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projective space, then there is a solution to our quotient1 such that classification problem: the Hilbert scheme. In this GIT M := ss// SL (C) = P1 case, we need to fix an invariant known as the 1,1 U 3 ∼ Hilbert polynomial which records geometric infor- is a complex projective variety compactifying M1,1. mation such as their dimension and degree. It was Moreover, there is a unique “minimal” closed shown in 1961 by Grothendieck that there exists a GIT p(m) SL3(C)-orbit associated to the point M1,1 M1,1: projective scheme Hilbr which parametrizes all \ namely, the orbit of the plane cubic x0x1x2 =0 . the closed complex solutions of polynomial equa- Turning to a complex-analytic{ perspec-} r tions in P with Hilbert polynomial p(m). In our tive, we can view a nonsingular cubic particular example, the Hilbert polynomial of plane C := F (X ,X ,X ) = 0 P2 as a com- 2 0 1 2 cubics in P is equal to p(m)=3m, and the associ- pact Riemann{ surface, with} ⊂ homology basis ated Hilbert scheme is P9. α, β H1(C, Z) (oriented so that α β = 1). However, the Hilbert scheme is not the solu- Up to∈ scale, there is a unique holomorphic· form tion that we are looking for. The reason is that ω Ω1(C), with period ratio τ := R ω/R ω in the ∈ β α given an elliptic curve defined by the equation upper-half plane H. This τ is well-defined modulo F (x0, x1, x2)=0 , we can use a linear change of the action of γ = ( a b ) Γ = SL (Z) through {coordinates x a} x + a x + a x with a C, c d ∈ 2 i i0 0 i1 1 i2 2 ij fractional-linear transformations γ(τ) = aτ+b to obtain another7→ equation G(x , x , x ) =∈ 0 . cτ+d 0 1 2 induced by changing the homology basis. We claim The elliptic curve defined by{ this second equation} that the resulting (analytic) invariant [τ] H/Γ is isomorphic to the first one, and yet the Hilbert captures the (algebraic) isomorphism class of∈ C. scheme tells us that they are different objects. This is closely related to the classical theory of The critical point here is that if we are modular forms. The (biperiodic) Weierstraß ℘- parametrizing varieties X Pn with a fixed Hilbert ⊂ function associated to the lattice Λ = Z 1, τ , polynomial, then we want to account for the au- h i 2 2 2 tomorphisms of the ambient projective space. In ℘(u) := u− + Pλ Λ 0[(u + λ)− λ− ], our example, the ambient space of elliptic curves ∈ \ − 2 3 F (x , x , x )=0 is P2, and the group associated satisfies (℘′) =4℘ g2(τ)℘ g3(τ), where g2(τ) := { 0 1 2 } 4 − − 6 to linear transformations x a x + a x + a x 60 Pλ Λ 0 λ− and g3(τ) := 140 Pλ Λ 0 λ− are i 7→ i0 0 i1 1 i2 2 ∈ \ ∈ \ has a dimension equal to 8. Therefore, of the 9 de- modular forms Mk(Γ) of weights k = 4 resp. 6. grees of freedom associated with the coefficients of That is, they transform by the automorphy factor k the equation F (x0, x1, x2), only one is intrinsic to (cτ + d) under pullback by γ Γ, which makes 3 ∈ g2 C the geometry of the elliptic curve, while the other  := g3 27g2 : H/Γ into a well-defined function. 2− 3 → eight are related to the linear change of coordinates Evidently, the image E of the map W : C/Λ ֒ P2 τ → in the ambient space. sending u [1 : ℘(u): ℘′(u)] is a Weierstraß cubic We arrive at our first (and almost correct!) defini- (1) 7→ tion of a moduli space: let P9 be the open locus 2 3 2 3 3 X2 X0 = 4X1 g2X1X0 g3X0 = 4Qi=1(X1 eiX0), parametrizing smooth ellipticU ⊂ curves, and SL(3, C) − − − the group associated to linear change of coordinates with period ratio τ. The key point is that any C can be brought into among the variables x0, x1 and x2. The quotient this form (without changing [τ]) through the action M := / SL(3, C) = C 1,1 U ∼ of SL3(C) on coordinates. Fix a flex point o C ˇ∈ is the “moduli space” of elliptic curves up to iso- (i.e. (C ToC) = 3); since the dual curve C has · 3 morphism. It will be tempting to consider a naive degree 6, there are 3 more tangent lines Tpi C i=1 9 { } quotient P / SL (C) for constructing a compactifi- passing through o. The pi are collinear, since 3 { } cation of M . However, these naive quotients are otherwise one could construct a degree-1 map C 1,1 → usually either of the wrong dimension or yield a P1. So we may choose coordinates to have o = [0 : 3 non-Hausdorff topological space. 0 : 1], ToC = X0 = 0 , X2 = 0 C = P pi, { } { }· i=1 The correct framework for constructing quotients and 3 X1 (p ) = 0, which puts us in the above Pi=1 X0 i within algebraic geometry is given by Geometric form (1). In fact, if  := (τ) / 0, 1, =: Σ, Invariant Theory (GIT), initiated by Mumford in ∈ { ∞} 1969 [17]. A key result from GIT is the existence of 1U/G denotes a geometric quotient while U//G denotes a a larger open locus ss P9 and a well-defined categorical one. U ⊂ U ⊂ ALGEBRAIC AND ANALYTIC COMPACTIFICATIONS OF MODULI SPACES 3

then rescaling yields a member of the family 1.2. Picard curves and points in a line. Next, we’ll visit ideas that are central for constructing (2) y2 =4x3 27 x 27 compact moduli spaces. They include the use of fi- −  1 −  1 − − nite covers to associate periods to varieties without over P1 Σ, whose period map [τ]: P1 Σ H/Γ periods and the first example of “stable pairs.” composed\ with  extends to the identity\ C→ C. We begin this time with the complex-analytic → point of view. Though ordered collections of n Hence C ∼= C′ = [τ] = [τ ′] =  = ′ = SL3 ⇒ ⇒ ⇒ points in P1 do not themselves have periods, we C = C′ yields the claimed equivalence of analytic ։ 1 SL∼3 can consider covers C P branched over such and algebraic “moduli”, and  is an isomorphism. collections, generalizing the n = 4 case of Legendre While the choice of o does not refine the moduli elliptic curves. For n = 5, compactifying problem, keeping track of the ordered 2-torsion sub- 3 4 2 4 (3) y = x + G2x + G3x + G4 = Qi=1(x ti) group o,p1,p2,p3 does. In (1), this preserves the − { } 2 ordering of the ei , which are parametrized by the to C P yields a genus-3 curve with cubic au- { } 1 τ τ+1 ⊂ weight-2 modular forms ℘( ), ℘( ), ℘( ) with re- tomorphism µ: y ζ3y and (up to scale) unique 2 2 2 7→ dx ¯ spect to Γ(2) := ker Γ SL (Z/2Z) . The roles of holomorphic differential ω = with µ∗ω = ζ3ω. { → 2 } y e3 e2 = 1 P  and (2) are played by ℓ := − : H/Γ(2) ∼ P Σ The moduli spaces Mord := (P ti = 0) i

the meaning of φ˜∗, notice that colliding two ti (We have to blow up at the 4 triple-intersection in (3) and normalizing yields a genus 2 curve points to make a2/a1 well-defined.) This is a first with cubic automorphism, whose (single) period example of using limiting mixed Hodge structures ratio is parametrized by one of the disk-quotients (LMHS) to extend period maps to a toroidal com- previously omitted. When 3 t collide in (3), pactification “ ” refining the Baily-Borel “ ” com- i ∗∗ ∗ the normalization has genus 0 and thus no mod- pactification. uli, which explains the 4 boundary points in This close relationship between moduli of n = 5 (B2/Γ(√ 3))∗. The unnormalized scenarios are points, Picard curves, and ball quotients has a depicted− in the left and middle degenerations in beautiful extension to configurations of up to 12 Fig. 1. points by Deligne and Mostow. The geometric meaning of the corresponding “*” and “**” compactifications will be addressed in §3. Going back to the moduli of n ordered points in P1, and adopting a geometric viewpoint, we should phrase the problem in terms of objects up to an equivalence relation. An “object” here is an n- 1 q1 pointed curve (P , (p1,...,pn)), which is equivalent E q2 P1 q to ( , (q1,...,qn)) if g(pi) = qi (1 i n) for 3 some g Aut(P1). The resulting moduli≤ space≤ ti = tj ti = tj = tk ti = tj = tk ∈ 1 n with blowup M0,n = (P ) S ∆(ik)/ SL2(C) \ i

