Lecture given by Evgeny Shinder, notes taken by Edward Pearce, 30/09/2019, 3pm-4:30pm at The University of Sheffield, based on Daniel Huybrechts' book Lectures on K3 surfaces.
A (generalized) Calabi-Yau variety is a variety \(X\) with a non-vanishing global holomorphic top-form \(\sigma\in\Omega_{X}^{n}(X)\), where \(\Omega_{X}^{1}=T^{\ast}\) is the cotangent bundle and \(\Omega_{X}^{j}=\bigwedge^{j}\Omega_{X}^{1}\), and \(n=\dim X\)
More generally, for any \(X\), let \(\sigma\) be a meromorphic top-form on \(X\), and let
\[\mathrm{div}(\sigma) = \sum_{D\subset X}\mathrm{ord}_{D}(\sigma)\cdot[D]\in\mathrm{Div}(X)\]
Where the sum is over irreducible divisors \(D\) in \(X\). Then \(\mathrm{div}(\sigma)\) is called a canonical divisor.
If \(\sigma'\) is another form, then \(\mathrm{div}(\sigma') - \mathrm{div}(\sigma)\) is a principal divisor, i.e. a divisor of a meromorphic function. Hence \(\mathrm{div}(\sigma)\) is well-defined in \(\mathrm{Pic}(X)=\mathrm{Div}(X)/\{\text{principal divisors}\}\). We write \(K_{X}=\mathrm{div}(\sigma)\).
The sheaf (or bundle) \(\omega_{X}=\Omega_{X}^{n}\) is called the canonical line bundle.
In this language \(X\) is Calabi-Yau if and only if \(K_{X}=0\in\mathrm{Pic}(X)\) if and only if \(\omega_{X}\cong\mathcal{O}_{X}\).
Let \(X\) be a curve and \(g=g(X)\) be its geometric genus. Then \(\Omega_{X}^{1}=\omega_{X}\), \(\Gamma(X, \Omega_{X}^{1})=\mathbb{C}^{3}\), and \(\deg K_{X}=2g-2\).
There are two types of CY surfaces: abelian surfaces and K3 surfaces.
A K3 surface is a CY surface with \(H^{1}(S, \mathbb{Z})=0\) (in this setting we can replace the last condition by \(\pi_{1}(S)=0\) simply connected, or \(H^{1}(S, \mathbb{Z})=0\).
Horizontal rows in the bottom half describe the decomposition \(H^{m}(X, \mathbb{C})=\bigoplus_{p=0}^{m} H^{p, m-p}(X)\) where \(m\) runs from \(0\) to \(n\).
\[[[h^{n,n}],\ldots, [h^{0,n}, h^{1,n-1}, \ldots, h^{n-1,1}, h^{n,0}], \ldots, [h^{0,1}, h^{1,0}], [h^{0,0}]]\]
For \(X\) a curve of genus \(g\), the Hodge diamond is \([[1], [g, g], [1]]\)
For \(X\) a connected surface, the Hodge diamond is \([[1], [q, q], [p, h, p], [q, q], [1]]\) where \(q=h^{1,0}\) is the number of 1-forms on \(X\) and \(p=h^{2,0}\) is the number of 2-forms on \(X\) and \(h=h^{1,1}\).
For \(A=E\times E'\), \([[1],[2,2],[1,4,1],[2,2],[1]]\).
For \(S\) a K3 surface, \([[1],[0,0],[1,20,1],[0,0],[1]]\).
For \(X=\mathbb{P}^{2}\), \([[1],[0,0],[0,1,0],[0,0],[1]]\).
For \(X=\mathbb{P}^{1}\times\mathbb{P}^{1}\), \([[1],[0,0],[0,2,0],[0,0],[1]]\).
To show that \(h^{1,1}=20\) for a K3 surface, we need to apply Noether's formula.
