--- created: 2023-03-29T11:24 updated: 2023-03-29T11:24 --- # Questions - Extending mirror symmetric compactifications to moduli of hyperkahlers? - See [[Vinberg algorithm]] - If $G = \gens{i_1, i_2}$, take invariant lattice $H^2(X; \ZZ)^G$? What is the induced decomposition on $S_X$ and $T_X$? Are these 2-elementary? - We're taking $L\dash$polarized K3s for $L$ some lattice -- which one? - Fix $h\in S_\RR$, very irrational and "maximal" with $h^2 > 0$, then $S\dash$quasipolarized K3s are pairs $(X, j:S\to \NS(X))$ where $j(h)$ is big and nef. - Set $\overline{\DD/\Orth(T)}^{\mathrm{BB} }$... - 0-cusps: orbits of primitive isotropic lines in $T$, something like $\Gr_1(T)^\isotrop/\Orth(T)$, Type III - Degenerations: $R \da R_1 \disjoint\cdots \disjoint R_n$ for $R_i$ ruled surfaces, $\Gamma(R) \cong \IAS^2$, double curves on $R_i$ are cycles of $\PP^1$. - 1-cusps: orbits of primitive isotropic planes in $T$, something like $\Gr_2(T)^\isotrop/\Orth(T)$, Type II - Degenerations: $R_1 \disjoint_{C_1} E_1 \disjoint_{C_2}\cdots \disjoint_{C_m} E_m \disjoint_{C_{m+1}} R_2$ where the $R_i$ are rational, $C_i$ are elliptic curves, $E_i$ are elliptic ruled. - Since we're in a Type IV domain, only need to worry about 1-cusps for toroidal compactification? Or both for semitoroidal? - Reduce to understanding isotropic vectors in $D_T$? - Goal: identify $\overline{F_S}^{\KSBA}$ with a semitoroidal compactification $\overline{F_S}^{\mcf}$? - 0-cusps classified by mirror moves on triangle: what is the analogue of the triangle? How to find mirror moves? - How to use IAS to construct semifans at the cusps? - How to prove normalization of KSBA is semitoroidal, given by these fans? - For $E\leq T$ a lattice, how to think about $\bar T \da E^\perp/E$? - Does $\Fix(G)$ contain a curve that can be contracted to get a KSBA stable pair? - Need to find a lattice $S$ to define $\DD_S/\Gamma_S$ where $\dim \DD_S = 20 - \rank_\ZZ S$. # Notes - $S(X)$ 2-elementary $\implies \exists \iota\in \Aut(X)$ nonsymplectic involution where $\Pic(X) = \NS(X) = S_X \oplus T_X$ where $S_X = \NS(X)^{+1}$ and $T_X = \NS(X)^{-1}$. - $\iota\actson D_S$ by $+1$ and $\iota \actson D_T$ by $-1$. - $\sgn S_X = (1, \rho-1)$ and $\sgn T_X = (2, 20-\rho)$, adding to $(3, 19)$. - $\size \Aut(X) < \infty \iff \Pic(X)$ is 2-reflective (generated up to finite index by $W_2$) - Reflective iff the fundamental chamber has finite volume. - Weyl groups: Weyl Group: $W(L)$ for the Weyl group of $L$, meaning the subgroup of $O(L)$ generated by reflections in roots. - When $L$ is Lorentzian, $O(L)\actson \HH^n \da$ the image of the negative-norm vectors of $L \otimes \mathbb{R}$ in $\PP(L \otimes \mathbb{R})$. - $G\actson H^2(X; \ZZ)$ yields $H^2(X; \ZZ) = \bigoplus_{\chi \in G\dual} V_\chi$ where $V_\chi = \ts{v\in H^2(X; \ZZ) \st g.v = \chi(g)v}$ where $G\dual \da \Hom(G, \CC\units)$. - Decompose as $H^2(X; \ZZ) = V_1 \oplus \bigoplus_{\chi\in G\dual\smts{1}} V_\chi$, so $V_1 = H^2(X; \ZZ)^G$. ## Baily Borel - $F_d \cong \DD/\Gamma_d$ where $\DD \da \ts{v\in \PP(\Lambda_d\tensor \CC)\st v^2 = 0, \norm v > 0}_0$. - Here $\Gamma_d \leq \Orth(\Lambda_d)$ is finite index - Here $\Lambda_d \da E_8\sumpower 2 \oplus H \sumpower{2} \oplus \gens{-d}$ is $H^2_\prim(X)$? - Idea: 1-cusps $\mapstofrom$ Type II components $\mapstofrom$ rank 2 isotropic planes $E \leq \Lambda_d$ modulo $\Gamma_d$. - Basic invariant of $E$ is $E^\perp/E$. - Coarser invariant: $R_E$ the root sublattice of $E$, sometimes determines $E$ up to isometry. - Type IV domains: ## Group actions ![](attachments/2023-03-08intersect.png) ![](attachments/2023-03-08refine.png) ![](attachments/2023-03-08compute.png) ![](attachments/2023-03-08lefschetz.png) # Extension from paper - $\rho: G\to \Orth(\lkt)$ a subgroup inducing an action $g.x \da \rho(g)(x)$. - $\chi: G\to \CC^*$ - $\psi: H^2(X; \ZZ)\to \lkt$ - $\Lambda_{\rho, \chi} = \ts{x\in \lkt \tensor \CC \st g.x = \chi(g) x}$ the joint eigenspaces for $g\in G$. - Contained in coinvariants $(\lkt)_G$? - $\Omega(\Lambda_{\rho, \chi}) \da \ts{x\in \PP(\Lambda_{\rho, \chi}\tensor \CC) \st \norm{x} > 0 }$. - Don't need $x^2 = 0$...? - Type IV if $\abs{\chi(G)} = 2$, Type 1 ball domain otherwise. - $\Gamma_\rho \da \ts{g\in \Orth(\lkt) \st g\circ \rho = \rho\circ g}$. - $\Delta_\rho \da \displaystyle\Union_{\delta\in (\lkt)_G,\,\,\delta^2 = -2}\delta^\perp$. - $$F_{\rho, \chi} \da \dcosetl{\Gamma_\rho}{\Omega(\Lambda_{\rho, \chi})\sm \Delta_\rho }$$ - 0-cusps $\mapstofrom \ts{e\in T\da (\lkt)_G \st e^2=0}$ defining isotropic lines? Degenerating on the $S$ side. - At a zero cusp $e$, set $S\dual \da \bar T \da e^\perp/e$.. - Look at $F_{S\dual}$? Their symplectic geometry? Lagrangian torus fibration and IAS? - Neither may be 2-elementary? - $\sgn S = ?, \sgn T = ?$. - Conversely, given $\lambda \in S\dual \intersect \mcc$ a monodromy invariant, construct a d-semistable Type II degeneration? - $\IAS^2$ with $C_2^2$ symmetry? Swapping L/R and N/S hemispheres respectively? How this works for involutions: - $G = \ts{1, \iota}$ - $S = (\lkt)^+ = \NS(X)$ and $T = S^\perp = (\lkt)^-$. - So $S = (\lkt)^G$ and $T = (\lkt)_G$. - $\Gamma_\rho$ defined the same way. - $$F_S \da \dcosetl{\Gamma_\rho}{\DD_S\sm \Delta}$$ How this works for involutions: - Hodge structure $V = H^2(X; \ZZ)$ with Hodge decomposition $V \tensor \CC = V^{2,0} \oplus V^{1,1} \oplus V^{0, 2}$. - $G = \ts{\id, \sigma}$. - $S_X \da \NS(X) = \Pic(X) = V \intersect V^{1, 1}$ and $T_X \da S_X^\perp$ in $V$. - So $V = S_X + T_X$ but the sum need not be direct - How does this interact with the Hodge decomposition...? - Note that $T_X$ usually intersects $V^{1,1}$; in fact in our case $$T_X = (V^{2,0})^\perp \intersect V$$ - $\sgn S_X = (1, \rho - 1)$ and $\sgn T_X = (2, 20-\rho)$. - $G\actson V \implies G\actson S_X, G\actson T_X$ since $S_X, T_X \subseteq V$. - Decomposes $V = V^+ \oplus V^-$ where $S_X \subseteq V^+$ and $T_X \subseteq V^-$. - So generally $S_X \subseteq V^G$ and $T_X \subseteq V_G \da (V^G)^\perp$ in $V$. - Pairs $(X, \sigma)$ are classified by $S\injects \lkt$ primitive 2-elementary sublattices with $\sgn S = (1, n)$ where $0\leq n \leq 20$? Classified by Nikulin. - $\DD_S \da \ts{x\in \PP(T\tensor \CC)\st x^2 = 0,\,\, \norm x > 0}$. - $\Delta \da \displaystyle\Union_{\delta\in S,\,\, \delta^2 = 0} \delta^\perp \intersect \DD_S$. - $\Gamma_\rho \da \ts{g\in \Orth(\lkt) \st g\circ \rho = \rho\circ g}$ is the centralizer in