3 1 3 1 the values of w induces birational transformations 4 2 4 2 among the GIT quotients. For example, for each n n 3 1 n 3 5 5 there are choices of w that yield P − and (P ) − as quotients. Figure 2. Generic limit Cubic surfaces. parametrized by D12 1.3. In our first higher dimensional example, we will visit the idea of K-stability, The reader may wonder if we can also compact- which leads to a big part of current moduli theory. 3 ify these M spaces by going down the same road Cubic surfaces SF P are cut out by ho- 0,n ⊂ as for elliptic curves. The answer is yes — there mogeneous polynomials F (X0,...,X3) of degree are indeed GIT compactifications — but with a three. The corresponding Hilbert scheme is equal P19 P 0 P3 C new twist. We determined already that M0,n is a to ∼= H ( , (3)) and the group SL4( ) as- 1 n O quotient of an open locus within (P ) by SL2(C). sociated with the linear change of coordinates has Geometric Invariant Theory and subsequent devel- dimension 15. Let PH0(P3, (3))sm P19 be the O ⊂ opments imply that there are finitely many open open locus parametrizing smooth cubic surfaces. ss loci w , depending on a collection of rational num- The four-dimensional quotient U bers w = (w1,...,wn) with 0 < wi 1 and 0 3 sm M := PH (P , (3)) /SL4(C) w + + w = 2, such that ≤ O 1 ··· n 1 n ss 1 n is our moduli space of smooth cubic surfaces. (P ) Si k ∆(ik) w (P ) \ ≤ ⊂ U ⊂ For compactifying M, we observe that smooth and the quotient cubic surfaces belong to a family known as Fano 1 n ss varieties. These are complex projective varieties (P ) //w SL2(C) := w // SL2 (C) U whose anticanonical bundle — the determinant of is a projective variety compactifying M0,n. For the the tangent bundle — is ample. A fundamental case n = 5 and depending on the choice of “weights” 1 5 2 problem in complex geometry is to classify the Fano w, the quotients (P ) //w SL2(C) can be either P , 1 1 2 varieties that admit a Kähler-Einstein (KE) met- P P , or a blow-up of P at k points in general ric: that is, a Kähler metric ω with Ricci curvature position× with 1 k 4. ≤ ≤ equal to the metric (Ric(ω)= ω). A celebrated re- These GIT quotients are philosophically differ- sult of Tian from 1990 implies that all smooth cubic ent from M0,n, because they parametrize different surfaces admit a KE metric. So we arrive at a ques- types of geometric objects. Remember that M0,n tion which leads to much of the moduli theory for 1 allows P itself to degenerate, so as to keep the Fano varieties: Can we construct a compactification points distinct (as in the previous figure). The GIT KE M M of M whose limiting objects admit a KE quotient, on the other hand, enables the points to metric⊂ as well? collide amongst themselves in a controlled manner, The answer revolves around the relatively new and P1 does not degenerate. This scenario is de- concept of K-(semi)stability first introduced in 1997 picted in Fig. 3. by Tian and then expanded in 2002 by Donaldson. Leaving definitions aside for the moment, we only 3 1 3 1 = 2 remark that K-(semi)stability induces substantial 4 2 4 restrictions on a Fano variety’s singularities. For 5 5 example, if X is a “reasonable” Fano K-semistable variety, then X should be normal. Figure 3. GIT limit when w + 1 In our particular case, results of Odaka, Spotti w < 1 2 and Sun from 2012 imply that the moduli of KE And so we arrive at one of the main questions cubic surfaces is isomorphic to the GIT quotient: within moduli theory. How are a priori different KE GIT 0 3 ss M = M := PH (P , (3)) // SL4(C) compactifications of a moduli space related to each ∼ O where (PH0(P3, (3)))ss PH0(P3, (3)) is the other? In our case, it is a theorem of Kapranov O ⊂ O that for every w as above there is a morphism open subset parametrizing semistable cubic sur- 1 n M0,n (P ) //w SL2(C). A framework known faces: those which are either smooth, or admit A1 or → 2 2 2 as “variation of GIT” (VGIT), developed by Dol- A2 singularities (of the local form x + y + z resp. GIT gachev, Hu, and Thaddeus, shows that changing x2 + y2 + x3). The compactification M contains 6 PATRICIO GALLARDO AND MATT KERR