For example, for \(X=\mathbb{P}^{2}\), \(\chi(X, \mathcal{O}_{X})=1\), \(K_{X}=-3H\) where \(H\) is the class of a line, \(K_{X}^{2}=9\), and \(\chi_{\text{top}}(X)=3\).
Exercise: Show that \(h^{1,1}=20\) for a K3 surface.
Exercise: Verify the adjunction formula for \(\mathbb{P}^{n-1}\subset\mathbb{P}^{n}\).
For \(X\subset\mathbb{P}^{2}\) a degree \(d\) curve, \(\omega_{X}=\mathcal{O}(-3-d)\).
Exercise: Deduce the genus-degree formula \(g=\frac{(d-1)(d-2)}{2}\)
For \(S\subset\mathbb{P}^{3}\) a degree \(d\) surface, \(\omega_{S}=\mathcal{O}(-4-d)\). Hence \(S\) is Calabi-Yau if and only if \(d=4\), \(\mathcal{O}(1)=\mathcal{O}(H)\), and \(H^{1}(S, \mathcal{O}_{S})=0\) by cohomology long exact sequence.
Exercise: Classify all K3 surfaces which are complete intersections of hypersurfaces. That is, \(S=H_{1}\cap\ldots\cap H_{r}\subset\mathbb{P}^{r+2}\), where \(H_{i}\) is a hypersurface of degree \(d_{i}\). Hint: Induction on adjunction to get equations for \(d_{1},\ldots,d_{r}\).
Given \(S\subset\mathbb{P}^{N}\), let \(H\in\mathrm{Pic}(S)\) be the hyperplane section divisor. Such an \(H\) is called a polarization of \(S\).
Let \(S\subset\mathbb{P}^{3}\) be a quartic surface, and \(H\subset S\) a quartic curve obtained as \(H=S\cap\mathbb{P}^{2}\subset\mathbb{P}^{3}\), then \(H^{2}=4\).
More generally, a polarization is the class of an ample divisor.
Let \(S\to\mathbb{P}^{2}\) be a 2-to-1 cover ramified in a degree 6 curve (sextic), then Riemann-Hurwitz formula and the vanishing of \(H^{1}(S, \mathcal{O}_{S})\) implies that \(S\) is a K3 surface, since \(K_{S}=\mathcal{O}(-3)\bigotimes\mathcal{O}(\frac{6}{2})=\mathcal{O}(S)\).
A complex-analytic K3 surface is a compact analytic surface such that \(\omega_{S}\cong\mathcal{O}_{S}\) and \(H^{1}(S, \mathcal{O}_{S})=0\).
Remark: There exist many more analytic K3 surfaces than algebraic K3 surfaces.
A complex torus \(A=\mathbb{C}^{2}/\Lambda\), \(\Lambda\cong\mathbb{Z}^{4}\). Note: most of these are not algebraic.
Let \(S'A/\pm\), i.e. quotient out 2-torsion points, that is, points (equivalence classes) for which \(\bar{z}=-\bar{z}\), or equivalently \(2\bar{z}=0\). This has 16 ordinary double points. Let \(S\to S'\) be the blow up of these ordinary double points, then \(S\) has \(\omega_{S}=0\) and \(H^{1}(S, \mathcal{O}_{S})=0\), so \(S\) is a K3 surface.
Hence \(S\) is algebraic if and only if \(A\) is algebraic, where algebraic means it can be embedded in \(\mathbb{P}^{N}\) for some \(N\). This \(S\) is called a Kunicov K3 surface.
The same definition over an arbitrary field \(k\) gives K3 surfaces over \(k\). Usually we will work with complex algebraic projective K3 surfaces.
Let \(\rho=\mathrm{rank}\,\mathrm{Pic}(S)\) noting that \(\mathrm{Pic}(S)\) is a torsion-free finitely generated abelian group.
Starting from \(S\), consider the bounded derived category \(\mathcal{D}^{b}(S)\) of \(S\) and the maps to: