangle}\right\}\).
It contains \(2\) and \(5\), but contains 2 more! So one might draw its
``graph'' in the following way:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-10_10-42.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Moreover one has
\({\mathcal{O}}_{\operatorname{Spec}{\mathbf{Z}}}(\left\{{2,3}\right\}^c) = \left\{{f/g {~\mathrel{\Big\vert}~}f,g\in {\mathbf{Z}}, g = 2^a3^b}\right\}\).

\end{example}

\begin{remark}

We can formulate manifolds and varieties in terms of transition
functions: for \(U, V \subseteq X\) and charts
\(\phi_U, \phi_V: X\to M\) for \(M\) some model space like
\({\mathbf{A}}^n\) or \({\mathbf{R}}^n\), we can require
\(t_{UV} = \phi_V \circ \phi_U^{-1}\mathrel{\Big|}_{\phi_U(U \cap V)}\)
be continuous, smooth, holomorphic, etc. For schemes, the gluing will be
by regular maps,
e.g.~\({\mathbf{P}}^1 = {\mathbf{A}}_1 \amalg_{t\mapsto s = {1\over t} } {\mathbf{A}}_1\)
where \(t,s\) are the coordinates on each factor.

\end{remark}

\hypertarget{thursday-january-12}{%
\section{Thursday, January 12}\label{thursday-january-12}}

\begin{remark}

Recall the definition of \({\mathbf{P}}^n_{/ {k}}\) as a variety: lines
in \({\mathbf{A}}^{n+1}_{/ {k}}\) passing through the origin. In the
classical topology, it is compact since it can be realized as a quotient
\(S^{2n+1}/S^1\). One can also cover it by affine charts
\(U_0 = \operatorname{Spec}{\mathbf{C}} { \left[ \scriptstyle {{x_1\over x_0}, {x_2\over x_0}} \right] }\)
and \(U_2, U_3\) defined similarly, using that
\({\left[ {x_0:\cdots x_k \cdots : x_n} \right]} = {\left[ {{x_0\over x_k}:\cdots 1 : \cdots {x_n\over x_k}} \right]}\)
on \(\left\{{x_k\neq 0}\right\}\). Recall that
\({\mathcal{O}}_{{\mathbf{P}}^n}(U) = \left\{{f\in {\mathcal{O}}_{U_i}(U \cap U_i) {~\mathrel{\Big\vert}~}{ \left.{{f}} \right|_{{U \cap U_i}} } = { \left.{{f}} \right|_{{U \cap U_j}} } }\right\}\).

\end{remark}

\begin{example}[?]

\({\mathcal{O}}_{{\mathbf{P}}^1}({\mathbf{P}}^1) = \left\{{ (f_0, f_1) \in k[s] \times k[t] {~\mathrel{\Big\vert}~}f_0(s) = f_1(s^{-1}) \text{ on } {\mathbf{A}}^1\setminus\left\{{0}\right\}}\right\}\)
using that \(t=s^{-1}\) on the overlap. This equals
\(k[s] \cap k[s^{-1}] = k\), so the only global regular functions are
constant.

\end{example}

\begin{proposition}[?]

Considering \({\mathbf{P}}^1_{/ {{\mathbf{C}}}}\) in the analytic
(classical) topology,
\begin{align*}
{\mathcal{O}}_{{\mathbf{P}}^1_{/ {{\mathbf{C}}}}}^\text{hol}({\mathbf{P}}^1_{/ {{\mathbf{C}}}}) = \left\{{f: {\mathbf{P}}^1_{/ {{\mathbf{C}}}}\to {\mathbf{C}}\text{ holomorphic}}\right\} = {\mathbf{C}}
.\end{align*}

\end{proposition}

\begin{proof}[?]

Since \({\mathbf{P}}^1_{/ {{\mathbf{C}}}}\) is compact, \(f\) achieves a
maximum value \(m\) at some point \(p\), so write \(f(p) = m\). Letting
\(U\ni p\) be a closed disk containing \(p\), then \(f\) has a maximum
on \(U\). By the maximum modulus principle,
\({ \left.{{f}} \right|_{{U}} }\) is constant, and so \(f\) is constant.

\end{proof}

\begin{remark}

Call \(k[x_0, \cdots, x_{n}]\) the \emph{projective} coordinate ring of
\({\mathbf{P}}^n_{/ {k}}\). Recall that \(f \in k[x_0, \cdots, x_{n}]\)
is not a regular function on \({\mathbf{P}}^n\), but if \(f\) is
homogeneous then \(V(f)\) is well-defined since
\(f(\lambda x_0, \cdots, \lambda x_n) = 0 \iff \lambda^n f(x_0, \cdots, x_n) = 0\).

\end{remark}

\begin{example}[?]

Consider \(V(s^2-st) \subseteq {\mathbf{P}}^1_{/ {k}}\), then \(s(s-t)\)
vanishes on \({\left[ {0: 1} \right]}\) and \({\left[ {1:1} \right]}\).

\end{example}

\begin{example}[?]

Consider \(V(x^3+y^3+z^3) \subseteq {\mathbf{P}}^2_{/ {{\mathbf{C}}}}\)
-- topologically this is \(S^1\times S^1\) and defines an elliptic curve
over \({\mathbf{C}}\).

\end{example}

\begin{example}[of a K3 surface]

\(V(x_0^4 + x_1^4 + x_2^4 + x_3^4) \subseteq {\mathbf{P}}^3_{/ {{\mathbf{C}}}}\)
is a K3 surface.

\end{example}

\begin{remark}

More generally, \(V(f_1,\cdots, f_m) \subseteq {\mathbf{P}}^n_{/ {k}}\)
with \(f_i\) homogeneous of degrees \(d_i\) is a projective scheme,
where we use the scheme structure to distinguish e.g.~\(V(s^2-st)\) and
\(V(s^3-s^2t)\).

\end{remark}

\begin{remark}

Some recollections:

\begin{itemize}
\tightlist
\item
  The definition of irreducibility: \(X\) is reducible if
  \(X = A\cup B\) for \(A, B\) proper nontrivial closed sets.

  \begin{itemize}
  \tightlist
  \item
    E.g. \(V(xy) = V(x) \cup V(y)\) is not irreducible in
    \({\mathbf{A}}^2\).
  \end{itemize}
\item
  Dimension is defined in terms of lengths of chains of closed
  irreducible subsets.
\item
  \(X\) is irreducible iff \(k[X] \coloneqq k[x_1, \cdots, x_{n}]/I(X)\)
  is a domain iff \(I(X)\) is prime
\item
  Krull's PID theorem, used to show
  \(\dim R/\left\langle{f}\right\rangle = \dim R - 1\) if \(f\) is not a
  zero divisor

  \begin{itemize}
  \tightlist
  \item
    To see why this is, consider \(R = {\mathbf{C}}[x,y]\), then
    \(\dim R/\left\langle{xy}\right\rangle = 1 = \dim {\mathbf{C}}[x] = \dim R/\left\langle{x}\right\rangle\).
  \end{itemize}
\item
  \(\operatorname{Spec}R\) is reduced iff \(R\) has no nilpotents

  \begin{itemize}
  \tightlist
  \item
    E.g.
    \(V(x^2) = \operatorname{Spec}k[x]/\left\langle{x^2}\right\rangle\)
    is not reduced, since \(x\) is nilpotent (\(x^2=0\) but
    \(x\neq 0\)).
  \end{itemize}
\item
  \(X \in {\mathsf{Sch}}\) is reduced iff \({\mathcal{O}}_X(U)\) has no
  nilpotents, so every regular function \(f\) satisfies \(f^n\neq 0\)
  for every \(n\).
\item
  \(\operatorname{Spec}R\) is quasicompact.

  \begin{itemize}
  \tightlist
  \item
    E.g. \({\mathbf{A}}^1\) in the Zariski topology is the cofinite
    topology, so if \({\mathcal{U}}\rightrightarrows{\mathbf{A}}^1\) the
    \(U_1\) covers all but finitely many points \(p_k\) and each \(p_k\)
    is in some \(U_k\).
  \end{itemize}
\item
  Open sets are big: they are essentially the whole space, minus lower
  dimensional things.
\item
  Completeness replaces compactness, where \(X\) is (universally)
  complete iff for all \(Y\), the projection \(X\times Y\to Y\) is a
  closed map.

  \begin{itemize}
  \tightlist
  \item
    \({\mathbf{A}}^1\) is not complete: take
    \(Y\coloneqq{\mathbf{A}}^1\), then
    \(V(xy-1) \mapsto {\mathbf{A}}^1\setminus\left\{{0}\right\}\) is a
    closed set mapping to an open set.
  \end{itemize}
\item
  Producing varieties that aren't manifolds:
  \(V(xy) \subseteq {\mathbf{A}}^2_{/ {{\mathbf{C}}}}\) is singular at
  the origin and has no local chart to \({\mathbf{C}}\) there.
\item
  \(V(f) \subseteq {\mathbf{R}}^n\) is a manifold when \(0\) is a
  regular value of \(f\), so
  \(df: T_p {\mathbf{R}}^n\to T_0 {\mathbf{R}}\) is surjective at all
  \(p\in V(f)\).

  \begin{itemize}
  \tightlist
  \item
    Can be formulated as
    \(Jf \coloneqq\qty{{\frac{\partial f_i}{\partial x_j}\,}}\) has
    maximal rank everywhere.
  \end{itemize}
\end{itemize}

\end{remark}

\hypertarget{tuesday-january-17}{%
\section{Tuesday, January 17}\label{tuesday-january-17}}

\begin{remark}

Recall
\(X = V(f_1,\cdots, f_m) = \operatorname{Spec}R \subseteq {\mathbf{A}}^n_{/ {k}}\)
where
\(R\coloneqq{\mathbf{C}}[x_1,\cdots, x_n]/\left\langle{f_1,\cdots, f_m}\right\rangle\)
is smooth if
\(\operatorname{Jac}\left\{{f_i}\right\} = \qty{{\frac{\partial f_i}{\partial x_j}\,}}\)
has maximal rank \(r = \operatorname{codim}_{{\mathbf{A}}^n} X\) at
\emph{all} points \(x\in X\), and we'll give a more intrinsic notion of
smoothness which does not depend on the choice of equations
\(\left\{{f_i}\right\}\). Over \(k={\mathbf{C}}\), if \(X\) is smooth it
is a complex manifold.

\end{remark}

\begin{example}[?]

For \(X\coloneqq V(x^2 + y^2 + 1)\), note
\(\nabla f = {\left[ {2x, 2y} \right]}\) has rank 1 everywhere except
\(0\), but since \(0\not\in X\), in fact \(X\) is smooth.

\end{example}

\begin{definition}[Kähler differentials]

Recall that \(\Omega^1{R/k} \coloneqq\bigoplus R\,dr/I\) where
\begin{align*}
I = \left\langle{ d(rs) = rds + sdr, d(cr) = cdr, d(r+s) = dr + ds {~\mathrel{\Big\vert}~}r,s\in R, c\in k}\right\rangle
.\end{align*}
Note that \(\Omega^1_{R/{\mathbf{C}}}\in {}_{R}{\mathsf{Mod}}\), while
\(\Omega^1_{X/{\mathbf{C}}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\) is a
sheaf.

\end{definition}

\begin{example}[?]

Example: for
\(R = {\mathbf{C}}[x,y]/\left\langle{x^2+y^2+1}\right\rangle\), we have
\(\Omega^1_{R/{\mathbf{C}}} = Rdx + Rdy/I\). Noting \(x^2+y^2+1=0\) in
\(R\), we have
\begin{align*}
0 = d(x^2+y^2+1) = 2xdx + 2y dy 
.\end{align*}

\end{example}

\begin{definition}[Smoothness]

\(X\) is \textbf{smooth} iff the rank of \(\Omega^1_{X/{\mathbf{C}}}\)
at \(p\) is \(\dim X\) for every \(p\in X\). For \(X\) a variety, point
\(p\in X\) correspond to \({\mathfrak{m}}_p \in \operatorname{mSpec}R\)
and
\(\Omega^{1}_{R/{\mathbf{C}}}/{\mathfrak{m}}_x \in \operatorname{mod}{R/{\mathfrak{m}}_p}\),
so we take
\begin{align*}
\operatorname{rank}_p \Omega^1_{X/{\mathbf{C}}} \coloneqq\dim_{R/{\mathfrak{m}}_p} (\Omega^1_{R/{\mathbf{C}}}/k)
\end{align*}
where \(R/{\mathfrak{m}}_p \cong k\) is a fixed field via the map
\(f\mapsto f(p)\).

\end{definition}

\begin{example}[?]

The previous example is still smooth: we have
\begin{align*}
\Omega^1_{R/{\mathbf{C}}}/{\mathfrak{m}}_p = {{\mathbf{C}}dx \oplus {\mathbf{C}}dy \over x(p)dx + y(p) dy}
\end{align*}
which has \({\mathbf{C}}{\hbox{-}}\)dimension 1 if we \emph{don't} have
\(x(p) = y(p) = 0\). This exactly recovers the Jacobi criterion.

\end{example}

\begin{definition}[$\oxmods$]

An \({\mathcal{O}}_X{\hbox{-}}\)module is a sheaf \({\mathcal{F}}\) on
\(X\) where

\begin{itemize}
\tightlist
\item
  \({\mathcal{F}}(U) \in {}_{{\mathcal{O}}_X(U)}{\mathsf{Mod}}\), so the
  sections are modules over regular functions, and
\item
  \({\mathcal{F}}(U) \xrightarrow{\mathop{\mathrm{Res}}_{UV}} {\mathcal{F}}(V)\)
  is compatible with the module structure and
  \({\mathcal{O}}_X(U) \xrightarrow{\mathop{\mathrm{Res}}_{UV}} {\mathcal{O}}_X(V)\),
  so
  \(\mathop{\mathrm{Res}}_{UV}(f.s) = \mathop{\mathrm{Res}}_{UV}(f) . \mathop{\mathrm{Res}}_{UV}(s)\)
  for \(f\in {\mathcal{O}}_X(U)\) and \(s\in {\mathcal{F}}(U)\).
\end{itemize}

\end{definition}

\begin{example}[?]

\(\Omega^{1}_{X/{\mathbf{C}}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\),
where the sections are 1-forms on open sets, as is \({\mathcal{O}}_X\)
itself. An example:
\(\Omega^1_{{\mathbf{A}}^1\setminus\left\{{0}\right\}} = {\mathbf{C}}[x,x^{-1}]dx\).

\end{example}

\begin{example}[?]

Let \(X = {\mathbf{A}}^1_{/ {{\mathbf{C}}}}\), then let
\({\mathcal{O}}_p\) be the skyscraper sheaf at \(p\). This can be made
into an \({\mathcal{O}}_X{\hbox{-}}\)module in the following way: for
\(f\in {\mathcal{O}}_X(U), s\in {\mathcal{O}}_p\), define
\(f.s = f(p) s\). How to visualize: think of \({\mathcal{O}}_X\) as a
trivial bundle.

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-17_10-09.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Compare to the skyscraper sheaf:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-17_10-11.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{example}

\begin{definition}[Morphisms in $\oxmods$]

If
\({\mathcal{F}}, {\mathcal{G}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\)
then \(\phi: {\mathcal{F}}\to {\mathcal{G}}\) is a morphism iff it is a
morphism of sheaves, so a collection
\(\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)\), which are compatible
with the module actions.

\end{definition}

\begin{example}[?]

There is a morphism \({\mathcal{O}}_X \to {\mathcal{O}}_p\) of sheaves
determined by
\begin{align*}
\phi(U): {\mathcal{O}}_X(U) &\to {\mathcal{O}}_p \\
f &\mapsto f(p)
,\end{align*}
which is a morphism in \({}_{{\mathcal{O}}_X}{\mathsf{Mod}}\) since
\((g\cdot f)(p) = g(p)\cdot f(p)\) is defined by pointwise
multiplication.

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-17_10-16.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Recall that the presheaf \(\ker \phi\) is a sheaf, and here
\(\ker \phi(U) = \left\{{f\in {\mathcal{O}}_X(U) {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\)
is the \emph{ideal sheaf} of \(p\), \(I_p\), so we get a SES
\begin{align*}
I_p \hookrightarrow{\mathcal{O}}_{{\mathbf{A}}^1_{/ {{\mathbf{C}}}}} \twoheadrightarrow{\mathcal{O}}_p
.\end{align*}

\end{example}

\begin{definition}[Ideal sheaf]

For \(V \subseteq W\) a subvariety, define
\begin{align*}
I_V(U) = \left\{{f\in {\mathcal{O}}_W(U) {~\mathrel{\Big\vert}~}{ \left.{{f}} \right|_{{V \cap U}} } = 0}\right\}
.\end{align*}

\end{definition}

\begin{example}[?]

Let \(X = {\mathbf{P}}^1_{/ {{\mathbf{C}}}}\), then recall
\({\mathcal{O}}_X(X) = {\mathbf{C}}\) in this case. Letting
\(p\neq q\in X\), there is a map
\begin{align*}
{\mathcal{O}}_X &\to {\mathcal{O}}_p \oplus {\mathcal{O}}_q \\
f &\mapsto f(p) \oplus f(q)
.\end{align*}
This is a surjection of sheaves, despite not being surjective on global
sections: \({\mathcal{O}}_X(X) = {\mathbf{C}}\), while
\(({\mathcal{O}}_p \oplus {\mathcal{O}}_q)(X) = {\mathbf{C}}\oplus {\mathbf{C}}\).
However, this is an open cover \({\mathcal{U}}\rightrightarrows X\)
where this is a surjection on sections: take
\(U_1 \coloneqq X\setminus\left\{{ p }\right\}\) and
\(U_2\coloneqq X\setminus\left\{{ q }\right\}\). Here we get a SES
\begin{align*}
I_{\left\{{p, q}\right\}} \hookrightarrow{\mathcal{O}}_{{\mathbf{P}}^1_{/ {{\mathbf{C}}}}}\twoheadrightarrow{\mathcal{O}}_p \oplus {\mathcal{O}}_q
.\end{align*}

\end{example}

\begin{example}[?]

For \(X = {\mathbf{A}}^1_{/ {{\mathbf{C}}}}\), there is an isomorphism
\begin{align*}
\Omega^1_{X} & { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X \in    {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\\
f &\mapsto f\,dx
.\end{align*}

\end{example}

\begin{remark}

For \(X\) not affine, what is \(\Omega^1_{X}\)? If
\(\omega = \sum f_i dx_i\) in one chart and \(\sum g_i dy_i\) in another
via charts \(\vec x, \vec y\), how are they related? One needs a notion
of pullbacks. We define \(\Omega^1_X(U)\) to be well-defined 1-forms
\(\omega_i \in \Omega^1_X(U \cap U_i)\) which are compatible on
overlaps.

\end{remark}

\begin{example}[?]

Let \(X = {\mathbf{P}}^1\), glued from affines
\(U_0 = \operatorname{Spec}{\mathbf{C}}[s]\) and
\(U_1 = \operatorname{Spec}{\mathbf{C}}[t]\) by
\begin{align*}
t_{01}: U_0 &\to U_1 \\
s &\mapsto t=s^{-1}
.\end{align*}

Take \(\omega_i \in \Omega^1_X(U_i)\), then

\begin{itemize}
\tightlist
\item
  \(\omega_0 = f_0(s)ds \in {\mathbf{C}}[s] ds\)
\item
  \(\omega_1 = f_1(t)dt \in {\mathbf{C}}[t] dt\)
\end{itemize}

Then the compatibility condition is that
\begin{align*}
t_{01}^*(\omega_1) = f_1(s^{-1})d(s^{-1}) = f_0(s) ds
.\end{align*}

This becomes
\begin{align*}
- {f_1(s^{-1}) \over s^2}ds &= f_0(s) ds \\
\implies f_1(s^{-1}) &= -s^2 f_0(s)\\
\implies c_0 + c_1 s^{-1}+ c_2 s^{-2} + \cdots + c_k s^{-k} &= d_0 s^2 + d_1 s^3 + \cdots + d_r s^r
,\end{align*}
which can only be true if \(f \equiv 0\). This implies that
\(\Omega^1_{X}\) is a line bundle.

\end{example}

\begin{definition}[Vector bundle]

A \textbf{line bundle} on \(X\) is
\({\mathcal{F}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\) where
\(\exists {\mathcal{U}}\rightrightarrows X\) where
\({ \left.{{{\mathcal{F}}}} \right|_{{U_i}} } = {\mathcal{O}}_{U_i}\). A
\textbf{vector bundle} of rank \(r\) is such an \({\mathcal{F}}\) where
\({ \left.{{{\mathcal{F}}}} \right|_{{U_i}} } = {\mathcal{O}}_{U_i}{ {}^{ \scriptscriptstyle\oplus^{r} } }\)
for some \(r\).

\end{definition}

\begin{example}[?]

\({\mathcal{O}}_p\) is not a vector bundle, since
\({\mathcal{O}}_p(U)\not { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X(U){ {}^{ \scriptscriptstyle\oplus^{r} } }\)
for any \(r\) or any affine open \(U\ni p\).

\end{example}

\begin{definition}[Divisors]

A Weil divisor on \(X\) is a \({\mathbf{Z}}{\hbox{-}}\)linear
combination of irreducible codimension 1 subvarieties. For
\(D = \sum n_i p_i\), its degree is \(\sum n_i\).

\end{definition}

\begin{example}[?]

The irreducible codimension 1 subvarieties of \({\mathbf{P}}^1\) are
points, so
\begin{align*}
\operatorname{WDiv}({\mathbf{P}}^1) = \bigoplus _{p\in {\mathbf{P}}^1} {\mathbf{Z}}[p]
.\end{align*}
For example,
\(\operatorname{WDiv}({\mathbf{P}}^1_{/ {{\mathbf{C}}}})\ni D \coloneqq 2[0] - [\pi] + 3[\infty]\)
and \(\deg D = 4\). Similarly,
\(\operatorname{WDiv}({\mathbf{A}}^2)\ni [V(x)] - [V(y)]\).

\end{example}

\begin{definition}[Divisors of functions]

Let \(X\) be irreducible and \(f\in {\mathcal{O}}_X(U)\) with
\(U \subseteq X\) Zariski open, and define
\(\operatorname{div}(f) \coloneqq\sum n_i [D_i]\) where \(n_i\) is the
order of zeros/poles of \(f\) along \(D_i\).

\end{definition}

\begin{example}[?]

Let \(x/y^2 \in {\mathcal{O}}_{{\mathbf{A}}^2}(V(y)^c)\), then
\(\operatorname{div}(x/y^2) = [V(x)] - 2[V(y)]\). Similarly,
\(f\coloneqq{s^2-t^2\over st}\in {\mathcal{O}}_{{\mathbf{P}}^1}({\mathbf{P}}^1\setminus\left\{{ 0, \infty }\right\})\)
has divisor \(\operatorname{div}(f) = [1] + [-1] - [0] - [\infty]\).

\end{example}

\hypertarget{thursday-january-19-divisors}{%
\section{Thursday, January 19
(Divisors)}\label{thursday-january-19-divisors}}

\begin{remark}

Recall that \(\exp\) is surjective as a map of sheaves. On open
contractible subsets \(U \subseteq {\mathbf{C}}\), for any
\(g\in {\mathcal{O}}_{\text{hol}}^{\times}(U)\) there is an
\(f\coloneqq\log(g)\), but
\(z\mapsto \log(z)\not\in {\mathcal{O}}_{\text{hol}}({\mathbf{C}}^{\times})\).
Thus surjections of sheaves need not induce surjections on global
sections, the failure is measured by sheaf cohomology.

\end{remark}

\begin{definition}[Divisor class group]

Define the \textbf{principal Weil divisors} as
\begin{align*}
\mathop{\mathrm{Prin}}\mathrm{W}Div = \left\{{\operatorname{div}(f) {~\mathrel{\Big\vert}~}f\in K(X)}\right\}
,\end{align*}
divisors of nonzero rational functions. Here
\(\operatorname{div}(f) = \sum n_Y [Y]\) where \(n_Y\) is the order of
vanishing/poles along \(Y\). We then define the \textbf{(Weil) divisor
class group} as
\begin{align*}
\mathrm{W} \operatorname{Cl}(X) \coloneqq\operatorname{WDiv}(X) / \mathop{\mathrm{Prin}}\operatorname{Div}(X)
.\end{align*}

\end{definition}

\begin{example}[?]

On \({\mathbf{P}}^1\),
\(\operatorname{div}\qty{s^2-t^2\over st} = [1] + [-1] - [0] - [\infty]\),
regarding \(\infty = s/t\).

\end{example}

\begin{example}[?]

\(\operatorname{Cl}({\mathbf{A}}^1) = 0\) since
\(\sum n_p [p] = \operatorname{div}f\) where \(f = \prod (x-p)^{n_p}\).

\end{example}

\begin{example}[?]

There is an isomorphism
\begin{align*}
\deg: \mathrm{W} \operatorname{Cl}({\mathbf{P}}^1) & { \, \xrightarrow{\sim}\, }{\mathbf{Z}}\\
\sum n_p [p] &\mapsto \sum n_p
.\end{align*}
E.g. considering \(\operatorname{div}\qty{}\)
\([1] + [-1] - [0] = [\infty]\) in
\(\operatorname{Cl}({\mathbf{P}}^1)\).

\end{example}

\begin{example}[?]

Consider \(\operatorname{Cl}(\operatorname{Spec}{\mathbf{Z}})\):
principal divisors are primes, so
\(\operatorname{WDiv}(\operatorname{Spec}{\mathbf{Z}}) = \left\{{\sum_{p\text{ prime}} n_p [p] }\right\}\).
Rational functions on \(\operatorname{Spec}{\mathbf{Z}}\) are identified
with \({\mathbf{Q}}\), and if \(r = \prod p_i^{n_i}\in {\mathbf{Q}}\)
then \(\operatorname{div}(r) = \sum n_i [p_i]\), so
\(\operatorname{Cl}(\operatorname{Spec}{\mathbf{Z}}) = 0\) since every
\(\sum_{p \text{ prime}} n_p [p]\) is the divisor of some
\(r\in {\mathbf{Z}}\subseteq {\mathbf{Q}}\).

\end{example}

\begin{example}[?]

For \(X = \operatorname{Spec}R\) for
\(R\coloneqq{\mathbf{Z}} { \left[ \scriptstyle {\sqrt{-5}} \right] }\),
we have
\(\operatorname{WDiv}(X) = \sum_{p\neq 0 \text{ prime ideals}} n_p [p]\),
and the rational functions on \(X\) are \({\mathbf{Q}}(\sqrt{-5})\).
Since \(R\in \mathbb{D}\), there is unique factorization of (fractional)
ideals, so writing \((r) = \prod p_i^{n_i}\) we have
\(\operatorname{div}r = \sum n_i [p_i]\). However, \(R\) is not a UFD,
considering \(2\cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})\).

\begin{quote}
Consider \(r = 1\cdot [p]\) for \(p = (2, 1 + \sqrt{-5})\)? Ask.
\end{quote}

\end{example}

\begin{definition}[Cartier divisors]

A \textbf{Cartier divisor} is a collection of rational functions \(f_i\)
on \(U_i\) such that
\(\operatorname{div}(f_i) = \operatorname{div}(f_j)\) on \(U_{ij}\).
These form a group \(\operatorname{CDiv}(X)\), and there is a
corresponding class group
\(\mathrm{Ca} \operatorname{Cl}(X) \coloneqq\operatorname{CDiv}(X) / \mathop{\mathrm{Prin}}\operatorname{CDiv}(X)\).