an open subset Ms = (PH0(P3, (3)))s / SL (C) B /Γ by allowing A singularities in S . Next, re- O 4 4 1 F called the stable locus. In our case, each point in calling that any smooth SF has 27 lines, we may re- the stable locus corresponds to a unique isomor- fine our moduli (in analogy with the level-structures phism class of cubic surfaces with at worst A1 sin- of §1.1) by an ordered choice of 6 disjoint lines. gularities. Our GIT quotient compactifies the open Writing Mord (։ M by forgetting the lines) for ˜ ֒ stable locus by adding a single point ω. Although the resulting moduli space, φ lifts to φ: Mord 19 → there is not a unique SL4(C)-orbit within P as- B /Γ(√ 3), where Γ(√ 3) := U(h, √ 3 ) and 4 − − − O sociated to ω, there is a unique closed one! In Γ/Γ(√ 3) is again W (E6). (Notice the analogy this case, the associated surface is the cubic sur- between− the role of the marking and of the level 3 face x0x1x2 + x =0 with three A2 singularities. structures in §1.1.) We postpone discussing the { 3 } We should be proud of progress until now, and compactifications, but note that Allcock and Fre- yet there is an aspect missing from the above story. itag have given an automorphic description of Cay- 1 It is known since work of Cayley and Salmon circa ley’s cross-ratios (hence of φ˜− ). 1850 that every smooth cubic surface has 27 lines on it. Can we produce a compactification that 1.4. K3 surfaces. Named in honor of Kummer, keeps tracks of such lines and their limits? With Kähler and Kodaira, these are simply connected that purpose, we choose in order 6 disjoint lines compact complex 2-manifolds X with Ω2 = out of the 27. A smooth cubic surface together X ∼ , hence (up to scale) a unique and nowhere- with such a choice is called a marked cubic sur- X Ovanishing holomorphic 2-form ω Ω2(X). Alge- face. The resulting open moduli space M has a ord braic (quasi-polarized) K3s have∈ countably many finite map to M defined by forgetting the labeling K3 (19-dimensional) coarse moduli spaces Mg , one of the lines. The associated finite field extension K3, C C for every “genus” g 2. An open subset Mg ◦ (M) (Mord) induces a unique compactifici- ≥ g −→GIT parametrizes surfaces X P of degree 2g 2 M ation ord such that for g 3, and double covers⊂ X ։ P2 branched− along≥ a sextic curve for g = 2. Given a hyper- / GIT Mord Mord plane section h X of one of these polarized K3s, ⊂ VZ := H2(X, Z)/[h] is a lattice of rank 21 (and sig- /W (E6) /W (E6) nature (2, 19) under the intersection form), com- / GIT M M prising classes of topological 2-cycles. As Rh ω = 0, integrating ω over (some basis of) these yields a where we used the classical fact that the symmetric P P20 well-defined period point [ω] VC∨ ∼= , which is group of the 27 lines is the Weyl group of E6. ∈ further restricted by RX ω ω = 0 (and RX ω ω¯ > 0) Returning to a Hodge-theoretic perspective, we to lie in a 19-dimensional∧ submanifold D ∧called a note that SF has no holomorphic 1- or 2-forms; type-IV (Hermitian symmetric) domain. By a the- so the situation is analogous to that of point- orem of Piatetskii-Shapiro and Shafarevich, this pe- configurations on P1, with no “straightforward” pe- K3, = riod map φ embeds Mg ◦ ֒ D/Γg (where Γg riods. Recycling the cyclic-cover trick from §1.2 → O(VZ)) as the complement of a “hyperplane config- yields cubic threefolds uration”. To explicitly describe geometric compactifica- T := X3 = F (X ,...,X ) P4 F { 4 0 3 } ⊂ tions of the moduli space of K3 surfaces of a given degree is a complicated and mostly open problem with automorphism µ: X4 ζ3X4. Up to scale, 7→ for many cases. We illustrate the case of degree TF has a unique closed 3-form ω of type (2, 1) with equal to 2, since generically those K3 surfaces are µ∗ω = ζ¯3ω, whose periods over a basis of the - O double covers of P2 branched along a smooth plane lattice H3(TF , Z) produce a point [τ] in a 4-ball B4. We now summarize some results of Allcock, Carl- sextic. Then one can be tempted to use the GIT son and Toledo. Note that the intersection form on quotient parametrizing such plane curves — a tech- nique used in the case of cubic surfaces. Indeed, H3(TF , Z) gives rise to a Hermitian form h of signa- ture (4, 1). It turns out that the monodromy group GIT guarantees the existence of a semistable locus h PH0(P2, (6))ss such that the quotient (through which π1(M) acts on H3) is Γ := U( , ), O the resulting period map φ: M B /Γ is an em-O 4 K3,GIT 0 2 ss → s M := PH (P , (6)) //SL (C) bedding, and φ extends to an isomorphism M ∼= 2 O 3 ALGEBRAIC AND ANALYTIC COMPACTIFICATIONS OF MODULI SPACES 7

is a compact projective variety of the correct dimen- acts, and which is always a ball or type-IV do- K3,GIT main). It follows from Lefschetz’s (1,1)-theorem sion. However, we should only think of M2 as 1 a first approximation to our desired moduli space. that its preimage φ− (DL/ΓL) =: ML parametrizes The reason is that MK3 contains an 18-dimensional L-polarized K3s (X,ℓ), where ℓ: L ֒ Pic(X) is an 2 → subfamily of unigonal K3s whose map3 to P2 in- embedding of L into the classes of line bundles (or duced by the quasi-polarization is not surjective; divisors) on X. For instance, for L = H E8 E7 ⊕ ⊕ instead, it produces an elliptic fibration of X over (ρ = 17), Clingher and Doran have shown that the 2 2 points a smooth conic (x0x1 x2) = 0 P . The {K3,GIT − } ⊂ compact space M does not account for such WP 2 [α2:α3:α5:α6] ML ∼= [2:3:5:6] α5=α6=0 K3s. Instead, it has an unique point ω parametriz- ∈ \{ } ing the sextic (x x x2)3 = 0 . In 1980 Shah parametrize K3s Xα given by the minimal resolu- 0 1 2 3 showed that the{ blow-up− at ω yields} a compact tion of the quartic hypersurface in P with affine K3,GIT K3 equation space Blω(M2 ) containing M2 as an open K3,GIT 2 3 2 1 2 subset. The exceptional locus Bl (M ) y z 4x z +3α2xz +α3z +α5xz (α6z +1)=0. E ⊂ ω 2 − − 2 is isomorphic to The period map φ restricts to an isomorphism 12 8 P Sym (st) Sym (st) // (3,2)SL2(C) = × O M ∼ H Z L DL/ΓL ∼= 2/Sp4( ) where st is the standard SL2(C) representation. → 4 Geometrically, one interprets a general point in which is inverted by the Siegel modular forms αj E ∈ as a pair of collections of 12 resp. 8 distinct points M2j(H2, Sp4(Z)). 1 in P up to an action of SL2(C). To see the relation What about the blowing-up of periods? For a with unigonal K3 surfaces, note that the latter may 1-parameter degeneration of K3 surfaces ∆, m X → be written in the form after a finite base-change (t t ) there are two possibilities: (a) two periods7→ blow up like log(t); z2 y3 yg (x , x , x ) g (x , x , x )= x x x2 =0. − − 4 0 1 2 − 6 0 1 2 0 1− 2 or (b) one period blows up like log(t), and another We recover collections of 12 resp. 8 points by con- like log2(t). Geometrically, after birational mod- sidering the intersections g4(x0, x1, x2) = x0x1 ifications the singular fiber in (a) looks like two 2 { 2 − x2 =0 and g6(x0, x1, x2)= x0x1 x2 =0 . rational surfaces glued along an elliptic curve E Our} visit{ to the case of points− in the} line in (or a more complicated variant of this), while in §1.2 suggests that there should be another type of (b) we get a “sphere-like” configuration of rational compactification based on “stable” degenerations, surfaces glued along P1’s (and the log2(t) reflects a topic to be elaborated on in later sections. Here the “triple points” in such a configuration). In the we only remark that the analogous compactification WP Z compactification [2:3:5:6] ∼= (H2/Sp4( ))∗, the for degree-2 K3s is significantly more complicated point [1:1:0:0] corresponds to (b), and the rest of than the GIT one, and has only been completed in α5=α6=0 to (a); the latter must be 1-dimensional 2019 by Alexeev, Engel, and Thompson. to{ keep track} of the modulus of E, after all. The increasing complexity of the moduli phe- As in the case of Picard curves, we can refine nomena makes some previously discussed questions, the “limiting period map” by computing additional such as inversion of the period map, impracticable periods that remain finite as t 0. The idea for arbitrary degree. Therefore, a common strategy is to look at the intersections of→ certain divisors is to consider a smaller moduli problem for which on the rational surfaces with the rational or ellip- one is able to give a more complete description. For tic curves along which they are glued. One then example, one can consider either K3 surfaces whose gets semiperiods by integrating (a) ωE (on E) or Picard group is isometric to a particular lattice, or dz P1 (b) z (on 0, ) between the intersection surfaces with a given finite automorphism group. points. In the\ example, { ∞} this contributes 1 modu- From a Hodge-theoretic perspective, the vanish- lus to (a) and 2 moduli to (b), making them into ing of periods on a sublattice L H2(X, Z) = divisors and yielding a “toroidal” compactification 3 2 ⊂ ∼ H⊕ E8⊕ of rank ρ cuts out a subdomain DL D ⊕ ⊂ 4 t of dimension 20 ρ (on which ΓL := stabΓg (L) Here H2 := τ M2(C) τ = τ , Im(τ ) > 0 is Siegel’s − { ∈ A B | } upper half-space, γ = ( C D ) Sp4(Z) acts by γ(τ ) = (Aτ + 3 −1 ∈ The degree-2 linear system D has a base curve R, and B)(Cτ + D) , and f (H2) belongs to Mk(H2, Sp4(Z)) D =2E + R. The free part of D| |maps X to the conic. iff f(γ(τ )) = det(Cτ +∈D O) kf(τ ). | | { } 8 PATRICIO GALLARDO AND MATT KERR

