# Andreas Malmendier, On 2-elementary K3 surfaces and string dualities (Friday, July 22)

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For certain families of lattice polarized K3s (with 3 conditions) there exists a projective normal form.
This has something to do with heterotic string duality.
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Setup: $X$ a comple K3, $\NS(X) = H^{1, 1} \intersect H^2$ the Néron-Severi lattice, a sublattice of $\HH^3 \oplus E_8(-1)^{\oplus 2}$.
A $\Lambda\dash$lattice polarization is $\iota: \Lambda \injects \NS(X)$ whose image contains an ample class?
Two notions of equivalence: isomorphisms between K3s inducing a homothety $\beta\in \OO(\Lambda)$.
Torelli theorems: $\mcm_\Lambda$ is a moduli space, $\dcosetl{\Gamma_\Lambda}{\Omega_{\Lambda} }$ classifies something.
For $\Lambda = \HH$ (rank 2 hyperbolic), an $H\dash$polarization is equivalent to admitting a Jacobian elliptic fibration (elliptic fibration with section) $X \mapsvia{\phi} \PP^1$, so the K3 has a nice Weierstrass model.
One can interpret $\mcm_\HH$ as the coarse moduli of Jacobian elliptic K3s, an 18-dimensional quasiprojective variety.
Kodaira classification (well-known) describes singular fibers in terms of ADE singularities.
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Similar story for $\Lambda = H \bigoplus N$ for $N$ a Nikulin lattice corresponds to fibrations with an order 2 section.
There is a Mordell-Weil group of such sections.
Nikulin construction: $X$ carries a canonical symplectic involution $\iota$, so take $\hat{X/\iota}$ is an $\HH \bigoplus N\dash$polarized K3.
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Elliptic fibrations on K3s: there are only finitely many, up to automorphisms.
Even hyperbolic lattices $\Lambda$ of rank $r\geq 5$ such that for a K3 with $\NS(X) = \Lambda$ one has $\size \Aut(X) < \infty$ are classified; there is a finite list.
As a corollary, if a K3 $X$ admits an EFS (elliptic fibration with section) then $\MW(\phi, s) = 0$.

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There is a classification of all EFS on K3s $X$ with $\size \Aut(X) < \infty$ for $p\geq 5$.
Separate theorem: there is a classification of 2-elementary ($\disc_\Lambda = C_2^n$) lattices appearing as $\NS(X)$ for $X$ a K3 with finite automorphism group.

>  The discriminant group for a lattice $L$ is $\disc_L \da L\dual/L$.

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Kondo 2014: an interesting modular form picture for subfamilies of K3s coming from 8 points on $\PP^1$ with a non-symplectic involution.
Roulleau 2020: gives an algorithm for studying $\NS(X)$ for K3s with finite automorphisms?

Can cook up a moduli space of $P\dash$polarized K3s as an open subvariety of a weighted projective space.

Note: for any nodal curve, you can always consider the pencil of lines through the node.
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Heterotic $F\dash$theory duality: the heterotic string compactified on an elliptically fibred $(n-1)\dash$dimensional Calabi-Yau is equivalent to $F\dash$theory compactified on an $n\dash$dimensional K3-fibered CY.
Can interpret in the case of elliptic curves.

See Friedman-Morgan-Witten to identify moduli of $G\dash$bundles $\mcm_{E, G}$ as a projective variety?
Look at $\mcm_H = \dcosetl{\Gamma}{\Omega} \injects \bar{\dcosetl \Gamma \Omega}$ a toroidal compactification.
Realize $\mcm_{\mathrm{het}}$ as the total space of a $\CCstar$ bundle over $\mcm_{E, G}$.
So moduli space of $H\dash$polarized K3s (18 dim) $\mcm_H$ is roughly equivalent to $\mcm_{\mathrm{het}}$?
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See Camere, Garbagnati for $H \bigoplus E_8(-2)\dash$polarized K3s.
See "even eights".
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