\end{definition}

\begin{example}[?]

Write \({\mathbf{P}}^1 = U_0 \cup U_1\) and consider

\begin{itemize}
\tightlist
\item
  \(f_0 = s\) on \(U_0 = \operatorname{Spec}{\mathbf{C}}[s]\)
\item
  \(f_1 = 1\) on \(U_1 = \operatorname{Spec}{\mathbf{C}}[t]\)
\end{itemize}

Note \(\operatorname{div}f_0 = [0]\) on \(U_0\) and
\(\operatorname{div}f_1 = 0\) on \(U_1\), but
\({ \left.{{f_0}} \right|_{{U_{01}}} }\) has no poles or zeros and thus
\(\operatorname{div}{ \left.{{f_0}} \right|_{{U_{01}}} }= 0 = \operatorname{div}{ \left.{{f_1}} \right|_{{U_{01}}} }\).

\end{example}

\begin{fact}

If \(X\) is smooth then
\(\operatorname{WDiv}(X) = \operatorname{CDiv}(X)\). Note all Weil
divisors are Cartier: consider
\(X = V(xy-z^2) \subseteq {\mathbf{A}}^3\), which is a circular cone.
Note that \(V(z)\) is a union of two lines along the edge of the cone.
Consider \(D = V_X(z, y)\), an irreducible codimension 1 subvariety, so
\(D\in \operatorname{WDiv}(X)\). This is locally principal away from the
origin, since one can slice by the plane \(z=0\). Suppose \(f(x,y,z)\)
cuts out \(D\) at \(0\), then write
\(f = c_0 + c_1x + c_2y + c_3z + \cdots\). Since \(f(0) = 0\) we have
\(c_0 = 0\), and the remaining terms always cut out \emph{two} lines. On
the other hand, \(2D\) is Cartier and principal, since the tangent plane
along the cone \(V_X(y) = V_X(y, z^2)\) cuts out a doubled line.

\end{fact}

\begin{remark}

Recall that line bundles are
\({\mathcal{L}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\) locally
isomorphic to \({\mathcal{O}}_{U_i}\). Given
\(D\in\operatorname{CDiv}(X)\) with Cartier data
\(\left\{{(f_i, U_i)}\right\}\) with \(f_i\) rational on \(U_i\) and
\(\operatorname{div}(f_i) = \operatorname{div}(f_j)\) on overlaps.
Define \({\mathcal{O}}_X(D)\in\operatorname{Pic}(X)\) to be the sheaf
whose sections over \(U\) are
\(\left\{{ (s_i) \in {\mathcal{O}}_X(U \cap U_i) s_i f_i = s_j f_j }\right\}\),
so the sections are related by \(s_j = {f_i\over f_j} s_i\) on
\(U_{ij}\). Write
\(t_{ij} = {f_j\over f_i}\in {\mathcal{O}}_X(U_i \cap U_j)\) for the
transition functions.

\end{remark}

\begin{example}[?]

Write \({\mathbf{P}}^1 = U_0 \cup U_1\) and
\(D = \left\{{(s, U_0), (1, U_1)}\right\} \in \operatorname{CDiv}({\mathbf{P}}^1)\),
and consider \({\mathcal{O}}_{{\mathbf{P}}^1}(D)\). This is given by
\(\left\{{p\in k[s], q\in k[t] {~\mathrel{\Big\vert}~}p = sq}\right\}\).
Writing \(t=s^{-1}\), we have \(p(s) = sq(s^{-1})\), so if \(q=1\) then
\(p=s\) and if \(q(t) = t\) then \(p=1\). One can check
\({\mathcal{O}}_{{\mathbf{P}}^1}(D) = {\mathbf{C}}\left\langle{s, 1}\right\rangle \oplus {\mathbf{C}}\left\langle{1, t}\right\rangle\).

\end{example}

\begin{exercise}[?]

Show that if \(D = \left\{{(s^k, U_0), (1, U_1)}\right\}\) then
\({\mathcal{O}}_{{\mathbf{P}}^1}(D) = {\mathcal{O}}_{{\mathbf{P}}^1}(k)\),
whose global sections are homogeneous degree \(k\) polynomials on
\({\mathbf{P}}^1\).

\end{exercise}

\hypertarget{tuesday-january-24}{%
\section{Tuesday, January 24}\label{tuesday-january-24}}

\begin{remark}

Recall that
\(\operatorname{WDiv}(X) = \mathrm{Ca}\operatorname{Div}(X)\) if \(X\)
is smooth or if \(\operatorname{codim}X_{\mathrm{sing}}\geq 3\), and for
any \(D\in \operatorname{CDiv}(X)\),
\begin{align*}
{\mathcal{O}}_X(D)(U) \coloneqq\left\{{f\in k(U) {~\mathrel{\Big\vert}~}\operatorname{div}f + D \geq 0}\right\}\in \operatorname{Pic}(X)
\end{align*}
is a line bundle.

\end{remark}

\begin{example}[?]

Let \(D = V(xyz) \subseteq {\mathbf{P}}^2\), then \(D\) is 3 copies of
\({\mathbf{P}}^1\) linked in a triangle. Consider
\(f\in {\mathcal{O}}_{{\mathbf{P}}^2}(D)({\mathbf{P}}^2)\), so
\(\operatorname{div}f = -L_1 -L_2 -L_3 + \sum n_p P\) for some
\(n_p \geq 0\). Since \(f\in k({\mathbf{P}}^2)\implies f = a/b\) with
\(a,b\) homogeneous polynomials of the same degree, one example is
\(f(x,y,z) = a(x,y,z)/xyz\) with \(a\) homogeneous of degree 3. Thus
\({\mathsf{H}}^0({\mathcal{O}}_{{\mathbf{P}}^2}(D)) \cong k[x,y,z]^{3, { \mathrm{homog} }}\).

\end{example}

\begin{proposition}[?]

If \(D \equiv D'\in \operatorname{Cl}(X)\), so
\(D-D'= \operatorname{div}h\) for some \(h\), then
\({\mathcal{O}}_X(D)\cong {\mathcal{O}}_X(D')\) as sheaves.

\end{proposition}

\begin{proof}[?]

One needs a map \({\mathcal{O}}_X(D)(U) \to {\mathcal{O}}_X(D')(U)\) for
every open \(U \subseteq X\), so take the map
\begin{align*}
{\mathcal{O}}_X(D')(U) &\to {\mathcal{O}}_X(D)(U) \\
f' &\mapsto f\coloneqq f'\cdot h^{-1}
.\end{align*}
Since
\(\operatorname{div}(f' h^{-1}) = \operatorname{div}(f') - \operatorname{div}(h)\),
we have
\begin{align*}
\operatorname{div}f' + D' \geq 0 \iff \operatorname{div}f' + D' + \operatorname{div}h \geq \operatorname{div}h \iff \operatorname{div}f + D \geq 0
.\end{align*}

\end{proof}

\begin{remark}

Note that if \(D\geq 0\) then \({\mathcal{O}}_X(D)(X) \ni 1\), the
constant function, and this is a global section so
\(H^0({\mathcal{O}}_X(D)) > 0\).

\end{remark}

\begin{fact}

Any irreducible codimension 1 \(D \subseteq {\mathbf{P}}^n\) is of the
form \(V(f)\) for a single function \(f\), which follows from the fact
that any height 1 prime in \(k[x_0, \cdots, x_{n}]\) is principal. Thus
\(\operatorname{Div}({\mathbf{P}}^n) = \left\{{\displaystyle\sum_{f\in k[x_0,\cdots, x_n]^{ { \mathrm{homog} }, {\mathrm{irr}}} } n_f [V(f)] }\right\}\),
and if \(D = \sum n_f [V(f)]\) and \(D' = \sum n_{f'}[V(f')]\), then
\(D\equiv D' \iff \sum n_f \deg f = \sum n_{f'} \deg f'\), noting
\(D-D' = \operatorname{div}\qty{\prod f^{n_f} \over \prod (f')^{n_{f'}} }\)
which is a rational function on \({\mathbf{P}}^n\). So
\(\operatorname{Cl}({\mathbf{P}}^n)\underset{\deg}{ { \, \xrightarrow{\sim}\, }} {\mathbf{Z}}\)
where \(\sum n_f [V(f)] \mapsto \sum n_f \deg f\).

\end{fact}

\begin{definition}[?]

\(\operatorname{Pic}(X)\) is the group of line bundles on \(X\) up to
isomorphism, with group structure given by the following: for
\(L_1, L_2\in \operatorname{Pic}(X)\), define
\begin{align*}
(L_1\otimes L_2)(U) \coloneqq L_1(U)\otimes_{{\mathcal{O}}_X(U)} L_2(U)
.\end{align*}
Alternatively, the transition functions on the tensor product are
products of transition functions:
\begin{align*}
t_{UV}^{L_1\otimes L_2} = t_{UV}^{L_1}\cdot t_{UV}^{L_2}
.\end{align*}
The identity element is \({\mathcal{O}}_X\), since
\(L_1(U) \otimes_{{\mathcal{O}}_X(U)} {\mathcal{O}}_X(U) = L_1(U)\).
Inverses are given by
\(L^{-1}\coloneqq\mathop{\mathcal{H}\! \mathit{om}}(L, {\mathcal{O}}_X)\),
so
\(L^{-1}(U) = \mathop{\mathrm{Hom}}_{{\mathcal{O}}_X(U)}(L(U), {\mathcal{O}}_X(U))\)
on small enough open sets, and the transition functions are given by
\begin{align*}
t_{UV}^{L^{-1}} = (t_{UV}^{L})^{-1}
.\end{align*}
It can be checked that if \(L\) satisfies the cocycle condition iff
\(L^{-1}\) does, and similarly for \(L_1\otimes L_2\).

\end{definition}

\begin{proposition}[?]

If \(X\) is smooth then
\(\operatorname{Pic}(X)\cong \operatorname{Cl}(X)\) via
\(D\rightleftharpoons{\mathcal{O}}_X(D)\).

\end{proposition}

\begin{proof}[?]

This uses that
\({\mathcal{O}}_X(D)\cong{\mathcal{O}}_X(D')\iff D\equiv D'\), the
interesting part is to show surjectivity. Let
\(L\in \operatorname{Pic}(X)\), then
\({ \left.{{L}} \right|_{{U}} } = {\mathcal{O}}_U\) for some \(U\), and
we can consider \(1\in {\mathcal{O}}_U(U) \cong L(U)\). In any other
trivialization, \({ \left.{{L}} \right|_{{V}} } \cong {\mathcal{O}}_V\)
and there is a transition function \(t_{UV} \in k(V)\). Since
\(1\in {\mathcal{O}}_U(U)\), we have \(\operatorname{div}(1) = D\) where
the LHS is regarded as a rational section of \(L\), and
\(L \cong {\mathcal{O}}_X(D)\).

\end{proof}

\begin{definition}[Rational sections]

A \textbf{rational section} of \(L\) is a section of \(L\otimes k(X)\).

\end{definition}

\begin{remark}

This allows for a section \(s\in H^0(L)\) to have poles, and
\(L\cong {\mathcal{O}}(\operatorname{div}(s))\) for any section \(s\) of
\(L\). If \(s, s'\) are rational sections, then \(s/s'\) is a rational
\emph{function}. Concretely, if
\(s = \left\{{s_u \in k(U) {~\mathrel{\Big\vert}~}t_{UV} s_U = s_V}\right\}\)
and
\(s' = \left\{{s'_U \in k(U) {~\mathrel{\Big\vert}~}t_{UV} s'_U = s'_V}\right\}\).
Then \(s/s' = \left\{{s_U \over s_U' = s_V/s_V'}\right\} \in k(X)\), so
\(\operatorname{div}(s) = \operatorname{div}(s')\).

\end{remark}

\begin{remark}

The degree of any principal divisor on a curve is zero.

\end{remark}

\begin{example}[?]

More interesting examples come from elliptic curves. Write
\(X = {\mathbf{C}}/ \Lambda\) where
\(\Lambda = {\mathbf{Z}}\oplus {\mathbf{Z}}\tau\) with
\(\tau \in {\mathbb{H}}\) This yields a complex manifold, since the
transition functions are translations and thus holomorphic. We can write
\(\operatorname{Div}(X)\ni D = \sum n_p [p]\) where \(p\in X\) are
points. A meromorphic function is a rational function
\(f: X\dashrightarrow{\mathbf{C}}\) which extends to a holomorphic map
\(f:X\to {\mathbf{CP}}^1\) by mapping poles to \(\infty\) -- note that
this extension only works because \(X\) is complex dimension 1, and does
not work in higher dimensions. This pulls back to
\(\tilde f: {\mathbf{C}}\to {\mathbf{P}}^1\) which satisfies
\(\tilde f(z+ \lambda) = \tilde f(z)\) for all \(\lambda\in \Lambda\).
The Weierstrass \(\wp{\hbox{-}}\)function is defined by
\begin{align*}
\wp(z) \coloneqq{1\over z^2} + \sum_{\lambda\in \Lambda\setminus\left\{{0}\right\}}\qty{ {1\over (z - \lambda)^2 } - {1\over \lambda^2} }
,\end{align*}
which averages over the lattice and is thus periodic. Note
\begin{align*}
{1\over (z- \lambda)^2} - {1\over \lambda^2} = { \lambda^2 - (z- \lambda)^2 \over (z- \lambda)^2 \lambda^2}
.\end{align*}
where the denominator is
\(\geq C {\left\lvert { \lambda} \right\rvert}^4\) for
\({\left\lvert {z} \right\rvert} \gg 1\) and the numerator is
\(\leq c {\left\lvert { \lambda} \right\rvert}\) for
\({\left\lvert {z} \right\rvert} \gg 1\), so
\begin{align*}
\sum_{ \lambda \in \Lambda\setminus\left\{{0}\right\}} C{\left\lvert {\lambda} \right\rvert}^{-3} \leq C \int_{{\mathbf{R}}^2} {\left\lvert {\lambda} \right\rvert}^{-3} = C \iint r^{-3} r\,dr\,d\theta
\end{align*}
which converges. So the extra constant \({1\over \lambda^2}\) is
necessary to make the series converge. Since translating by
\(\lambda\in \Lambda\) rearranges the series, \(\wp(z)\) is a
well-defined rational function on \(X\) with a double pole at every
\(z\in \Lambda\), corresponding to \(0\in {\mathbf{C}}/ \Lambda\). So
\(\operatorname{div}\wp(z) = -2 [0]\), and it induces
\(X\xrightarrow{\pi} {\mathbf{P}}^1\) where we have
\(\deg \pi = \deg \pi^{-1}(0) = \deg \pi^{-1}(\infty)\).

\end{example}

\begin{proposition}[?]

Let \(f\in {\mathbf{C}}(X)\) be meromorphic and let
\(\operatorname{div}f = \sum n_p [p]\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\sum n_p = 0\)
\item
  \(\sum n_p [p] \equiv 0 \operatorname{mod}\Lambda\)
\end{enumerate}

\end{proposition}

\begin{proof}[?]

Let \(f\in {\mathbf{C}}(X)\) with some zeros and poles.

Take the following contour:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-01-24_10-44.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Note that
\(\int_\gamma \operatorname{dLog}(f) = \int_\gamma {f'(z) \over f(z)} \,dz= \sum_{p \text{ inside}} n_p\)
by the residue theorem. Recall that in local coordinates \(w\), if
\(f(w) = cw^{-k} + \cdots\) then
\(\operatorname{dLog}(f) = -k {dw \over w} + h(w)\) where \(h\) is
holomorphic. However, by periodicity, the edge integrals cancel and
\(\int_{C_1 + C_2} \operatorname{dLog}(f) = \sum_{p\text{ edge}} -n_p\),
forcing \(\sum n_p = 0\).

Now considering
\(\int_\gamma z\operatorname{dLog}(f) = \sum_{p \text{ inside}} n_p p\)
and
\(\mathop{\mathrm{Res}}_p z\operatorname{dLog}(f) = n_p \cdot z(p)\). On
the other hand,
\(-\sum_{p \text{ edge}} n_p p+ \int_{A\setminus A'} z\operatorname{dLog}(f)\).

\end{proof}

\hypertarget{thursday-february-02}{%
\section{Thursday, February 02}\label{thursday-february-02}}

\begin{theorem}[Riemann-Hurwitz]

If \(f:C\to D\) is a map of smooth complete curves then
\begin{align*}
2g(C) - 2 = \deg(f)\cdot\qty{ 2g(D) - 2 } + \deg R(f)
\end{align*}
where \(R(f) = \sum_{p \in C} (e_p - 1)[p]\) is the ramification
divisor.

\end{theorem}

\begin{example}[?]

Take
\begin{align*}
{\mathbf{C}}&\to {\mathbf{C}}\\
z &\mapsto z^{4}
\end{align*}
which is a 4-fold cover

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-02_09-43.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{example}

\begin{example}[?]

If \(f: C\to {\mathbf{P}}^1\) is degree 2 then \(f=a/b\) where
\(a,b\in H^0(C; K_C)\), and Riemann-Hurwitz gives \(\deg R(f) = 6\). If
\({\left[ {s: t} \right]}\) are homogeneous coordinates on
\({\mathbf{P}}^1\), then one can take an equation of the form
\(z^2 = f_6(s, t)\) for \(f_6(s,t) = \prod_{i=1}^6 (a_i s - b_i t)\) is
a homogeneous degree 6 polynomial, so
\(f\in H^0({\mathbf{P}}^1;{\mathcal{O}}_{{\mathbf{P}}^1}(6))\) and
\(z\in { \operatorname{Tot} }{\mathcal{O}}_{{\mathbf{P}}^1}(-3)\):

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-02_09-59.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{example}

\begin{exercise}[?]

Describe why
\(z\not\in { \operatorname{Tot} }{\mathcal{O}}_{{\mathbf{P}}^1}(3)\)
instead.

\end{exercise}

\begin{exercise}[?]

Recall that the normalization of a ring \(R\) is the integral closure of
\(R\) in \(\operatorname{ff}(R)\). Compute the normalization of
\(y^2=x^3\) using the algebraic definition.

\end{exercise}

\begin{example}[?]

An example of normalization:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-02_10-22.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{example}

\begin{fact}

Integrally closed and 1-dimensional implies smooth.

\end{fact}

\begin{exercise}[?]

Recall the definition of the Čech cochain complex and compute
\({\check{H}}(S^1; {\mathcal{F}})\) using an open cover of two sets.

\end{exercise}

\hypertarget{tuesday-february-07}{%
\section{Tuesday, February 07}\label{tuesday-february-07}}

\begin{remark}

Last time: Čech cochains defined as
\(C_{{\mathcal{U}}}^p(X; {\mathcal{F}}) \coloneqq\bigoplus_{i_0 < \cdots < i_p } {\mathcal{F}}(U_{i_0,\cdots, i_p})\)
with homology defined as
\({\check{H}}^*(X;{\mathcal{F}}) \coloneqq\colim_{{\mathcal{U}}} H^*(C_{\mathcal{U}}^*(X; {\mathcal{F}}))\),
and a SES of sheaves yields a LES in homology. Recall
\(\operatorname{Cl}(X) { \, \xrightarrow{\sim}\, }\operatorname{Pic}(X)\)
in our setting via \(D\mapsto {\mathcal{O}}_X(D)\) which sends \(U\) to
\(\left\{{f\in k(U) {~\mathrel{\Big\vert}~}\operatorname{div}f + D \geq 0}\right\}\).

\end{remark}

\begin{claim}

\begin{align*}
\operatorname{Pic}(X) & { \, \xrightarrow{\sim}\, }
H^1(X; {\mathcal{O}}_X^{\times}) \\
L &\mapsto [(t_{UV})]
.\end{align*}

\end{claim}

\begin{proof}[?]

Write
\(C_{{\mathcal{U}}}^1(X; {\mathcal{O}}_X^{\times}) = \left\{{ ( t_{UV} ) \in {\mathcal{O}}^{\times}( U \cap V) {~\mathrel{\Big\vert}~}U,V \in {\mathcal{U}}}\right\}\)
for some open cover \({\mathcal{U}}\rightrightarrows X\), and
\(Z_{{\mathcal{U}}}^1(X; {\mathcal{O}}_X^{\times}) = \left\{{(t_{UV}) \in {\mathcal{O}}^{\times}(U \cap V)}\right\}\)
with boundary \({{\partial}}^1(t_{UV}) = (t_{VW} t_{UW}^{-1}t_{UV})\).
Note that \(t_{UW} = t_{UV} t_{VW}\).

A line bundle \(L \xrightarrow{\pi} X\) has a trivialization, and we can
refine \({\mathcal{U}}\) to a cover that trivializes \(L\), so
\({ \left.{{L}} \right|_{{U}} } \cong {\mathcal{O}}_U\) as sheaves for
each \(U\). We have
\(\pi^{-1}(U) { \, \xrightarrow{\sim}\, }U\times {\mathbf{C}}\), and
\(h_V\circ h_U^{-1}: (U \cap V) \times {\mathbf{C}}\to (U \cap V)\times {\mathbf{C}}\)
is multiplication by \(t_{UV}(p)\) on the fiber over \(p \in U \cap V\).
These satisfy \(t_{UV} t_{VW} = t_{UW}\), so any \(L\) defines an
element of \(Z^1(X; {\mathcal{O}}_X^{\times})\) by sending \(L\) to its
transition function.

Conversely, given \((t_{UV})\), one can attempt to glue these to form a
line bundle, but which collections define the same bundle? Given two
line bundles, refine their trivializing covers so that they coincide.
Then any two trivializations
\(L_U \xrightarrow{h_U, h_{U}'} {\mathcal{O}}_U\) differ by an element
\(f_U \in {\mathcal{O}}^{\times}(U)\):

\begin{center}
\begin{tikzcd}
    {L_U} && {{\mathcal{O}}_U} \\
    \\
    && {{\mathcal{O}}_U}
    \arrow["{h_U}", from=1-1, to=1-3]
    \arrow["{\cdot f_U}", from=1-3, to=3-3]
    \arrow["{h_U'}"', from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJMX1UiXSxbMiwwLCJcXE9PX1UiXSxbMiwyLCJcXE9PX1UiXSxbMCwxLCJoX1UiXSxbMSwyLCJcXGNkb3QgZl9VIl0sWzAsMiwiaF9VJyIsMl1d}{Link
to Diagram}
\end{quote}

On overlaps, we have \(t_{UV}\mapsto f_U^{-1}t_{UV}\) and
\(t_{VU} \mapsto f_U t_{VU}\), so at the level of tuples,
\((t_{UV}) \mapsto (t_{UV}) \cdot {\partial}^0 (f_U)\) and thus
\((t_{UV})\) is uniquely defined up to
\({\partial}^0(C_{\mathcal{U}}^0(X; {\mathcal{O}}_X^{\times}))\).

\end{proof}

\begin{remark}

This works for higher rank vector bundles: one has
\(t_{VW} t_{UV} = t_{UW}\) in
\(\mathop{\mathrm{Hol}}(U \cap V \cap W, \operatorname{GL}_n({\mathbf{C}}))\),
however for \(n\geq 2\) this is a nonabelian group and order matters. In
this case we get
e.g.~\({\partial}^1(t_{UV}) = ( t_{UV} t{UW}^{-1}t_{VW} )\). One has
\begin{align*}
\left\{{\text{Vector bundles on } X}\right\}  { \, \xrightarrow{\sim}\, }H^1(X; \operatorname{GL}_n({\mathcal{O}}))
\end{align*}
where
\(\operatorname{GL}_n({\mathcal{O}})(U) = \left\{{\text{holomorphic functions } U\to \operatorname{GL}_n({\mathbf{C}})}\right\}\).

\end{remark}

\begin{remark}

Recall the exponential SES; taking the LES yields
\(c_1: \operatorname{Pic}(X) \to H^2(X; {\mathbf{Z}})\). For \(X\) a
smooth curve, \(c_1 = \deg\). For \(D\in \operatorname{Div}(X)\), one
can define the fundamental class in \(X\) by taking the fundamental
class
\([D]\in H_{\dim_{\mathbf{R}}D}(D; {\mathbf{Z}}) { \, \xrightarrow{\sim}\, }H_{\dim_{\mathbf{R}}D} (X; {\mathbf{Z}}) \underset{\mathrm{PD}}{ { \, \xrightarrow{\sim}\, }} H^2(X; {\mathbf{Z}})\).

\end{remark}

\begin{remark}

Why is \(c_1 = \deg\) true? Consider
\({\mathcal{L}}= {\mathcal{O}}_C(p)\) for \(p\in C\) a point on a curve.
One can take the \emph{point bundle construction}: let \(U\ni p\) be a
neighborhood of \(p\) and \(V\) the complement of a smaller neighborhood
of \(p\), so \(U \cap V\) is an annulus. For \(z: U\to {\mathbf{C}}\) a
local coordinate, one can form a Cartier divisor
\(\left\{{(U, z), (V, 1)}\right\}\) with transition function
\(t_{UV} = 1/z\). Note that
\(H^0({\mathcal{O}}_C(p)) \ni s = (s_U = z, s_V = 1)\) has a section
which vanishes precisely at \(p\).

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-07_10-19.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Refine the open cover to split \(U\) into two open subsets, then
\begin{align*}
c_1({\mathcal{O}}_C(p)) = ((z^{-1})_{U_1V}, (z^{-1})_{U_2V}, (1)_{U_1 U_2} ) \in Z^1(C; {\mathcal{O}}^{\times})
.\end{align*}
Lifting to \(C^1(C; {\mathcal{O}})\) using that exponential surjects on
sheaves yields \((-\log(z), -\log(z), 0) \in C^1(C; {\mathcal{O}})\).
Taking its boundary yields
\begin{align*}
{\partial}^1( (-\log z)_{U_1 V}, (-\log z)_{U_2 V}, (0)_{U_1 U_2} ) = ( (-\log z)_{U_2 V} + (\log z)_{U_i V} )_{U_1 U_2 V}
\end{align*}
which is \(0\) on the top component of \(U_1 \cap U_2 \cap V\) and
\(2\pi i\) on the bottom. This is an element of
\(\underline{2\pi i {\mathbf{Z}}}( U_1 \cap U_2 \cap V) \in Z^2(C; \underline{2\pi i {\mathbf{Z}}}) \to H^2(X; \underline{{\mathbf{Z}}} )\).
Thus \(c_1 ({\mathcal{O}}_C(p)) = [p]\) is the fundamental class of
\(p\).

\end{remark}

\begin{definition}[?]

\begin{align*}
{\operatorname{NS}}(X) \coloneqq\operatorname{im}c_1, \quad
\operatorname{Pic}^0(X) \coloneqq\ker c_1
.\end{align*}

\end{definition}

\begin{example}[?]