(H2/Sp4(Z))∗∗. (Of course, one does not get an ex- The resulting Torelli map Φg : Mg g (sending tension of the period map to this without first blow- C J(C)) is an injection, which is→ an A immersion 7→ ing up ML). off the hyperelliptic locus. Since the inequality (for g+1 g > 1) dimC Mg = 3g 3 2 = dimC g is 1.5. Cubic 4-folds. This case is similar to the case strict for g 4, most− PPAVs≤ are not Jacobians,A of K3 surfaces. A smooth cubic X P5 has (up to ≥ ⊂ and the period map for Mg does not have open scale) a unique closed (3, 1)-form, whose periods de- image. 0 5 sm fine a map φ from M := P (H (P , (3)) /SL6(C) to the quotient D/Γ of a 20-dimensionalO type-IV domain by the monodromy group. By work of Laza, 2. Aspects of the theory Looijenga and Voisin, φ is injective with image the Our tour has reached the base of the funicular, on complement of a hyperplane arrangement. Cubics which we now ascend to two different vantage points with action by a fixed finite group yield moduli for a brief theoretical overview. On the algebro- subspaces, which behave analogously to the M L geometric side, we refer the reader to [13], [18] and in §1.4: their images under φ turn out (by recent{ } references therein for the technical details; a good work of Laza-Pearlstein-Zhang and Yu-Zheng) to place to start for curves are [4], [10] and for higher be hyperplane-arrangement complements in type- dimensional cases are [2] and [15]. For background IV or ball quotients. on period maps in Hodge theory, we recommend [5]. 1.6. Abelian varieties. Given a full lattice Λ := Z λ(1),...,λ(2g) Cg, the compact complex torus 2.1. What is a moduli space? In moduli theory, h g i ⊂ g 1 T := C /Λ has bases dzi i=1 of Ω (T ) and γj := we want more than a space parametrizing objects, (j) 2g { } { such as smooth elliptic curves up to isomorphism. 0.λ j=1 of H1(T, Z), hence g 2g period ma- } (j) × We want to understand all possible families of the trix Π := ( dzi) = (λ ). By Kodaira’s em- Rγj i objects. For that purpose, we need to reformulate bedding theorem, T is algebraic5 (an abelian vari- our problem. ety) iff it possesses a closed, positive (1, 1)-form Ω Let Ω be a “reasonable” class of objects — for with [Ω] H2(T, Z). Equivalently, after changing example, the stable n-pointed curves of genus 0 de- ∈ both bases above, Π takes the form (δ τ ), where scribed in §1.2. The corresponding moduli functor 6 | δ = diag(δ1,...,δg) Mg(Z) (with δi δi+1) and τ ∈ | (B)= flat families B with fibers X Ω, belongs to the Siegel upper half space Hg := τ b t { ∈ M { X → ∈ Mg(C) τ = τ , Im(τ ) > 0 . modulo isomorphisms over B | } } If all δi = 1, then Ω is a principal polariza- maps schemes B to sets (B). For any scheme M, tion of T . The set of principally polarized abelian M M varieties (PPAVs) is then parametrized by Hg we can define another functor h from schemes to g ∋ τ Aτ := C /Λτ , where Λτ is the Z-span of sets by hM(B) := Mor(B, M). A moduli functor 7→ is represented by a scheme M if there is a naturalM columns of (1g τ ). In order that each isomorphism class occur| only once, we take the quo- isomorphism from to hM. In that event, M is A B called a fine moduliM space for M, and constructing tient g := Hg/Sp2g(Z), where γ = ( C D ) acts by A 1 a family of objects over a base B is equivalent to γ(τ )=(Aτ + B)(Cτ + D)− . This action on the defining a morphism B M. “period point” is equivalent to a change of inte- → gral basis preserving [Ω] (viewed as a nondegener- Looking back at the examples in §1, M0,n represents the moduli functor associated to stable n- ate alternating form on H1(Aτ , Z)). Siegel modular pointed curves of genus 0. Another case in point is forms embed g into projective space, yielding our algebraic moduliA space for PPAVs. The compacti- the Hilbert scheme, which represents the functor fication so obtained has a stratification of the form r (B)= flat families B such that Xb P g∗ = g g 1 0. M { X → ⊂ A A ∐A − ∐···∐A has Hilbert polynomial equal to p(m) . Given a projective algebraic curve C of genus g, } 1 the Jacobian J(C) ∼= Ω (C)∨/H1(C, Z) is princi- 6 pally polarized by the intersection form on H1(C). For the newcomer: “flatness” ensures reasonable behav- ior, such as constancy of the Hilbert polynomial in a pro- 5More concretely, T is embedded in projective space by jective family; if B is regular and Cohen-Macaulay, it is sections of powers of the theta line bundle. simply equivalent to equidimensionalityX of the fibers. ALGEBRAIC AND ANALYTIC COMPACTIFICATIONS OF MODULI SPACES 9

Unfortunately, most moduli functors are not repre- 1997 that there is a coarse moduli space M asso- sented by a scheme. Even the moduli functor as- ciated with the stack describing our moduli func- sociated to isomorphism classes of smooth elliptic tor . Recent results by Alper, Halpern-Leistner, M curves fails to be represented by M1,1. and Heinloth have generalized this result to a larger One option for moving forward is to weaken our class of stacks, whose parametrized objects can expectations. A scheme M is a coarse moduli space have positive-dimensional automorphism groups. A for if M is the scheme best approximating the caveat here is that the output M is an algebraic moduliM functor, and the geometric points of M are space rather than a scheme. in bijection with the equivalence-classes of objects The previous discussion leaves us with a delicate “parametrized” by . More precisely, there is a question. What are the “reasonable” classes of ob- M natural transformation hM, for which M jects for compactifying a given moduli space? Or, M −→ is universal and (Spec(K)) hM(Spec(K)) an more colloquially: what do we add in the boundary? isomorphism of setsM in K = K¯ .−→ But for B more gen- The answer depends on the types of varieties we are eral than a geometric point, (B) Mor(B, M) considering. We now visit two important cases. may be neither injective norM surjective:−→ distinct 2.2. Algebro-geometric compactifications. families over B may produce the same “classifying map” to M, and not every map to M (even 2.2.1. General type case. Let M be a moduli space idM) need give rise to a family. On the positive of pairs (X,D) where X is smooth projective, D is side, if a moduli functor has a coarse moduli space, a normal crossing effective divisor, and KX + D is then the latter is unique (up to canonical isomor- ample. Examples include (non-Eckardt) smooth cu- phism). For example, the GIT quotient M1,1 is the bic surfaces together with their 27 lines, or complex coarse moduli space for smooth elliptic curves up curves of genus g 2 (with D = 0). In terms of the to isomorphism, though it neither carries a univer- minimal model program,≥ the “correct” objects for sal family nor distinguishes (say) isotrivial families compactifying such M are the (more general) stable from trivial ones. pairs. These consist of a projective variety X and What if, in the absence of a fine moduli space, we an R-divisor D = P biDi on it, such that KX + D still want to keep track of the families? In that case, is R-Cartier and ample, and the pair has semi-log- we need new geometric objects known as stacks. canonical (slc) singularities, see [2, Def 1.3.1]. Introduced by Deligne, Mumford, and Artin in the One reason to add a divisor to X, is that KX may 1970s, they are (loosely speaking) enrichments of not be ample; in that case, without the “boost” schemes obtained by attaching an automorphism from D, the canonical model wouldn’t recover our group to every point. In our particular context, the original family, let alone compactify it. On the stack of objects Ω is a category whose objects are other hand, the choice of D may add dimensions to families T of our “reasonable” objects. A our moduli space; or there may be various choices {X → } morphism in this category T B of D that lead to distinct compactifications of the {X → } 7→ {Y → } is a pair of maps f : and g : T B such same M. X →Y → that the diagram We first illustrate the concepts surrounding slc singularities when D = 0 and X is singular but f normal. Let f : Y X be a resolution of singular- / → X Y ities such that the exceptional set j is a normal crossing divisor. Assume that the∪E canonical divi- g sor K is Q-Cartier; then we have the numerical T / B X equivalence

