# Tuesday, January 10

:::{.remark}
References:

- Beauville, "Complex Algebraic Surfaces"
- Huybrechts, "Lectures on K3 Surfaces"
- Gathmann, "Algebraic Geometry" (2002)

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:::{.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 $\CC^2$).
These can be studied topologically or using algebraic geometry.

A rough plan:

- Review algebraic varieties:
  - Riemann-Roch,
  - Curves,
  - Divisors,
  - Line bundles,
  - Picard group,
  - The canonical bundle.
- Complex analytic tools:
  - The exponential exact sequence,
  - Betti numbers,
  - Topological Euler characteristic.
- K3s:
  - Examples of K3s,
  - Enriques-Kodaira classification,
  - The intersection form.
- Hodge theory:
  - Periods, etc.

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:::{.remark}
Recall the definition of an affine variety, e.g. $V(x_1, x_2) \subseteq \AA^2\slice k$ the union of the coordinate axes, or the cone $V(x_1^2-x_2^2-x_3^2) \subseteq \AA^3\slice k$.
Alternatively, view them as schemes: $X = V(f_1,\cdots, f_m) = \spec k[X]$ where $k[X] \da k[x_1,\cdots, x_n]/\gens{f_1, \cdots, f_m}$ 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 $\gens{x_1-a_1,\cdots, x_n-a_n}$, this is a bijection by the Nullstellensatz.
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:::{.remark}
An example of why schemes are useful: consider $V(y)$ and $V(y-x^2)$ in $\AA^2\slice k$.
The set-theoretic intersection is $X\da V(y, y-x^2) = \ts{0} \in \AA^2$, but note that $k[X] = [x,y]/\gens{y, x^2} \neq k = k[x,y]/\gens{x,y}$, 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.
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:::{.remark}
Recall that varieties carry the Zariski topology: the closed sets are of the form $V_X(I)$ for $I \normal k[X]$.
From a scheme-theoretic perspective, 
\[
V(I) \da \ts{\text{prime ideals } p\in X \st p \contains I}
.\]
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:::{.exercise title="?"}
Consider $X\da V(xy) \subseteq \AA^2$, one has $k[X] = k[x,y]/\gens{xy}$ and $I=\gens{y-x-1}$ corresponding to the line $y=x+1$.
What are the closed sets?
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:::{.remark}
Note that $\AA^1\slice \CC$ with the Zariski topology differs from $\AA^1\slice \CC$ with the analytic topology.
The closed sets are of the form $V(I)$, and since $\CC[x]$ has GCDs every ideal is principal and $I = \gens{f} \subseteq k[X]$ 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 $\CC$.
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:::{.remark}
Affine varieties/schemes form a category
and there is an equivalence $\opcat{\Aff\Sch} \iso \CRing$.
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:::{.example title="?"}
An example of a morphism:
\[
\phi: \AA^1 &\to \AA^2 \\
t &\mapsto (t^2, t^3)
.\]
This induces a map on regular functions $\phi^*: \CC[\AA^2]\to \CC[\AA^1]$ which is of the form
\[
\phi^*: \CC[x,y] &\to \CC[t] \\
x &\mapsto t^2 \\
y&\mapsto t^3
.\]
One could similarly define $\phi$ with codomain $V(y^2-x^3)$.

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:::{.remark}
What are the regular functions on *open* sets?
Let $U \subseteq X$ in the Zariski topology, then regular functions on $U$ are ratios $f/g$ of polynomials.
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:::{.example title="?"}
Let $U \da \AA^1\smts{0, 1} \subseteq \AA^1$, then regular functions include ${1\over x}$ and ${1\over x-1}$.
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:::{.remark}
Recall the definition of a sheaf; we'll write $\OO_X$ for the structure sheaf and regard $\OO_X(U)$ as the $k\dash$algebra of functions on $U$, satisfying the sheaf axioms of existence and uniqueness of gluing.
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:::{.remark}
Write $\OO_\CC$ for the sheaf of **regular** functions, then e.g. $\OO_\CC(\ts{a_1,\cdots, a_n}^c) = \CC[x]\adjoin{{1\over x-a_1}, \cdots, {1\over x-a_n}}$, and more generally $\OO_X(V(f)^c) = \CC[x]\adjoin{1\over f}$.
We'll sometimes distinguish $\OO_\CC^{\hol}$ which is defined on $X^\an$ instead (in the Euclidean topology), which is a priori different as a ringed space.
Later we'll use this in the exponential exact sequence 
\[
2\pi i \ul{\ZZ} \to \OO &\mapsvia{\exp} \OO\units \\
f&\mapsto e^f
.\]
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:::{.example title="?"}
Schemes are useful in number theory: consider $X \da \spec \ZZ$, then $\OO_X(X) = \ZZ$.
There is a point $p$ for every prime, and a generic point $0$.
Note that e.g. $20\in \OO_X(X)$ can be regarded as a function on $\spec \ZZ$, and $V(20) \da \ts{p \st p\contains\gens{20}}$.
It contains $2$ and $5$, but contains 2 more!
So one might draw its "graph" in the following way:

\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}

Moreover one has $\OO_{\spec \ZZ}(\ts{2,3}^c) = \ts{f/g \st f,g\in \ZZ, g = 2^a3^b}$.
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:::{.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 $\AA^n$ or $\RR^n$, we can require $t_{UV} = \phi_V \circ \phi_U\inv \mid_{\phi_U(U \intersect V)}$ be continuous, smooth, holomorphic, etc.
For schemes, the gluing will be by regular maps, e.g. $\PP^1 = \AA_1 \Disjoint_{t\mapsto s = {1\over t} } \AA_1$ where $t,s$ are the coordinates on each factor.
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