For \(X = E\) an elliptic curve,
\(\operatorname{Pic}X = E\times {\mathbf{Z}}\) where
\(D \mapsto (D, \deg D)\). Thus
\({\operatorname{NS}}(E) = {\mathbf{Z}}\) and
\(E = \operatorname{Pic}^0(E) = \left\{{[p] - [0] {~\mathrel{\Big\vert}~}p\in E}\right\}\).
Note that \(\operatorname{Pic}^0(X) \cong \operatorname{Jac}(X)\) in
this case.

\end{example}

\begin{remark}

If \(X\) is smooth projective, global holomorphic functions are
constant, so part of the LES breaks into an exact piece:
\begin{align*}
H^0({\mathbf{Z}}) = {\mathbf{Z}}^n \hookrightarrow H^0({\mathcal{O}}) = {\mathbf{C}}^n \twoheadrightarrow H^0({\mathcal{O}}^{\times}) = ({\mathbf{C}}^{\times})^n \qquad n = {\sharp}\pi_0 X
.\end{align*}
Thus
\(\operatorname{Pic}^0(X) = H^1(X; {\mathcal{O}})/ H^1(X; {\mathbf{Z}})\),
and Hodge theory shows
\(\operatorname{rank}H^1(X; {\mathbf{Z}}) = 2\dim_{\mathbf{C}}H^1(X; {\mathcal{O}})\)
and the image of \(H^1(X;{\mathbf{Z}})\to H^1(X; {\mathcal{O}})\) is
discrete. This yields
\({\mathbf{Z}}^{2r} \hookrightarrow{\mathbf{C}}^r\) with image
\(\Lambda\) a lattice and
\(\operatorname{Pic}^0 X \cong {\mathbf{C}}^r/\Lambda\). In particular,
for \(C\) a smooth genus \(g\) curve,
\(\operatorname{Pic}^0 X \cong {\mathbf{C}}^g/ \Lambda\). Note that
\(H^1(C; {\mathbf{Z}})\) carries the intersection pairing, which induces
a symplectic form and thus a polarization.

\end{remark}

\hypertarget{thursday-february-09}{%
\section{Thursday, February 09}\label{thursday-february-09}}

\begin{remark}

Last time:
\(\operatorname{Pic}(X) \cong H^1(X; {\mathcal{O}}_X^{\times})\) and
\begin{align*}
c_1: \operatorname{Pic}(X) &\to H^2(X; {\mathbf{Z}}) \\
{\mathcal{O}}(D) &\mapsto [D]
\end{align*}
with \(\operatorname{im}c_1 = {\operatorname{NS}}(X)\) and
\(\ker c_1 = \operatorname{Pic}^0(X) \cong {\mathbf{C}}^g/ \Lambda\)
where \(H^1(X; {\mathcal{O}}) \cong {\mathbf{C}}^g\). Today: consider
the cohomology of vector bundles on a complex manifold \(X\).

\end{remark}

\begin{definition}[$(p,q)\dash$forms]

A \textbf{smooth \((p, q){\hbox{-}}\)form} is locally of the form
\begin{align*}
\sum_{ {\left\lvert {I} \right\rvert} = p, {\left\lvert {J} \right\rvert} = q} a_{I, \overline{J}} \,dz_{i_1}\wedge\cdots \,dz_{i_p} \wedge \,d\overline{z}_{j_1}\wedge \cdots \wedge \,d\overline{z}_{j_q}
.\end{align*}
Let \(A^{p, q}\) be the sheaf of smooth \((p, q){\hbox{-}}\)forms

\end{definition}

\begin{example}[?]

Some examples:

\begin{itemize}
\tightlist
\item
  \(A^{0, 1}({\mathbf{C}}) \ni \omega \coloneqq z\overline{z} \,d\overline{z}\).
\item
  \(A^{1, 0}({\mathbf{C}}^{\times}) \ni \alpha \coloneqq\log{\left\lvert {z} \right\rvert} \,dz\)
\item
  \(A^{1,1}({\mathbf{C}}^2)\ni \alpha \coloneqq e^{z_1}\,dz_1 \wedge \,d\overline{z}_2 + \overline{z}_2 \,d\overline{z}_1 \wedge \,dz_2\).
\item
  An example of differentiation:
  \(d(e^{z_2} \,dz_1 + \,d\overline{z}_1 \,dz_2) = e^{z_2} \,dz_2\wedge \,dz_1 + \,d\overline{z}_1 \wedge \,dz_2\).
\end{itemize}

\end{example}

\begin{remark}

Let \(\Omega^{p}\) be the holomorphic \((p, 0)\) forms, noting that
differentiation \(d\) on smooth forms is not a map of
\(C^\infty(X, {\mathbf{C}}){\hbox{-}}\)modules since
\(d(f\alpha) = fd(\alpha) + df\wedge \alpha\) for
\(f \in C^\infty(U, {\mathbf{C}})\) and \(\alpha A^k(U)\). There is a
decomposition
\begin{align*}
A^k(U) = \bigoplus _{p+q=k} A^{p, q}(U)
,\end{align*}
which leads to a decomposition of sheaves. Define
\({\partial}= \pi_{p+1, q}(d)\) and
\({ \overline{{\partial}}}: \pi_{p, q+1}(d)\), this yields a complex
\begin{align*}
0 \to {\mathcal{O}}_X \to A^{0,0} \xrightarrow{{ \overline{{\partial}}}} A^{0, 1} \xrightarrow{{ \overline{{\partial}}}} A^{0, 2} \to \cdots \to A^{0, \dim X} \to 0
,\end{align*}
which is an exact sequence of sheaves. Noting that \(d^2 = 0\), one has

\begin{itemize}
\tightlist
\item
  \({\partial}^2 = 0\),
\item
  \({ \overline{{\partial}}}^2 = 0\),
\item
  \({\partial}{ \overline{{\partial}}}+ { \overline{{\partial}}}{\partial}= 0\).
\end{itemize}

See the Poincaré \({ \overline{{\partial}}}\) lemma.

\end{remark}

\begin{remark}

More generally, for a vector bundle
\(E \in {}_{{\mathcal{O}}_X}{\mathsf{Mod}}\), note that
\({\mathcal{O}}\hookrightarrow A^{0, 0}\) yields
\({\mathcal{O}}\hookrightarrow{\mathbf{C}}^\infty\), so can form
\(E\otimes_{{\mathcal{O}}} C^\infty\). Locally,
\(E\cong {\mathcal{O}}{ {}^{ \scriptscriptstyle\oplus^{r} } }\) on
\(U\), so one has
\(E\otimes A^{0,0} \cong (C^\infty){ {}^{ \scriptscriptstyle\oplus^{r} } }\)
on \(U\). This yields \(0\to E\hookrightarrow E \otimes A^{0,0}\), and
the claim is that there is a well-defined map
\(E\otimes A^{0,0}\to E\otimes A^{0, 1}\). Locally this is given by
\({\left[ {f_1,\cdots, f_r} \right]} \mapsto {\left[ {{ \overline{{\partial}}}f_1,\cdots, { \overline{{\partial}}}f_r} \right]}\).
In a different trivialization, \(s_V = t_{UV}(f_1,\cdots, f_r)\) where
\(t_{UV}\) is a holomorphic function valued in
\(\operatorname{GL}_r({\mathbf{C}})\). One has
\({ \overline{{\partial}}}(t_{UV} \circ (f_1,\cdots, f_r)) = t_{UV} ({ \overline{{\partial}}}f_1, \cdots, { \overline{{\partial}}}f_r)\)
since \({ \overline{{\partial}}}(t_{UV}) = 0\), noting that in the first
expression one is carrying out matrix multiplication.

\end{remark}

\begin{definition}[Dolbeault complex]

\begin{align*}
0 \to E \xrightarrow{i} E\otimes A^{0,0} \xrightarrow{{ \overline{{\partial}}}} E\otimes A^{0, 1} \xrightarrow{{ \overline{{\partial}}}} \cdots \xrightarrow{{ \overline{{\partial}}}} E\otimes A^{0, \dim X} \to 0
,\end{align*}
and \(H^*(X; E)\) can be computed from the homology of this complex.

\end{definition}

\begin{fact}

For any smooth vector bundle \(V\to M\) over a manifold \(M\),
\({\check{H}}^{i \geq 1}(M; V) = 0\) since \(M\) admits partitions of
unity. Moreover if
\({\mathcal{F}}\hookrightarrow{ {I}_{\scriptscriptstyle \bullet}}\) with
\({ {I}_{\scriptscriptstyle \bullet}}\) acyclic, so
\(H^{i\geq 1}({ {I}_{\scriptscriptstyle \bullet}} ) = 0\), then
\(H^k(X; F)\) is computed as the homology of
\({ {I}_{\scriptscriptstyle \bullet}}\).

\end{fact}

\begin{remark}

Since \(\Omega^p\) is a holomorphic vector bundle on \(X\), this yields
the Dolbeault resolution
\begin{align*}
0\to \Omega^p \to A^{p, 0} \to A^{p, 1} \to \cdots
,\end{align*}
and \(H^{p, q} \coloneqq H^q(X; \Omega^p)\) is the homology of this
complex. Define \(h^{p, q} \coloneqq\dim_{\mathbf{C}}H^{p, q}\) -- note
that this forms a diamond since for \(p,q\geq \dim X\) there are no
\(p{\hbox{-}}\)forms or \(q{\hbox{-}}\)forms whatsoever.

\end{remark}

\begin{theorem}[Hodge decomposition and symmetry theorems]

There is a decomposition
\begin{align*}
H^k(X; \underline{{\mathbf{C}}}) \cong \bigoplus _{p+q=k} H^{p, q}
,\end{align*}
and a symmetry
\begin{align*}
H^{p, q}(X) \cong \overline{H^{q, p}(X)}
.\end{align*}

\end{theorem}

\begin{remark}

Note that \(\overline{V}\) doesn't necessarily make sense yet for
\(V = H^{p, q}(X)\), since we don't know that it is a subspace of some
real vector space \(W\) -- here we'll take
\(H^k(X; \underline{{\mathbf{C}}}) = H^k(X; \underline{{\mathbf{R}}}) \otimes{\mathbf{C}}\).
E.g. writing
\({\mathbf{C}}^2 = {\mathbf{R}}^2 \otimes_{\mathbf{R}}{\mathbf{C}}\), if
\(V = \left\langle{u, v}\right\rangle_{\mathbf{C}}\) then
\(\overline{V} = \left\langle{\overline{u}, \overline{v}}\right\rangle_{\mathbf{C}}\).

\end{remark}

\hypertarget{tuesday-february-14}{%
\section{Tuesday, February 14}\label{tuesday-february-14}}

\begin{proposition}[Riemann-Roch for curves]

\begin{align*}
h^0(C, L) - h^0(C, K_C \otimes L^{-1}) = \deg L + 1-g
.\end{align*}

\end{proposition}

\begin{remark}

Recall:

\begin{itemize}
\tightlist
\item
  \(h^i(X, F) \coloneqq\dim_k H^i(X; F)\).
\item
  \(\chi(X, F) \coloneqq\sum_{i\geq 0}h^i(X, F)\) for
  \(F\in {\mathsf{Sh}}(X, {}_{k}{\mathsf{Mod}})\), provided these
  numbers are finite.
\item
  \(H^0({\mathbf{A}}^1_{/ {k}}; {\mathcal{O}}) = k[x] t^0\), and note
  \(\dim_k k[x] = \infty\).
\item
  For \(X\) an irreducible Noetherian topological space with
  \(\dim X = d\), \(H^i(X, F) = 0\) for \(i > d\).
\item
  For \(X\in \mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\) and \(F\) a
  finitely presented \({\mathcal{O}}_X{\hbox{-}}\)module, i.e.~there is
  an exact sequence
  \({\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{m} } }\to {\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{n} } }\twoheadrightarrow F\),
  we have \(h^i(X, F) < \infty\).
\item
  Finitely presented sheaves are coherent. An analytic coherent sheaf is
  defined in the same way with respect to
  \({\mathcal{O}}_X^{\mathrm{an}}\) (the sheaf of holomorphic
  functions).
\item
  \(h^0(X, K_C\otimes L^{-1}) = h^1(C, L)\).
\end{itemize}

\end{remark}

\begin{theorem}[Serre duality]

Let \(X\) be a compact complex manifold and let \(E\to X\) be a
holomorphic vector bundle. Then
\(H^i(X, E) { \, \xrightarrow{\sim}\, }H^{\dim_{\mathbf{C}}X - i}(X, E {}^{ \vee }\otimes K_X) {}^{ \vee }\)
where \(K_X = \operatorname{det}\Omega_X \coloneqq\Omega_X^{\dim X}\).

\end{theorem}

\begin{proof}[?]

Regard \(s\in H^i(X, E)\) as an element in Dolbeault cohomology,
\begin{align*}
H^i(X, E) \cong { \ker \qty{ E\otimes A^{0, i}(X) \xrightarrow{{ \overline{{\partial}}}} A^{0, i+1}(X) } \over \operatorname{im}\qty{E\otimes A^{0, i-1}(X) \xrightarrow{{ \overline{{\partial}}}} E\otimes A^{0, i}(X) } }
.\end{align*}
Note that \(K \otimes_{\mathcal{O}}C^\infty = A^{n, 0}\). Let

\begin{itemize}
\tightlist
\item
  \(t\in H^{n-i}(X, E {}^{ \vee }\otimes K)\)
\item
  \(\tilde s\in E\otimes A^{0, i}(X)\)
\item
  \(\tilde t\in E {}^{ \vee }\otimes K \otimes A^{0, n-i}(X) \cong E {}^{ \vee }\otimes A^{n, n-i}(X)\).
\end{itemize}

One can then pair
\({\left\langle {\tilde s},~{\tilde t} \right\rangle}\in A^{n, n}(X)\),
and
\(\int_X {\left\langle {\tilde s},~{\tilde t} \right\rangle} \in {\mathbf{C}}\)
is a perfect pairing.

\end{proof}

\begin{remark}

Upshot: the LHS in RR is \(\chi(C, L)\).

\end{remark}

\begin{proposition}[?]

The following is an important exact sequence of sheaves: for any
\(D\in \operatorname{CDiv}(X)^\mathsf{eff}\), one has
\begin{align*}
{\mathcal{O}}_X(-D) \hookrightarrow{\mathcal{O}}_X \twoheadrightarrow{\mathcal{O}}_D \qquad \in {\mathsf{Sh}}(X)
\end{align*}
where \({\mathcal{O}}_D \coloneqq\iota_* {\mathcal{O}}_X\) for
\(\iota: D\hookrightarrow X\) the inclusion.

\end{proposition}

\begin{proof}[?]

Note
\({\mathcal{O}}_X(D)(U) = \left\{{f\in {\mathcal{O}}_(U) {~\mathrel{\Big\vert}~}\operatorname{div}f \geq D}\right\}\),
so \({\mathcal{O}}_X(D) = I_D\) is the ideal sheaf of \(D\). If \(D\) is
cut out by a single function on \(U\), we have
\(I_D(U) = (f) \subset {\mathcal{O}}_X(U)\). This yields an inclusion
\({\mathcal{O}}_X(-D) = I_D \hookrightarrow{\mathcal{O}}_X\). By
definition, the quotient \({\mathcal{O}}_X/I_D\) are functions defined
on \(D\), at least on affine opens \(U\). Since exactness of sheaves is
local, this check suffices.

\end{proof}

\begin{remark}

In particular, on a curve one has
\begin{align*}
{\mathcal{O}}_C(-p) \hookrightarrow{\mathcal{O}}_C \twoheadrightarrow{\mathcal{O}}_p
\end{align*}
where \(p\in C\) is a point. This can be tensored with any vector bundle
\(L\) to get
\begin{align*}
L(-p) \coloneqq L\otimes{\mathcal{O}}_C(-p) \hookrightarrow L \twoheadrightarrow{ \left.{{L}} \right|_{{p}} }
,\end{align*}
which is exact since \(L\) is locally free. For
\(s_p\in H^0({\mathcal{O}}_C(p))\), we have \(V(s_p) = [p]\) as a
divisor:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-14_10-11.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{remark}

\begin{proposition}[?]

For \(F_1\hookrightarrow F_2 \twoheadrightarrow F_3\),
\begin{align*}
\chi(F_2) = \chi(F_1) + \chi(F_3)
.\end{align*}

\end{proposition}

\begin{proof}[?]

Take the LES in cohomology, where \(H^n(F_i) = 0\) for large enough
\(n\). Now for a LES of vector spaces
\(V_1\hookrightarrow V_2\to \cdots \twoheadrightarrow V_n\), one has
\(\sum (-1)^i \dim_k V_i = 0\).

\end{proof}

\begin{claim}

\({ \left.{{L}} \right|_{{p}} } \cong {\mathcal{O}}_p\) satisfies
\(H^i(C, {\mathcal{O}}_p) = {\mathbf{C}}t^0\), which is more generally
true for a skyscraper sheaf at a point.

\end{claim}

\begin{remark}

Take a fine enough open cover (e.g.~an affine cover) so that \(p\)
appears in only one set, and use Čech cohomology. That
\({ \left.{{L}} \right|_{{p}} } \cong {\mathcal{O}}_p\) follows from the
fact that this holds on a small enough open \(U\) and both are
identically zero away from \(p\).

\end{remark}

\begin{remark}

Now use that \(\chi(C, {\mathcal{O}}_p) = 1\), we then claim that
\(\chi(L) = \chi(L(-p)) + 1\) and thus
\(L\cong {\mathcal{O}}_C(\sum n_p [p])\) and
\(\chi(L) = \sum n_p + \chi({\mathcal{O}}_C)\) by repeatedly applying
this fact. Note \(\sum n_p = \deg L\), so
\begin{align*}
\chi(L) = \deg L + \chi({\mathcal{O}}_C)
.\end{align*}
We have
\(\chi({\mathcal{O}}_C) = h^0({\mathcal{O}}_C) - h^1({\mathcal{O}}_C) = 1- h^0(K)\)
by Serre duality. Applying the above version of RR to \(K\) yields
\(\chi(K) = \deg K + \chi({\mathcal{O}}) = 2g-2 + \chi({\mathcal{O}})\).
On the other hand, this equals
\(h^0(K) - h^0(K {}^{ \vee }\otimes K) = h^0(K) - 1\). Combining these
yields
\begin{align*}
\chi({\mathcal{O}}) = -(2g-2 + \chi({\mathcal{O}})) \implies 2\chi({\mathcal{O}}) = 2-2g \implies \chi({\mathcal{O}}) = 1-g
.\end{align*}
Plugging this back into the first equation yields
\begin{align*}
\chi(L) = \deg L + 1-g
.\end{align*}

\end{remark}

\begin{remark}

Note that \(\chi^{\mathsf{Top}}(C) = g\) was defined as the index of a
vector field, and this shows that also \(g = h^1({\mathcal{O}}_X)\).

\end{remark}

\begin{remark}

An application of Serre duality: the Hodge diamond. Let
\(n\coloneqq\dim_{\mathbf{C}}X\) and recall
\(h^{p, q} \coloneqq\dim_{\mathbf{C}}H^q(X, \Omega^p_X)\). By duality,
\(h^{p, q} = \dim_{\mathbf{C}}H^{n-q}(X, (\Omega_X^p) {}^{ \vee }\otimes\Omega^n)\).
We first claim
\begin{align*}
(\Omega^p_X) {}^{ \vee }\otimes\Omega^n \cong \Omega^{n-p}
.\end{align*}
Note that sections of \(\Omega^p\) are of the form
\(\sum a_I \,dz_{i_1}\wedge \cdots \wedge \,dz_{i_p}\) with \(a_I\)
holomorphic functions on \(U\), and sections of
\((\Omega^p) {}^{ \vee }\) look like
\(\sum a_I {\frac{\partial }{\partial z_{i_1}}\,} \wedge \cdots \wedge {\frac{\partial }{\partial z_{i_p}}\,}\)
whose transition functions are the inverse of those for \(\Omega^p\).
Noting that \(\Omega^n\) is a line bundle with local sections of the
form \(f \,dz_1\wedge\cdots \,dz_n\), one can contract forms (interior
multiplication) to obtain
\begin{align*}
\qty{\sum a_I {\frac{\partial }{\partial z_I}\,}} \otimes\qty{f \,dz_I} = f \sum_{j\in I^c} \,dz_{j_1}\wedge\cdots \wedge \,dz_{j_{n-p}} 
.\end{align*}
Thus \(h^{p, q} = \dim H^{n-q}(X, \Omega^{n-p}) = h^{n-p, n-q}\).

\begin{center}
\begin{tikzcd}
    && {h^{n,n}} \\
    & {h^{n-q, n-p}} && {h^{n-p, n-q}} \\
    {h^{n, 0}} &&&& {h^{0, n}} \\
    & {h^{p, q}} \\
    && {h^{0,0}} \\
    && {}
    \arrow[dashed, no head, from=1-3, to=5-3]
    \arrow[curve={height=-12pt}, Rightarrow, no head, from=4-2, to=2-4]
    \arrow[dashed, no head, from=3-1, to=3-5]
    \arrow[Rightarrow, no head, from=2-4, to=2-2]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

This yields \(K_4\) symmetry, and we'll see that for a Calabi Yau there
is a \(D_4\) symmetry.

\end{remark}

\hypertarget{tuesday-february-21}{%
\section{Tuesday, February 21}\label{tuesday-february-21}}

\begin{remark}

Last time: \(M\) a compact manifold, \(\dim_{\mathbf{R}}M = 2n\), there
is a perfect pairing
\begin{align*}
H^k(M;{\mathbf{Z}})/{\operatorname{tors}}\otimes_{\mathbf{Z}}H^k(M; {\mathbf{Z}})/{\operatorname{tors}}&\to {\mathbf{Z}}\\
\alpha \otimes\beta &\mapsto \int_M \alpha\vee\beta
.\end{align*}
Interpret \(\alpha . \beta\) as
\([N_1] . [N_2] = \sum_{p\in N_1 \pitchfork N_2} \pm 1\) where the
\(N_i\) are Poincaré duals. Satisfies
\(\alpha . \beta= (-1)^{n} \beta \alpha\) for
\(\alpha, \beta\in H^{n}(X; {\mathbf{Z}})/{\operatorname{tors}}\) where
\(n\coloneqq\dim_{\mathbf{C}}M = {1\over 2}\dim_{\mathbf{R}}M\).

\end{remark}

\begin{definition}[Lattices]

An (orthogonal) lattice is a free abelian group
\(\Lambda \cong {\mathbf{Z}}^k\) of finite rank, together with an
integral symmetric bilinear form
\begin{align*}
\cdot: \Lambda\otimes\Lambda\to {\mathbf{Z}}
.\end{align*}

\begin{itemize}
\item
  \(\Lambda\) is \textbf{symplectic} if \(\cdot\) is alternating, so
  \(\alpha.\beta = -\beta . \alpha\).
\item
  \(\Lambda\) is \textbf{unimodular} if for all primitive nonzero
  vectors \(x\in \Lambda\), \(\exists y\in \Lambda\) such that
  \(x.y = 1\), where \(x\) is primitive if \(x\neq \lambda z\) for any
  \(z\in \Lambda\)
\item
  \(\Lambda\) is nondegenerate if \(\forall x\in \Lambda\),
  \(\exists y \in \Lambda\) with \(x.y \neq 0\).
\item
  The \textbf{Gram matrix} of a basis \(\left\{{e_i}\right\}\) for a
  lattice \((\Lambda, \cdot)\) is \(M_{ij} \coloneqq e_i . e_j\). \(M\)
  is symmetric for orthogonal lattices and skew-symmetric for symplectic
  lattices. One can write \(v.w = v^t M w\).
\item
  If \(\Lambda_i\) are lattices, so is \(\bigoplus_{i} \Lambda_i\).
\end{itemize}

\end{definition}

\begin{example}[?]

The standard example: \(\Lambda \in {\mathbf{Z}}^2\) with
\({\left[ {x_1, y_1} \right]}.{\left[ {x_2, y_2} \right]} \coloneqq x_1 x_2 + y_1 y_2\)
is nondegenerate and unimodular. Here
\({\left[ {2,2} \right]} = 2{\left[ {1,1} \right]}\) is not primitive,
but \({\left[ {2,1} \right]}\) is primitive. Proving unimodularity: if
\(\gcd(m, n) = 1\), one just needs to solve \(mx + ny = 1\) for
\({\left[ {x,y} \right]}\in {\mathbf{Z}}^2\). This has Gram matrix
\({ \begin{bmatrix} {1} & {0} \\ {0} & {1} \end{bmatrix} }\).

\end{example}

\begin{example}[?]

A degenerate lattice:
\(({\mathbf{Z}}^2, \mathbf{x} . \mathbf{y} \coloneqq x_1 x_2)\). This is
symmetric, but \(\mathbf{x} . {\left[ {0, 1} \right]} = 0\) for every
\(\mathbf{x} \in \Lambda\). The Gram matrix is
\({ \begin{bmatrix} {1} & {0} \\ {0} & {0} \end{bmatrix} }\).

\end{example}

\begin{example}[?]

A symplectic lattice:
\(({\mathbf{Z}}^2, \mathbf{x} . \mathbf{y} \coloneqq x_1 y_2 - x_2 y_1)\),
which has Gram matrix
\({ \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} }\).

\end{example}

\begin{example}[?]

There is a \(2g{\hbox{-}}\)dimensional symplectic lattice for every
\(g\geq 0\) given by
\({\mathbf{Z}}^{2g}_\mathrm{symp} \coloneqq\bigoplus_{i=1}^{g} ({\mathbf{Z}}^2, \cdot_\mathrm{symp})\),
which has Gram matrix comprised of \(g\) diagonal blocks of
\({ \begin{bmatrix} {0} & {1} \\ {-1} & {0} \end{bmatrix} }\). This is
the only unimodular symplectic lattice up to isomorphism, and is the
intersection form on \(H^1(\Sigma_g; {\mathbf{Z}})\). The vectors here
can be represented by fundamental classes of 1-dimensional submanifolds,
i.e.~real curves:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-21_10-03.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

More generally, if \(\dim_{\mathbf{R}}M = 4k+2\), then
\(H^{2k+1}(M; {\mathbf{Z}}) \cong {\mathbf{Z}}^{2g}_\mathrm{symp}\) for
some \(g\).

\end{example}

\begin{definition}[Orthogonal complements]

For \(M \subseteq \Lambda\), define
\(M^\perp \coloneqq\left\{{x\in \Lambda{~\mathrel{\Big\vert}~}x.m =0 \, \forall m\in M}\right\}\).

\end{definition}

\begin{example}[?]

For
\({\mathbf{Z}}^2_\mathrm{symp} = {\mathbf{Z}}\alpha \oplus {\mathbf{Z}}\beta\),
we have \({\mathbf{Z}}\alpha^\perp = {\mathbf{Z}}\alpha\). For
\({\mathbf{Z}}^2_\text{std}= {\mathbf{Z}}\alpha \oplus {\mathbf{Z}}\beta\),
one instead has \({\mathbf{Z}}\alpha^\perp = {\mathbf{Z}}\beta\).

\end{example}

\begin{remark}

Over \({\mathbf{R}}\), symmetric bilinear forms are classified by their
signature: they can all be diagonalized to
\(\operatorname{diag}(1,\cdots, 1, -1,\cdots,-1,0,\cdots, 0)\) for some
multiplicities \(n_+, n_-, n_0\) where
\(n_+ + n_- + n_0 = \operatorname{rank}_{\mathbf{Z}}\Lambda\). Any
lattice can be extended via
\(\Lambda_{\mathbf{R}}\coloneqq\Lambda \otimes_{\mathbf{Z}}{\mathbf{R}}\),
so define \(\operatorname{sgn}\Lambda \coloneqq(n_+, n_-, n_0)\).

\end{remark}

\begin{example}[?]

Let
\(\Lambda = {\mathbf{Z}} { \left[ \scriptstyle {\sqrt{5}} \right] }\)
and let \(u.v = \Re(x\overline{y})\), so
\begin{align*}
{\left\lVert {u} \right\rVert} = u.u = (a+b\sqrt{5})(a-b\sqrt{5}) = a^2 - 5b^2
.\end{align*}
Note that \(\Lambda_{\mathbf{R}}\cong {\mathbf{R}}^{1,1}\), which in the
standard basis has Gram matrix
\({ \begin{bmatrix} {1} & {0} \\ {0} & {-1} \end{bmatrix} }\). This can
be visualized as a 2-dimensional subspace of \({\mathbf{R}}^2\) spanned
by \(1, \sqrt{5}\). Note that
\({\left\lVert {\sqrt 5} \right\rVert} = -5\).

\end{example}

\begin{remark}

Define the \emph{hyperbolic signature} as \((1, n)\) for any
\(n\geq 0\). One can visualize positive/negative norm vectors using the
light cone: for \({\mathbf{R}}^{1, n}\), solving \(v.v=0\) to get
\(x_1^2 = x_2^2 + \cdots + x_{n_1}^2\). This is a cone over \(S^1\) at
height 1 in \({\mathbf{R}}^3\):

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-21_10-27.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Note that \(\left\{{v.v > 0}\right\}\) has two connected components.

\end{remark}

\begin{definition}[Hyperbolic planes]

A \emph{hyperbolic cell/plane} is the lattice \(H\) defined as
\({\mathbf{Z}}^2\) with pairing given by the Gram matrix
\({ \begin{bmatrix} {0} & {1} \\ {1} & {0} \end{bmatrix} }\) and
signature \((1, 1)\). This admits an orthonormal basis
\({e_1 + e_2\over \sqrt 2}, {e_1 - e_2 \over \sqrt 2}\), and
\(H_{\mathbf{R}}\cong {\mathbf{R}}^{1, 1}\).

\end{definition}

\begin{remark}

Recall the ADE Dynkin diagrams:

\includegraphics{figures/2023-02-21_10-32-44.png}

One can build a root lattice out of each diagram:

\begin{itemize}
\tightlist
\item
  Take one basis vector \(e_i\) for each node,
\item
  \(e_i^2 = -2\),
\item
  \(e_i.e_j = 1 \iff\) nodes \(i, j\) are connected, and zero otherwise.
\end{itemize}

The lattices will be negative definite, i.e.~of signature \((0, n)\)
with \(n\) the number of nodes. The only unimodular such lattice
corresponds to \(E_8\).

\end{remark}

\begin{example}[?]

\(A_1\) corresponds to the matrix \([-2]\), and thus \({\mathbf{Z}}\)
with bilinear form \(-2n^2\). \(A_2\) yields
\({ \begin{bmatrix} {-2} & {1} \\ {1} & {-2} \end{bmatrix} }\), and
\(A_3\) yields
\({ \begin{bmatrix}  {-2} & {1} & {0} \\  {1} & {-2} & {1} \\  {0} & {1} & {-2} \end{bmatrix} }\).

\end{example}

\begin{question}

How does one check that a lattice is unimodular?

\end{question}

\begin{definition}[?]

Let \((\Lambda, \cdot)\) be a nondegenerate lattice, so
\(\Lambda\hookrightarrow\Lambda_{\mathbf{R}}\cong {\mathbf{R}}^{a, b}\).
Define
\begin{align*}
\Lambda {}^{ \vee }\coloneqq\left\{{ y\in \Lambda_{\mathbf{R}}{~\mathrel{\Big\vert}~}x.y\in {\mathbf{Z}}\, \forall x\in \Lambda}\right\}
.\end{align*}

\end{definition}

\begin{example}[?]

Consider
\(\Lambda \coloneqq\sqrt{2}{\mathbf{Z}}\hookrightarrow{\mathbf{R}}^1\)
with the standard pairing, then
\(1\over \sqrt{2}\in \Lambda {}^{ \vee }\).

\end{example}

\begin{remark}

One can always find a basis of \(\Lambda {}^{ \vee }\) given by
\(e_i {}^{ \vee }\) where \(e_i {}^{ \vee }e_j = \delta_{ij}\). Since
\(e_i {}^{ \vee }M e_j = \delta_{ij}\) for \(M\) the Gram matrix of a
form, one finds that \(e_i {}^{ \vee }\) is the \(i\)th row of
\(M^{-1}\). Why: letting \(N\) be the matrix with rows
\(e_i {}^{ \vee }\), one has \(NM = I\).

\end{remark}

\begin{proposition}[?]

\(\Lambda\) is unimodular iff \(\Lambda {}^{ \vee }= \Lambda\).

\end{proposition}

\begin{proof}[?]

Note \(x^2\in {\mathbf{Z}}\) by definition, so
\(\Lambda \subseteq \Lambda {}^{ \vee }\). If
\(v\in \Lambda {}^{ \vee }\setminus\Lambda\), then one can show that the
minimal \(n\) such that \(nv\in \Lambda\) yields a primitive element of
\(\Lambda\). Since \(Nv.w \in n{\mathbf{Z}}\) for all \(w\), so can't
pair to 1.

\end{proof}

\begin{remark}

So
\(\Lambda {}^{ \vee }\coloneqq\bigoplus {\mathbf{Z}}e_i {}^{ \vee }\subseteq \Lambda \implies e_i {}^{ \vee }\in \Lambda \implies M^{-1}\in \operatorname{GL}_n({\mathbf{Z}})\),
and applying the same argument to duals yields
\(\operatorname{det}M = \pm 1\). In general,
\(\operatorname{det}M = {\sharp}( \Lambda {}^{ \vee }/ \Lambda)^2\) is
the covolume. So

\begin{itemize}
\tightlist
\item
  \(\operatorname{vol}({\mathbf{R}}^n / \Lambda) = \operatorname{det}M\)
\item
  \(\operatorname{vol}({\mathbf{R}}^n / \Lambda {}^{ \vee }) = \operatorname{det}M^{-1}\),
  which is the Gram matrix of \(\Lambda {}^{ \vee }\).
\item
  \({\sharp}(\Lambda {}^{ \vee }/ \Lambda)^2 = \operatorname{covol}( \Lambda)^2/ \operatorname{covol}( \Lambda {}^{ \vee })^2 = \operatorname{det}(M)^2\).
\end{itemize}

\end{remark}

\hypertarget{tuesday-march-14}{%
\section{Tuesday, March 14}\label{tuesday-march-14}}

\begin{remark}

Today: Hirzebruch-Riemann-Roch and Chern classes. Let \(E\to X\) be a
smooth \({\mathbf{C}}{\hbox{-}}\)vector bundle, then
\(\exists f: Y\to X\) such that \(f^* E\) splits (as a smooth
\({\mathbf{C}}{\hbox{-}}\)vector bundle) into a direct sum of line
bundles,
i.e.~\(f^* E { \, \xrightarrow{\sim}\, }\bigoplus _{i\leq r} L_i\) where
\(r\coloneqq\operatorname{rank}E\). One can ensure that
\(f^*: H^*(X; {\mathbf{Z}})\hookrightarrow H^*(Y; {\mathbf{Z}})\) is
injective.

This allows us to build \(c_k(E)\) and thus \(c_k(E)\) from the Chern
roots \(x_i \coloneqq c_1(L_i)\) of \(E\). Set
\(c_k(f^* E) \coloneqq\sigma_k(x_1,\cdots, x_r)\), the \(k\)th
elementary symmetric polynomial in the \(x_i\). Note that
\(\prod_{i\leq r}(t + x_i) = \sum_{i\leq r} s_{r-i} x^i\) where e.g.~

\begin{itemize}
\tightlist
\item
  \(s_1(x_1,\cdots, x_r) = \sum x_i\)
\item
  \(s_2(x_1,\cdots, x_r) = \sum_{i < j} x_i x_j\)
\item
  \(s_r(x_1,\cdots, x_r) = \prod x_i\)
\end{itemize}

Define
\begin{align*}
c(E, t) \coloneqq\prod_{i\leq r} (t+x_i),\qquad c(E) \coloneqq c(E, 1) = \sum c_i
\end{align*}
where \(c_i = H^{2i}(Y; {\mathbf{Z}})\), and set
\(f^*(c_k E) \coloneqq c_k (f^* E)\); this uniquely defines \(c_k(E)\).

\end{remark}

\begin{remark}

Note on proving the splitting principle: for
\(A\hookrightarrow B\twoheadrightarrow C\) smooth vector bundles,
putting a Hermitian metric on \(B\) yields \(A^\perp\) in \(B\) and thus
a (smooth) splitting. Set
\(Y\coloneqq\left\{{x\in X{~\mathrel{\Big\vert}~}V_1 \subseteq V_2 \subseteq \cdots \subseteq V_r = E_x,\, \dim_{\mathbf{C}}V_i = i}\right\}\),
where \(f: Y\to X\) by forgetting the flag. Then
\(\dim Y = \dim X + \dim \operatorname{Fl}({\mathbf{C}}^r)\), then
\(f^* E\) admits a filtration \(F^i\) where \(F^1\) is a line bundle.
This yields SESs \(F^{i-1} \hookrightarrow F_i \twoheadrightarrow L_r\)
which split.

\end{remark}

\begin{remark}

Define the total Chern character as
\(c(E) \coloneqq\sum_{i\leq r} e^{x_i}\). Note that
\({\mathbf{C}}[t_1,\cdots, t_r]^{S_r} = {\mathbf{C}}[s_1, \cdots, s_r]\),
and
\begin{align*}
\sum_{i\leq r} \qty{\sum_{k\geq 0} { x_i^k \over k!} } 
&= r + (x_1+\cdots + x_r) + \qty{{x_1^2\over 2} + \cdots + {x_r^2\over 2}} + \cdots \\
&= r + c_1 + \qty{{c_1^2\over 2} - c_2} + \cdots
,\end{align*}
noting that e.g.~\(c_1^2 = \sum_i x_i^2 + 2 \sum_{i < j} x_i x_j\).
Define the total Todd class as
\begin{align*}
\operatorname{Td}(E) \coloneqq\prod_i {1\over 1-e^{-x_i}} = 1 + {c_1\over 2} + {c_1^2 + c_2 \over 12} + {c_1c_2 \over 24}
.\end{align*}
Note this is holomorphic at each \(x_i\) by L'Hopital, and moreover
symmetric, and each term is a generating function for Bernoulli numbers.

\end{remark}

\begin{remark}

Recall that RR says that for a holomorphic line bundle of \(L\), one can
compute \(\chi(L)\) in terms of \(\deg L = \int c_1(L)\), a purely
topological invariant. The following theorem generalizes this:

\end{remark}

\begin{theorem}[Hirzebruch-Riemann-Roch]

Let \(E\to X\) be a holomorphic vector bundle over a compact complex
manifold. Defining \(\chi(E) \coloneqq\sum (-1)^i h^0(X; E)\),
\begin{align*}
\chi(E) = \int_X \mathrm{Chern}(E) \operatorname{Td}({\mathbf{T}}_X )
\end{align*}
where the multiplication is in \(H^*(X;{\mathbf{Z}})\), noting that both
classes are supported in \(H^\text{even}(X;{\mathbf{Z}})\) and the
integration means taking the top degree part in
\(H^{2\dim_{\mathbf{C}}X}(X; {\mathbf{Z}}) \cong {\mathbf{Z}}\).

\end{theorem}

\begin{remark}

Recovering RR: for \(L\to X\) a line bundle on a curve, one has
\(\mathrm{Chern}(L) = e^{c_1(L)} = 1 + c_1(L)\) and
\(\operatorname{Td}(T) = {c_1(T) \over 1 - e^{-c_1(T)}} = 1 + {c_1(T)\over 2}\).
Thus
\begin{align*}
\chi(L) = \int_X (1 + c_1(L) )\qty{1 + {c_1(T)\over 2}} = \int_X c_1(L) + {c_1(T)\over 2} = \deg L + {1\over 2}\int_X c_1(T)
.\end{align*}
Note that \(c_1(T)\) is the fundamental class of the zeros of some
section of \(T\), i.e.~the number of zeros of a vector field, which by
Chern-Gauss-Bonnet yields \(\int_X c_1(T) = \chi_{\mathsf{Top}}(X)\).
Thus
\begin{align*}
\chi(L) = \deg L + {1\over 2}\chi_{\mathsf{Top}}(X)
.\end{align*}
For a curve, \(\chi_{\mathsf{Top}}(X) = 2-2g\), so
\(\chi(L) = \deg L + (1-g)\).