(4) K f ∗(K )+ a commutes and is isomorphic to the pullback of Y Q X Pj j j X ∼ E via the map T B. of Q-Weil divisors. The coeﬃcients aj are called Y These two roads,7→ stacks and coarse moduli the discrepancies, and they measure how bad our spaces, converge to yield a rich picture. For a singularities are. The singularities of X are called well-behaved moduli problem (for example, when terminal (the best case) if aj > 0, canonical (still automorphisms of all parametrized objects are ﬁ- very good) if a 0 and log-canonical (tolerable) if j ≥ nite and the limits are uniquely determined, i.e. aj 1 for all j. In dimension 2, the canonical hy- the stack is separated), Keel and Mori showed in persurface≥ − singularities are precisely the ADE ones, 10 PATRICIO GALLARDO AND MATT KERR

while the log-canonical ones include all quotient sin- the points Di may coincide with each other, 1 C2 • gularities m (1,r) := /µm with gcd(m, r) = 1; but they should be different from the nodes, th µm is the group of m -roots of unity acting via and ρ(x, y) (ρx, ρrx) with ρm = 1. the sums of multiplicities of coincident 7→ • The defintion of discrepancies extends to pairs points must not exceed 1: P bi 1 i: Di=p ≤ with D = 0 by replacing K in (4) by K + D. We ( p C). 6 X X ∀ ∈ say a pair is klt (resp. log-canonical) if all its dis- It follows from works of Deligne, Mumford, Knud- crepancies are strictly larger than 1 (resp. 1). sen, and Hassett that for any n, b, and g 0 The slc singularities are those that− become≥ “log- ≥ the moduli stack g,b of weighted stable curves canonical after normalization.” They include (for of arithmetic genusMg is a smooth Deligne-Mumford 2 2 D = 0) the pinch point x + y z =0 and degen- stack with a projective coarse moduli space M b. 2 2 q {2 } g, erate cusps y (z + y − )=0 (with q 3). { } ≥ The explicit description of higher dimensional Even if we know that stable pairs are the right ob- pairs parametrized by compact moduli spaces is a jects for compactifying our moduli, defining families busy industry nowdays. A quite combinatorially of them is a tricky business that leads to two pos- rich example is the moduli of hyperplane arrange- sible functors, due to Kollár and Viehweg. More- ments in projective space [2]. We illustrate in Fig. over, if D = 0, the “boundary fibers” may exhibit 6 4 two stable pairs (from [op. cit.]) found in the phenomena like non-reduced points, see [16, Sec compactification of the moduli space of six labelled 1.1]. So here we shall limit ourselves to an oversim- lines in P2. plified7 definition of the moduli functor that produces the so-called slc or KSBA compactification 6 1 1 2 (after Kollár, Shepherd-Barron, and Alexeev) when 2 D = 0: 3 3 6 slc(B)= flat proper morphisms B whose M { X → 4 5 fibers Xs are n-dimensional stable 5 1 4 varieties with fixed Kdim(X) Q , Xs >0 2 2 2 2 2 ∈ Blpt P P Bl2pts P P P satisfying Kollár’s condition ∪ ∪ ∪ } This functor8 is in fact coarsely represented by a Figure 4. Examples of stable pairs projective scheme. This last claim is also true for the generalizations involving non-zero divisors D, see [16, Corollary 6.3] and [2, Sec 1.6]. 2.2.2. Fano case. The above theory does not work We now return to one of the main landmarks in when KX + D is not ample, so a new perspective algebraic geometry, of which the reader caught a is necessary. We will discuss the case where X is glimpse in §1.2. Fix n real numbers 0 < bi Fano (i.e. KX is ample) and D = 0. Many of the 1. A weighted stable curve for the weight b ≤= ideas can be− extended to the case when K D − X − (b1,...,bn) is a marked pair (in the sense of [13]) is ample, and we encourage the reader to consult (C, (D1,...,Dn)) comprising a reduced connected [18] for the details. projective curve C together with a collection of n Here, the concept of K-(semi)stability is central. points D C, such that the divisor K + D := Introduced in 1997 by Tian, it became a central con- i ∈ C KC + Pi biDi is ample. The slc condition trans- jecture — now theorem — that the existence of a lates into: KE metric on a smooth Fano variety X is equiv- C is either smooth or has at worst sin- alent to satisfying a K-stability condition. Key • gularities locally analytically isomorphic to ingredients are test configurations for (X, L), i.e. certain one-parameter degenerations C with xy =0 , and X → { } fibers Xt ∼= X ( t C∗) and a C∗-action, and the Donaldson-Futaki∀ (DF)∈ invariant of such test con- 7see [2, §1.5] and [16, Sec 5] for a more precise statement figurations. A Fano variety X is K-semistable iff 8Here “Kollár’s condition” requires that for any m Z the the DF invariant is non-negative for all test config- [m] ∈ reflexive power ωX /S commutes with arbitrary base change. urations for (X, L). ALGEBRAIC AND ANALYTIC COMPACTIFICATIONS OF MODULI SPACES 11

