We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,epsilon R)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifica
tions of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${bf A_n}$ ($nge1$), ${bf C_n}$ ($nge0$) and ${bf E_n}$ ($nge0$).
Let $F$ be a moduli space of lattice-polarized K3 surfaces. Suppose that one has chosen a canonical effective ample divisor $R$ on a general K3 in $F$. We call this divisor recognizable if its flat limit on Kulikov surfaces is well defined. We prove
that the normalization of the stable pair compactification $overline{F}^R$ for a recognizable divisor is a Looijenga semitoroidal compactification. For polarized K3 surfaces $(X,L)$ of degree $2d$, we show that the sum of rational curves in the linear system $|L|$ is a recognizable divisor, giving a modular semitoroidal compactification of $F_{2d}$ for all $d$.
We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus. We prove that the growth is bounded by a polynomial function of degree $14+varepsilon$ for any $varepsilon>0$ and, for three sets of i
nfinitely many genera, the bounds can be improved to degree 10. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperkaehler fourfolds.
We generalize the multiple cover formula of Y. Toda (proved by Maulik-Thomas) for counting invariants for semistable coherent sheaves on local K3 surfaces to semistable twisted sheaves over twisted local K3 surfaces. As applications we calculate the
$SU(r)/zz_r$-Vafa-Witten invariants for K3 surfaces for any rank $r$ defined by Jiang for the Langlands dual group $SU(r)/zz_r$ of the gauge group $SU(r)$. We generalize and prove the S-duality conjecture of Vafa-Witten of K3 surfaces for all higher ranks based on the result of Tanaka-Thomas on the $SU(r)$-Vafa-Witten invariants.
We show that the K-moduli spaces of log Fano pairs $(mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$ varies in
the interval $(0,1)$. We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-OGradys prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $mathbb{P}^3$.