\end{remark}

\begin{remark}

Let \(E\to S\) now be a line bundle over a surface, then
\(\chi(L) = h^0(L) - h^1(L) + h^2(L)\) since \(L\) is a coherent sheaf
and \(\dim X = 2\). HRR yields
\begin{align*}
\chi(L) = \int_S \qty{1 + c_1(L) + {c_2(L)\over 2!} }\qty{1 + {c_1(T)\over 2} + { c_1(T)^2 + c_2(T) \over 12}}
.\end{align*}
First consider the special case \(L = {\mathcal{O}}_S\) so
\(c_1(L) = 0\) and
\(\chi({\mathcal{O}}_S) = \int c_1({\mathbf{T}}_S)^2 + c_2({\mathbf{T}}_S)\).
By the splitting principle, if \({\mathbf{T}}_S = L_1 \oplus L_2\) then
\(\operatorname{det}{\mathbf{T}}_S = L_1 \otimes L_2\), so
\begin{align*}
c_1({\mathbf{T}}_S) = c_1(\operatorname{det}{\mathbf{T}}_S) = c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2) = x_1 + x_2
.\end{align*}
Note also that
\(c_1(\operatorname{det}{\mathbf{T}}_S) = c_1(-K_S {}^{ \vee }) = -c_1(K_S)\)
and so \(c_1({\mathbf{T}}_S) = K_S^2\).

For the second term, note that the top Chern class of \(E\) is always
the fundamental class of \(V(s)\) for \(s\) a generic smooth section of
\(E\). In particular,
\(\int_S c_2({\mathbf{T}}_S) = \chi_{\mathsf{Top}}(S)\) is the number of
zeros of a smooth vector field. This yields \textbf{Noether's formula}
\begin{align*}
\chi({\mathcal{O}}_S) = {1\over 12}\qty{K_S^2 + \chi_{\mathsf{Top}}(S)}
.\end{align*}

\end{remark}

\begin{remark}

Let \(X = {\mathbf{P}}^2\), so \(K = {\mathcal{O}}(-3)\) and
\(\chi_{\mathsf{Top}}(X) = 3\), so \(K^2 = (-3H)^2 = 9H^2 = 9\) so
\begin{align*}
\chi({\mathcal{O}}_{{\mathbf{P}}^2}) = {1\over 12}\qty{ 9 + 3} = 1
.\end{align*}
Since \(h^0({\mathcal{O}}) = 1\) and
\(h^1({\mathcal{O}}) = {1\over 2}\beta_1 = 0\), this yields
\(h^2({\mathcal{O}}) = 0\).

\end{remark}

\begin{remark}

RR for surfaces:
\begin{align*}
\chi(S, L) = \chi({\mathcal{O}}_S) + {1\over 2}(c_1({\mathbf{T}}_S) c_1(L)) + {1\over 2}c_1(L))^2
.\end{align*}
Reworking this, note \(c_1({\mathbf{T}}_S) = -c_1(K_S)\) and thus
\begin{align*}
\chi(S, L) = \chi({\mathcal{O}}_S) + {1\over 2}\qty{L^2 - L\cdot K_S}
.\end{align*}

\end{remark}

\begin{remark}

Computing the Hodge diamond of a K3: recall \(h^1(X) = 0\) and
\(K_X = {\mathcal{O}}_X\).

\begin{itemize}
\tightlist
\item
  \(H^0: 1\).
\item
  \(H^1: (0, 0)\).
\item
  \(H^2: (1, N, 1)\) by Serre duality for some \(N\).
\item
  \(H^3: (0, 0)\) by Poincare duality.
\item
  \(H^4: 1\) by Poincare duality.
\end{itemize}

Computing \(N\): \(\chi({\mathcal{O}}) = {1\over 12}(K^2 + \chi(X))\)
where \(\chi({\mathcal{O}}) = 1-0+1 = 2\) and \(K^2 = 0\), this yields
\(\chi(X) = 24\) and \(N = 22\).

\end{remark}

\hypertarget{thursday-march-16}{%
\section{Thursday, March 16}\label{thursday-march-16}}

\begin{definition}[Linear systems]

For \(X\) smooth projective, \(L\in \operatorname{Pic}(X)\),
\(V \subseteq H^0(L)\), define \({\mathbf{P}}V\) to be a linear system
(a collection of linearly equivalent divisors) and
\({\left\lvert {L} \right\rvert} \coloneqq{\mathbf{P}}H^0(L)\) to be a
complete linear system. If \(s\in H^0(L)\setminus\left\{{0}\right\}\),
then \(V(s)\in \operatorname{Div}(X)\) and \({\mathcal{O}}(V(s)) = L\)
(noting that all sections are linearly equivalent). Here we projectivize
since \(V( \lambda s) = V(s)\).

\end{definition}

\begin{example}[?]

Let \(X = {\mathbf{P}}^1\) and \(L = {\mathcal{O}}(1)\), then
\begin{align*}
H^0(L) = \left\{{ f\in k[x,y] {~\mathrel{\Big\vert}~}f \text{ homogenous },\, \deg f = n}\right\}
.\end{align*}
As divisors,
\begin{align*}
{\mathbf{P}}H^0({\mathcal{O}}(1)) = \left\{{\sum a_i p_i {~\mathrel{\Big\vert}~}a_i > 0,\, \sum a_i = n}\right\}
,\end{align*}
corresponding to the zeros and multiplicities of \(f\).

\end{example}

\begin{example}[?]

Let \(E = {\mathbf{C}}/ \Lambda\) be elliptic and
\(L = {\mathcal{O}}(D)\) for \(D = 3[0]\). Then
\({\left\lvert {L} \right\rvert} = \left\{{ [p] + [q] + [r] {~\mathrel{\Big\vert}~}p+q+r =0 \operatorname{mod}\Lambda }\right\}\).
Note that \({\left\lvert {L} \right\rvert} \cong {\mathbf{P}}^2\) since
\(r\) is determined by \(p, q\).

\end{example}

\begin{example}[?]

Let \(C\) be a curve with \(g\geq 2\) and \(L = K_C = \Omega^1_C\). Then
by RR \(h^0(K_C) = g\) and
\({\left\lvert {K_C} \right\rvert} \cong {\mathbf{P}}^{g-1}\) is called
the \textbf{canonical linear system}.

\end{example}

\begin{definition}[?]

Let \(V \leq H^0(L)\) be a subspace and
\(\left\{{s_0, \cdots, s_k}\right\}\) be a basis. Then there is a map
\begin{align*}
\phi_V: X &\dashrightarrow{\mathbf{P}}^k \\
x &\mapsto [s_0(x):\cdots : s_k(x)]
.\end{align*}
Defining the \textbf{base locus} as
\(\operatorname{Bs}(L) \coloneqq\left\{{x\in X{~\mathrel{\Big\vert}~}s_i(x) = 0 \, \forall i}\right\}\),
note \(\phi_{{\mathbf{P}}V}\) is not well-defined for any
\(x\in \operatorname{Bs}(L)\).

\end{definition}

\begin{proposition}[?]

For \(C\) a curve of \(g\geq 2\),
\(\operatorname{Bs}(K_X) = \emptyset\).

\end{proposition}

\begin{proof}[?]

STS \(\forall p\in C\) there is some \(s\in H^0(K_C)\) with
\(s(p) \neq 0\). Letting \(s_p\) be a section vanishing only at \(p\),
multiplication by \(s_p\) induces
\(H^0(K_C(-p)) \hookrightarrow H^0(K_C)\) with image the sections of
\(K_C\) vanishing at \(p\). Thus STS this is not surjective by showing
\(h^0(K_C) > h^0(K_C(-p))\). Apply RR and Serre duality:
\begin{align*}
h^0(K_C(-p)) - h^1(K_C(-p))
&= h^0(K_C(-p)) - h^0({\mathcal{O}}(p))\\
&= \deg K_C(-p) + (1-g)\\
&= (2g-3) + (1-g)\\
&= g-2
.\end{align*}
Note that if \(s\in H^0({\mathcal{O}}(p))\) then \(V(s) = [q]\) must be
a single point, but if \(p\neq q\) then \([p]-[q] = 0\) and
\(\exists f: C\to {\mathbf{P}}^1\) with \(f^{-1}(0) = p\) and
\(f^{-1}(\infty) = q\) with \(\deg f = 1\), forcing
\(C \cong {\mathbf{P}}^1\) and contradicting \(g \geq 2\). So \(p=q\),
and \(h^0({\mathcal{O}}(p)) = 1\), and thus
\begin{align*}
h^0(K_C(-p)) - 1 = g-2 \implies h^0(K_C(-p)) = g - 1 < g = h^0(K_C)
.\end{align*}

\end{proof}

\begin{exercise}[?]

Show that if \(C\) is not hyperelliptic
(\(\exists f: C\to {\mathbf{P}}^1\) 2-to-1) then \(\forall p,q\in C\)
one can find \(s \in H^0(K_C)\) with \(s(p) = 0, s(q)\neq 0\), so they
are separated by linear forms on \({\mathbf{P}}^{g-1}\). This yields an
actual morphism
\(\phi_{{\left\lvert {K_C} \right\rvert}}: C\to {\mathbf{P}}^{g-1}\)
where \(p, q\) are not mapped to the same point. This is the canonical
embedding of a curve, which only works when \(g\geq 3\) and \(C\) is
non-hyperelliptic. If \(g = 2\) or \(C\) is hyperelliptic, \(3K_C\)
yields an (tricanonical) embedding.

\end{exercise}

\begin{example}[?]

If \(C\) is not hyperelliptic and \(g = 3\), then
\(C\hookrightarrow{\mathbf{P}}^2\) by the canonical embedding. This
yields an element in \(\operatorname{Div}({\mathbf{P}}^2)\), which is a
smooth quartic.

\end{example}

\begin{proposition}[Canonical of $\PP^n$.]

\begin{align*}
K_{{\mathbf{P}}^n} = {\mathcal{O}}(-n-1)
.\end{align*}

\end{proposition}

\begin{proof}[?]

Take coordinates \([x_0:\cdots : x_n]\) and take
\(\omega \coloneqq{\,dx_1\over x_1} \wedge \cdots \wedge {\,dx_n\over x_n}\),
noting that this omits \(x_0\). This has poles along each \(V(x_i)\) for
\(i\neq 0\) and in fact a pole at \(x_0\), since these are \(n+1\)
distinct spaces. E.g. for \(n=1\), since \(x_1 = x_0^{-1}\) we have
\({ d(x^{-1})\over x^{-1}} = - {\,dx_1\over x_1}\).

\end{proof}

\begin{proposition}[Adjunction]

If \(X\) is smooth and \(D\in \operatorname{Div}(X)\) then
\begin{align*}
K_D = { \left.{{\qty{K_X \otimes{\mathcal{O}}(D)}}} \right|_{{D}} }
.\end{align*}

\end{proposition}

\begin{proof}[?]

Omitted, take residues.

\end{proof}

\begin{remark}

Applying this to the previous curve situation: note
\(\deg K_C = 2g-2 = 4\), which counts
\(\phi_{{\left\lvert {K_C} \right\rvert}}(C) \cap V(x_0)\) and yields a
quartic.

\end{remark}

\begin{proposition}[?]

For \(C\) a degree \(d\) curve in \({\mathbf{P}}^2\),
\begin{align*}
g = {d-1\choose 2}
.\end{align*}

\end{proposition}

\begin{proof}[?]

Take degrees in the adjunction formula and apply Bezout's theorem:
\begin{align*}
K_{C} = K_{{\mathbf{P}}^2} + C \mathrel{\Big|}_C \implies
2g(C) - 2 = = \deg(-3H + dH)\mathrel{\Big|}_C = d (d-3)
.\end{align*}
More generally if \(C \subseteq S\) a curve in a surface,
\(\deg L \mathrel{\Big|}_C = c_1(L) . [C]\) is an intersection number.
Expanding this yields \(g = {d^2 - 3d +2 \over 2} = {d-1\choose 2}\).

\end{proof}

\begin{definition}[Kodaira dimension]

\begin{align*}
\kappa(X) \coloneqq\max_{n > 0} \left\{{\dim \operatorname{im}\phi_{nK_X} }\right\}
\end{align*}
where \(\dim \emptyset \coloneqq-\infty\) and
\(\kappa(X) \in \left\{{-\infty,0,\cdots, \dim X}\right\}\).

\end{definition}

\begin{remark}

Note that since \(3K_C\) yields an embedding for a curve with
\(g\geq 2\), \(\kappa(C) = 1\). Also note that
\(\kappa(X) = -\infty \iff h^0(nK_X) = 0\) for all \(n\). For curves:

\begin{longtable}[]{@{}ll@{}}
\toprule
\(g(C)\) & \(\kappa(C)\) \\
\midrule
\endhead
\(0\) & \(-\infty\) \\
\(1\) & \(0\) \\
\(2\) & \(1\) \\
\(3\) & \(1\) \\
\(4\) & \(\vdots\) \\
\bottomrule
\end{longtable}

Here we've used that \(K_{{\mathbf{P}}^1} = {\mathcal{O}}(-2)\) has no
sections and \(K_E = {\mathcal{O}}_E\) is trivial. Fanos are
\(\kappa = -\infty\) and general type (\(\kappa(X) = \dim X\)) are
\(g\geq 2\).

\end{remark}

\begin{remark}

For smooth projective surfaces:
\(\kappa(S) \in \left\{{ -\infty,0,1,2 }\right\}\). See Beauville for
the Enriques-Kodaira classification due to the Italian school:

\begin{itemize}
\tightlist
\item
  \(\kappa(S) = -\infty\): ruled and rational.
\item
  \(\kappa(S) = 0\): K3, Abelian, Enriques, Bi-elliptic.
\item
  \(\kappa(S) = 1\): Elliptic.
\item
  \(\kappa(S) = 2\): General type.
\end{itemize}

\end{remark}

\begin{definition}[?]

\(S\) is \textbf{ruled} if \(\exists \pi: S\to C\) with generic fiber
\(\cong {\mathbf{P}}^1\):

\includegraphics{figures/2023-03-16_10-50-54.png}

\(S\) is \textbf{rational} if
\(S\overset{\sim}{\dashrightarrow}{\mathbf{P}}^2\).

\end{definition}

\addsec{ToDos}
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\addsec{Theorems}
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\addsec{Exercises}
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\printbibliography[title=Bibliography]


\end{document}