It is hard to check K-(semi)stability from its def- locus (depending on L), such that the categori- ss inition. However, there is a local-to-global inter- cal quotient (L)// SLN (C) is a projective vari- play that restricts the geometry of K-semistable ety compactifyingU our moduli space M. Moreover, varieties. By work of Odaka, a reasonable9 K- there is an intermediate stable locus s(L) (also semistable Fano variety has at worst klt singular- depending on L) for which the geometricU quotient ities. Moreover, Liu showed in 2018 that for an s(L)/ SL (C) is well-behaved, and U N n-dimensional K-semistable Fano variety X, a lo- s ss M (L)/ SLN (C) (L)// SLN (C). cal invariant at p X, denoted by vol(p,X),10 ⊆ U ⊆ U ∈ c is bounded by global invariants of the variety: We recover our initial moduli problem because M ∼= n n+1 n / SLN (C). vol (p,X) ( KX ) . These relations play c n U a central role in finding≥ − singular K-semistable vari- The choice of the embedding and of the line eties. bundle L mean that we have many possible GIT The construction of the moduli space of K- compactifications for M. In particular, Gieseker semistable varieties is ongoing work of many people showed that if we use the pluricanonical embedding with m 5 for complex curves of genus g 2, including Alper, Blum, Halpern-Leistner, Li, Liu, ≥ ≥ Wang, Xu, Zhuang, among others. The moduli then we can construct a GIT quotient that recov- functor KE sends the scheme B to ers Mg. Other GIT compactifications of Mg and M Mg,b have been the subject of work by Hassett, KE(B)= flat proper morphisms B whose Hyeon, Fedorchuk, Lee, Li, Schubert, Swinarski, M { X → fibers are n-dimensional K-semistable Smyth, Jensen, Moon, and others. A different ex- varieties with volume V , satisfying ample where multiple GIT quotients come into play, 0 this time involving cubic surfaces, was described by Kollár’s condition . } the first author with Martinez-Garcia [8]. This functor is represented by an Artin stack of 2.2.4. Remarks on other cases. We can use some of finite type and admits a projective coarse moduli the above techniques for compactifying moduli of MKE space . We remark that it is possible to extend varieties for which neither KX nor KX are ample. the above concepts to pairs. Particular cases are For instance, if X is a Calabi-Yau− variety, we can described by work of the first author with Martinez- select an ample divisor D, so that the pair (X, ǫD) Garcia and Spotti; while a more general setting is is stable. Kollár and Xu have recently determined given by Ascher, Liu and DeVleming. But this is that the irreducible components of the resulting another story and shall be told another time. moduli space are projective. Other compactifications may be obtained by using either GIT or the 2.2.3. A brief stop at GIT. If our moduli space GKZ secondary fan when X is a complete intersec- parametrizes, say, cubic surfaces up to isomor- tion in a toric variety. phism, shouldn’t the limit also be a cubic surface? KSBA or K-stability techniques do not guarantee 2.3. Hodge-theoretic compactifications. or even prioritize such a condition. However, Geo- metric Invariant Theory (GIT) does. 2.3.1. Period maps. A weight-n Hodge structure Q The main idea is that, given a moduli space M for on a -vector space V is a decomposition VC = p,q q,p p,q smooth varieties or pairs, we can select an embed- p+q=nV with V = V . By the Hodge theorem,⊕ the classes Hp,q(X) Hn(X, C) of closed ding that realizes such objects inside a projective ⊂ N 1 C∞ forms of type (p, q) yield a HS for any com- space P − . We can then parametrize those embed- pact Kähler manifold X. Equivalent data are given ded objects by an open subset in a Hilbert scheme k U by the decreasing filtration F • defined by F V = (or proxy). Under appropriate circumstances, there p,n p p kV − or homomorphism exists a line bundle L with an SLN (C)-linearization, ⊕ ≥ and a larger open ss(L)( ) called the semistable ϕ: S1 = z C z =1 SL(V ) U ⊃ U { ∈ | | }→ p q 9 defined by ϕ(z) V p,q = z − .IdV p,q . The Hodge num- X is equidimensional, reduced, S2, the codimension one | p,q p,q points of X are Gorenstein, and K is Q-Cartier bers of V are h =(h ) := (dimC V ), and its level X p,n p p ,n p 10 m is max p p′ h − =0 = h ′ − ′ . Let (X,p) = (Spec(R), ), where R is a local ring es- {| − | 6 6 } sentially of finite type and m is the maximal ideal; then A polarization of the HS (V, F •) is a nondegener- vol (p,X) = inf lct(a)n mult(a) a is m primary ate, ( 1)n-symmetric bilinear form Q: V V Q c { | − } − ⊗ → 12 PATRICIO GALLARDO AND MATT KERR

satisfying the Hodge-Riemann bilinear relations where φ(M) is an integral manifold of the G(R)- invariant horizontal distribution TD equiv- p p′ (5a) Q(F , F ) = 0 if p + p′ > n alent to (6). We want to use φW, together ⊂ with p q (5b) √ 1 − Q(v, v¯) > 0 if v V p,q 0 . a (possibly partial) compactification of D/Γ, to − ∈ \{ } obtain an algebraic compactification of M with n If the manifold X is projective, then H (X) admits (mixed) Hodge-theoretic modular meaning. This a polarization by the hard Lefschetz theorem. seems most plausible if (i) φ is injective (i.e. a The classifying spaces for HSs on V with D(h,Q) global Torelli theorem holds), (ii) D/Γ is algebraic, given h and polarized by Q are called period do- and (iii) φ is onto a Zariski open subset — as p p′,n p′ mains. Writing f := Pp p h − , (5a) defines a in the examples of §1. The bad news is that, if ′≥ projective subvariety is not TD, then it obstructs (iii), and D/Γ W ˇ p is usually non-algebraic (Griffiths-Robles-Toledo). (h,Q) Q Grass(f ,VC) D ⊂ p Though the situation is better than that makes it called the compact dual, inside which (5b) de- sound, we will start by discussing compactifications scribes (h,Q) as an analytic open subset. Writing in the “classical” case where = TD, which is to := Aut(D V, Q) (an orthogonal resp. symplectic say, where the family of HSsW parametrized by D/Γ QG -algebraic group for n even resp. odd), (R) acts satisfies (6) (yielding a tautological VHS). G transtitively on (h,Q) and (C) on ˇ(h,Q). There is D G D 2.3.2. The classical case. Given a HS ϕ D, ϕ(z) a plethora of (Hodge or Mumford-Tate) subdomains ∈ acts on the Lie algebra gC End(VC) of G(C) ⊂ D (h,Q), by conjugation, inducing a Hodge decomposition ⊂D p, p p constructed as follows: the Hodge group of a PHS gC = pgϕ − =: pgϕ of weight 0. Since TϕD is ⊕ ⊕ p 1 identified with g and with g− , we see (V, F •, Q) is the smallest Q-algebraic subgroup G ⊕p<0 ϕ Wϕ ϕ 1 + ≤ gp with ϕ(S ) G(R), and we set D := G(R) .F • that = TD forces ϕ = 0 for p = 1, 0, 1. G ≤ W { } 6 − (and Dˇ := G(C).F •). For instance, these may pa- So the “symmetry” ϕ(√ 1): D D fixing ϕ has − → + 0 rametrize HSs stabilized by a fixed cyclic automor- differential IdTϕD at ϕ, and D = G(R) /G (R) = − ∼ ∼ phism of V , as we saw in several examples in §1 Gad(R)+/K is a Hermtian symmetric domain (of (with D a complex ball and G(R) indefinite uni- noncompact type). Conversely, the irreducible tary). HSDs are classified by special nodes on connected Now let Xm m M be a smooth algebraic fam- Dynkin diagrams, which leads to the list { } ∈ ily of smooth projective varieties. We may iden- An: D = SU(p, q)/S(U(p) U(q)) n ∼ tify H (Xm, Q) with a fixed V up to the action of • (p+q = n+1; p,q > 0) × monodromy, and consider the variation of the struc- B + n: D ∼= SO(2, 2n 1) /(SO(2) SO(2n 1)) tures on VC induced by the Hodge theorem. Write • C R − × − n: D ∼= Sp2n( )/U(n) D for the smallest Hodge subdomain containing all • D + n: D ∼= SO(2, 2n 2) /(SO(2) SO(2n 2)) these structures, and Γ G(Q) for the monodromy • − × − ≤ or SO∗(2n)/U(n) group. The key observation, due to Griffiths, is that E6, E7: exceptional HSDs. p,q p while Vm varies nonholomorphically, FmV varies • C They are realized as Hodge domains (in some con- holomorphically and satisfies the infinitesimal pe- nected component + of a period domain) by tak- riod relation (a.k.a. Griffiths transversality) h ing a representationD V of G, and considering the p 1 p 1 (6) dFm ΩM Fm− . tautological VHS (of some weight n) induced by the ⊂ ⊗ 1 0 1 decomposition gC = g− g g (and polarized ϕ ⊕ ϕ ⊕ ϕ Abstractifying this yields the notion of a (polarized) by the Killing form) as ϕ varies over D. In partic- variation of HS M, consisting of a Q-local V → ular, taking V to be the standard representation in system V and a filtration of V M by (sections B D C + ⊗ O case k+1 / k +1 resp. g yields D = (1,k,1) resp. of) holomorphic subbundles, such that the fibers 2 2 D ( H ); and these are the only full period do- are PHSs, and the flat connection annihilating V (g,g) ∼= g Dmains with = T . Of course, for Hodge num- satisfies (6). bers differentW from theseD (i.e. (a, b, a) with a 2, Any PVHS yields a (locally liftable, holomorphic) or level 3), we can still have “classical” period≥ period map map targets;≥ they just have to be proper Hodge (7) φ: M D/Γ, subdomains. → ALGEBRAIC AND ANALYTIC COMPACTIFICATIONS OF MODULI SPACES 13

Now assume that Γ G(Q) stabilizes a lattice the z ) and Γ := stab(σ) Γ in a Hodge-theoretic ≤ { i} σ ∩ VZ V (as monodromy groups do), and that fur- boundary component B(σ)/Γσ, see [12]. Passing ⊂ W thermore Γ is of finite index in G(Q) Aut(VZ) to the Gr of the LMHS maps this down to a B- ∩ • (i.e. Γ is arithmetic). The Baily-Borel theorem B stratum, forgetting “extension classes” (like the then guarantees the existence of enough Γ-invariant semi-periods in the examples). 11 sections of powers of the canonical bundle KD Given a suitable Γ-compatible collection of such (i.e. modular forms) to present D/Γ as a quasi- cones (closed under taking intersections and faces) projective variety, which is smooth for Γ torsion- called a projective fan Σ in gR, D/Γ and the free (and called a locally symmetric variety). The B(σ)/Γσ σ Σ glue together into a smooth projec- { } ∈ resulting compactification tive toroidal compactification (D/Γ)Σ, which is a 0 k (D/Γ)∗ := Proj( H (D/Γ,K⊗ )) resolution of singularities of (D/Γ)∗ [1]. Such a k D/Γ 14 ⊕ fan always exists. In general, φ∗ factors through is (as a set) a disjoint union of D/Γ and finitely (D/Γ)Σ only after birationally modifying M along many Baily-Borel (B-B) boundary strata B /Γ , ∗ B∗ M M. where B∗ runs over Γ-equivalence classes of rational Finally,\ if D is a ball or type-IV domain, holomorphic path components in ∂D Dˇ. ⊂ and a Γ-invariant hyperplane configuration, These strata have a Hodge-theoretic characteri- thereH is an important semitoroidal compactification zation. Given a period map (7) with locally sym- ((D )/Γ)∗ due to Looijenga. Roughly speaking, metric target, and M M any good compactifica- it involves\ H blowing up the B-B boundary where /Γ 12 ⊃ tion, Borel’s extension theorem yields a holomor- meets it, and then blowing down /Γ. H phic map φ∗ : M (D/Γ)∗ restricting to φ. (By H → GAGA, φ∗ hence φ is algebraic; and it follows that 2.3.3. The general case. When D/Γ is not alge- the above algebraic structure on D/Γ is unique.) braic, the obvious question is “what about the The key point is then that φ∗ M M simply records image of the period map?” While Griffiths and | \ the limit of the flag F • V , up to Γ-equivalence. Sommese were able to settle the cases of M com- ⊂ This has the effect (seen in the examples of §1) of pact and φ(M) smooth (respectively), the general killing off all finite limits of periods of a form ω if problem remained open until 2018, when Bakker, one of its periods blows up, since these periods are Brunebarbe and Tsimerman [3] used algebraization projective coordinates of a line C[ω] F n V . results in o-minimal geometry to show that φ(M) is ⊆ ⊂ Hodge-theorists express limits of periods in terms always quasi-projective.15 They also show that the of limiting mixed Hodge structures (LMHS). Pass- restriction of the Griffiths line bundle LD/Γ to φ(M) ing to a finite cover, we may assume Γ is neat.13 is ample. Ongoing work of Green, Griffiths, Laza loc Let z1,...,zd be local coordinates at x M M = and Robles aims to construct a projective compact- ∈ \ z1 zk =0 . Then the corresponding local mon- ification φ(M)∗ to which L has an ample exten- { ··· } D/Γ odromy generators T1,...,Tk Γ commute and sion, and which is also Hodge-theoretically modular ∈ W are unipotent, so that the Ni = log(Ti) generate in the sense of recording the Gr (LMHS) with re- • a rational nilpotent cone σ = σx gR. There is spect to a good compactification M M. So this a unique increasing filtration W ⊂of V satisfying ⊃ • would yield a generalization of the B-B compactifi- k W ∼= W NW W 2 and N : Grn+kV Grn kV for any cation to the nonclassical case. • ⊂ •− → − N int(σ). Setting An earlier construction of Kato and Usui [11] ∈ z P log( i) N extends toroidal compactifications to the general i 2π√ 1 i Flim• := lim e− − Fz•, z 0 setting. The object DΣ/Γ which they associate → to a Γ-compatible fan Σ in gR (by “gluing in” the LMHS at x is the triple (V, W , Flim• ), which we Cσ • the B(σ)/Γσ σ Σ/Γ) is, outside the classical case, record modulo e (to eliminate the dependence on only{ a partial\}compactification∈ (as a “log mani- 2 11 fold”) with “slits” (imagine “compactifying” C One can also use the Griffiths line bundle LD/Γ := p,n−p ⊗p \ p(det V ) , which is always a rational tensor power ⊗ 14 of KD/Γ in the classical case. For ball quotients, the cones are rays and it is unique. 12 This means that M is smooth and proper, and M M On the other hand, for quotients of Hg, the reduction theory is a simple normal crossing divisor. \ of quadratic forms provides several options (cf. §3.5). 13i.e., its eigenvalues generate a torsion-free subgroup of 15The assumption in [3] that Γ is arithmetic is easily re- C∗. moved by appealing to Griffiths’s Properness Theorem. 14 PATRICIO GALLARDO AND MATT KERR

x =0 by adding in just the point (0, 0) ). How- Hodge numbers (1, n 3); and this is what fails for { } { } − ever, if M M is a good compactification, and ev- n> 12. ⊃ ery monodromy cone σx (x M M) of the period In a complementary direction, using the stable map φ is contained in a cone∈ of Σ,\ then they never- pairs described in §2.2, we obtain the Hassett mod- theless obtain a compactification φ : M D /Γ uli space M0,w+ǫ of n-pointed rational curves with Σ → Σ of φ by its “LMHS mod Γ”, with image a sepa- weights w + ǫ. This compactification of M0,n is rated compact algebraic space.16 So if one has a a smooth projective variety, and it admits a mor- 1 n Torelli theorem for φ, and can prove (say) that phism to (P ) //w SL2(C). We also have a unique tor Γ. σx x M M is a fan, then one obtains toroidal compactification, Bn 3/Γw , discussed in { | ∈ \ } − a Hodge-theoretically modular compactification of §2.3. Recent work of the authors with L. Schaf- M. While we are only aware of generic Torelli re- fler [7] found that there is an isomorphism between sults in the nonclassical case so far (such as those M0,w+ǫ/Sm and the toroidal compactification. Our of Donagi and Voisin for projective hypersurfaces), route leads us to the following commutative dia- this appears to be a promising line of inquiry for gram for n 12: future research. ≤

tor / ∼= / M0,n M0,w+ǫ/Sm Bn 3/Γw 3. Compactifying the examples − We are near the end of our tour. But we did make a promise about new routes to old haunts, or 1 n ∼= / (P ) //w SL2 Sm (Bn 3/Γw)∗ something like that. According to legend, a certain × − speakeasy in a certain city had two doors to two different streets: an entrance for the police, and an exit for the patrons (address ‘86’). Without ‘taking 3.2. Cubic surfaces. In §1.3, we learned that ֒ out’ our readers, we shall finally try to describe how there are injective period maps M ord → the different compactifications and methods come B4/Γ(√ 3) and M ֒ B4/Γ. Allcock, Carlson together for our examples from §1. and Toledo− showed in→ 2000 that these maps extend to isomorphisms between the respective GIT and 3.1. Beyond Picard curves. Deligne and Baily-Borel compactifications. Mostow showed in 1986, with additional con- Along the MMP route, Hacking, Keel, and tributions by Doran in 2004, that there are Tevelev considered the moduli space parametrizing isomorphisms between certain GIT completions pairs (X,D) where X is a smooth non-Eckardt cu- of M0,n and Baily-Borel compactifications of bic surface and D is the sum of its 27 lines Li (which (n 3)-dimensional ball quotients. From their then intersect transversely). The divisor is neces- − theorems one obtains a collection of weights sary because KX + D is an ample divisor but KX n w Q>0 and integers m N such that is not. The moduli space of such pairs is open in ∈ 1 n ∈ B w P w ( n 3/Γ )∗ ∼= ( ) // SL2 Sm (where Sm is Mord, and we can compactify it by adding stable the −mth symmetric group).× For a list of these pairs. Their work shows that the resulting compact cases see Tables 2 and 3 in [7]. The isomorphisms KSBA moduli space Mord is a smooth projective variety M B w GIT . compactify period maps 0,n/Sm ֒ n 3/Γ admitting a morphism to M P1 → − ord associated to cyclic covers of branched in a Another approach is to consider weighted pairs manner dictated by the configuration of weighted 27 1 (X, Pi=1( 9 + ǫ)Li), where the coefficient is the points. In each case, the cyclic automorphism of 27 1 1 “smallest” one for which KX + Pi=1( 9 + ǫ)Li is the covering curve has an eigenspace in H with W ample. Let Mord be the KSBA moduli space 16The original hope in [11] was to have a complete fan, parametrizing such weighted pairs and their stable such that DΣ/Γ would compactify images of every period degenerations. The authors’ work with L. Schaffler map φ into D/Γ; but by work of Usui’s student Watanabe, W ν [7] shows that its normalization (Mord) is isomor- these need not exist as soon as has integral manifolds phic to both the unique toroidal compactification of dimension > 1. However, it stillW seems likely that, for each φ, one can produce a suitable fan; and this is perfectly of the ball quotient and a compactification con- consistent with the classical case, since there (and only there) structed by Naruki in 1982 using Cayley’s cross- IdD/Γ is a period map. ratios. ALGEBRAIC AND ANALYTIC COMPACTIFICATIONS OF MODULI SPACES 15

We arrive then at the following diagram. singularities is GIT-stable iff Y has at worst ADE MADE MGIT KSBA W tor singularities. Thus, there is an open / ν ∼= / ⊂ M (M ) B4/Γ √ 3 parametrizing them. The period map described in ord ord − §1.5 extends (by Griffiths properness [5]) to a reg- ADE ular morphism M D/Γ whose image is the GIT → ∼= / complement of an arrangement known as . Sub- M (B4/Γ(√ 3))∗ ord − sequent work of Laza shows that thereH is∞ an iso- /W (E6) /W (E6) morphism from the GIT quotient to the Looijenga KE GIT compactification associated to . ∼= / ∼= / M M (B4/Γ)∗ In another direction, a compactificationH∞ can also be obtained via K-stability because smooth cubic K3 surfaces. 3.3. We are entering a region where fourfolds are Fano varieties admitting a KE metric. much less is known, so we will focus on the case of KE Liu showed in 2020 that the moduli space M is degree two. For constructing a functorial compact- isomorphic to the GIT quotient. Thus, we arrive via ification in terms of stable pairs (X,D), we need three different roads to the same compactification: a choice of an ample divisor D. Which one shall we use? One option described by Laza is to use a KE GIT M ∼= M ∼= ((D )/Γ)∗. divisor in the degree 2 linear system. However, this \ H∞ increases the dimension of the moduli problem by 3.5. Abelian varieties. We will not do justice to two. our last stop. Yet, we include it because g ad- A A natural choice which does not increase the di- mits several well-studied toroidal compactifications. mension is available for polarized K3 surfaces: the They depend on an admissible fan covering of the ramification divisor R X for the 2:1 map X P2 rational closure of the space of positive definite sym- branched over C ⊂P2. To describe the stable→ metric g g-matrices. There are three classical fans: 6 × degenerations for the⊂ pairs (X, ǫR) is the same the second Voronoi, the perfect cone, and the cen- 2 1 tral cone decomposition. They lead to three com- as describing them for the pairs P , ( 2 + ǫ)C6. pactifications of which coincide for g 3 but This last moduli space (of plane curves) was stud- Ag ≤ ied for arbitrary degree by Hacking in 2003 and are different in general. Among those, the second partially described in the case of sextic plane Voronoi compactification has a modular interpreta- curves. In 2019, Alexeev, Engel, and Thomp- tion due to Alexeev: it is the normalization of the K3,slc main irreducible component of the moduli space of son completely described the moduli space M2 of pairs (X, ǫR). Furthermore, they showed that stable semi-abelic pairs (see also subsequent work there exists a toroidal compactification (for the so- by Olsson in the context of log-geometry). called Coxeter fan Σcox) mapping to the normal- For which choices of a fan does the Torelli map slc M extend to a regular map M ( ) ? M ν g g g g Σ ization ( 2 ) . Along with the work of Looi- For→ the A 2nd Voronoi fan, a positive answer→ A was jenga and Shah, which extends the period map from K3,GIT given by Mumford and Namikawa in 1976. In 2011, Blω(M2 ) to the Baily-Borel compactification, Alexeev and Brunyate give a positive answer for this results in the diagram: the perfect cone, and a negative one for the central cone when g 9. The reader wishing to learn more ≥ K3,slc ν o about these developments, and moduli of abelian (M2 ) (D/Γ2)Σcox ◗◗◗ varieties more generally, may consult [20] and the ◗◗◗ ◗◗◗ references therein. ◗◗◗ ◗◗( K3,GIT / Blω(M2 ) (D/Γ2)∗ References ◗◗◗ ◗◗◗ ◗◗◗ [1] A. Ash, D. Mumford, M. Rapoport and Y. Tai, ◗◗◗ ◗◗( “Smooth compactifications of locally symmetric K3,GIT ∼= / varieties (2nd Ed.)”, Cambridge University Press, M2 ((D ω)/Γ2)∗ \ H 2010. MR2590897 3.4. Cubic 4-folds. The GIT compactification [2] V. Alexeev Moduli of weighted hyperplane ar- GIT rangements, with applications Advanced Course M was described by Laza in 2007. In particu- on Compactifying Moduli Spaces, 1, 2013. lar, he showed that a cubic fourfold Y with isolated MR3380944 16 PATRICIO GALLARDO AND MATT KERR

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