##
**Compactifications defined by arrangements. II: Locally symmetric varieties of type IV.**
*(English)*
Zbl 1079.14045

Two decades separate the origins of this article from its publication, during which time the techniques it describes have acquired renewed relevance and further applications. Many examples are given here, and thus it goes well beyond (although it contains) a revision of the Nijmegen preprint of 1985 [Semi-toric partial compactifications I, report 8520, University of Nijmegen; per bibl.] which was its first version.

The main results concern the construction and description of a new class of compactifications of (open parts of) locally symmetric varieties. Here they are worked out for type IV locally symmetric varieties, which together with the ball quotient case studied in the first part of the paper [Duke Math. J. 118, No. 1, 151–187 (2003; Zbl 1052.14036)] covers all the applications found so far. The compactifications are defined in terms of arithmetic arrangements \({\mathcal H}\) of hyperplane sections of the symmetric domain \({\mathbb D}\): that is, those that are locally finite and invariant under some arithmetic group \(\Gamma\). Thus \({\mathcal H}\) defines a hypersurface \(D\subset X=\Gamma\backslash{\mathbb D}\). But the closure of \(D\) in the Baily-Borel compactification \(X^{\text{bb}}\) of \(X\) will not in general be \({\mathbb Q}\)-Cartier. In particular it is not, in general, defined by an automorphic form. Therefore one should choose a compactification adapted to the pair \((X,D)\), i.e. to \(X^\circ=X\setminus D\), in as natural a way as possible. Such a compactification \(\widehat{X^\circ}\) of \(X^\circ\) is what is constructed here.

To construct \(\widehat{X^\circ}\) one first blows up \(X^{\text{bb}}\) along the boundary so as to make each irreducible component of the closure of \(D\) be a Cartier divisor (or at least \({\mathbb Q}\)-Cartier). To do so both economically and constructively one makes use of combinatorial data supplied by \(D\). This part of the procedure is a special case of semitoric partial compactifications, described originally in the Nijmegen preprint and here in §§3–6. These compactifications include, at one extreme, the Baily-Borel compactification, and at the other, toroidal compactifications in the sense of Mumford. An arithmetic hyperplane arrangement \({\mathcal H}\) induces such a compactification \(X^{\Sigma({\mathcal H})}\), given by some cones \(\Sigma({\mathcal H})\) satisfying compatibility conditions weaker than those of the cones of a fan. Trivial data (a single cone, \(\{0\}\)) recover the Baily-Borel compactification. The construction is described for \(1\)-dimensional boundary components in §3 and for \(0\)-dimensional ones in §4: there are no other cases to be considered in the type IV situation.

The proofs that \(X^\Sigma\) has the required properties are not given in full. Instead, for reasons of space and clarity, the author gives an outline and refers to the unpublished Nijmegen preprint for details. The outline will be sufficient for experts, however, and the details of the Nijmegen preprint have been checked independently by several people, the reviewer among them.

The other part of the construction of \(\widehat{X^\circ}\) follows in §5. It takes place within \(X\), not at the cusps. One makes some birational modifications on \(D\), in a way not very different to the ball quotient (unitary) case of the first part. The reason for wanting to do this is that one then has an ample line bundle \(\widehat{\mathcal L}\) on \(\widehat{X^\circ}\) that agrees with the bundle \({\mathcal L}\) of automorphic forms in \(X^\circ\). Some consequences of this are described below.

Even before the construction is given the paper contains much of interest. The first two sections are an introduction to boundary components of locally symmetric varieties and the Baily-Borel compactification. In §3, beyond what has already been mentioned, a simple criterion is given (Proposition 3.4) for the closure of \(D\) to be a Cartier divisor in \(X^{\text{bb}}\) (similar results can be found elsewhere, for instance J. H. Bruinier and E. Freitag [Ann. Inst. Fourier 51, No. 1, 1–27 (2001; Zbl 0966.11021)]). Using this the author gives a proof that the moduli space of amply polarised \(K3\) surfaces of genus \(g>2\) without double points cannot be affine, in contrast to the \(g=2\) case described by R. E. Borcherds et al. [J. Algebr. Geom. 7, No. 1, 183–193 (1998; Zbl 0946.14021)]. In other words, the discriminant locus in the moduli space of \(K3\) surfaces of genus \(g\) is not definable by an automorphic form. This was proved by Nikulin for \(g\gg 0\). However, the author asserts that the discriminant locus is irreducible for \(g>2\), although he does not use this nor attempt to prove it. This is a slip: it is false if \(g\equiv 0\pmod 4\).

The analysis of arrangements in tube domains in §5 includes a subsection on arrangements defined by product expansions. In this case there are strong restrictions on the Weyl group, and this leads to a proof of the first part of the mirror symmetry conjecture of V. A. Gritsenko and V. V. Nikulin [Int. J. Math. 9, No. 2, 153–199 (1998; Zbl 0935.11015)]. In the case of the even unimodular lattice of signature \((2,26)\) (two hyperbolic planes plus the Leech lattice) one verifies that a product formula defining the hyperplanes orthogonal to roots should exist: of course it does, being Borcherds’ famous denominator formula.

The main results about arithmetic arrangements are assembled and stated in §7. In particular, under rather weak conditions, one has that the ring \(R=\bigoplus H^0({\mathbb D}^\circ, {\mathcal O}({\mathbb L}^k))^\Gamma\) is finitely generated (in positive degree), and \(X^\circ=\text{Proj}R\). Often this Proj arises as a GIT quotient. Some examples of this, all related to \(K3\) surfaces, are listed in the introduction.

The paper concludes with three cases described in more detail, in reverse order to their historical development: \(K3\) surfaces, Enriques surfaces and the moduli of triangle singularities. The results on Enriques surfaces are mostly due to H. Sterk [Math. Z. 220, No. 3, 427–444 (1995; Zbl 0841.14031); ibid. 207, No. 1, 1–36 (1991; Zbl 0736.14017)]. The work on triangle singularities gave the first motivation for the author to consider this circle of ideas, as long ago as 1980.

The main results concern the construction and description of a new class of compactifications of (open parts of) locally symmetric varieties. Here they are worked out for type IV locally symmetric varieties, which together with the ball quotient case studied in the first part of the paper [Duke Math. J. 118, No. 1, 151–187 (2003; Zbl 1052.14036)] covers all the applications found so far. The compactifications are defined in terms of arithmetic arrangements \({\mathcal H}\) of hyperplane sections of the symmetric domain \({\mathbb D}\): that is, those that are locally finite and invariant under some arithmetic group \(\Gamma\). Thus \({\mathcal H}\) defines a hypersurface \(D\subset X=\Gamma\backslash{\mathbb D}\). But the closure of \(D\) in the Baily-Borel compactification \(X^{\text{bb}}\) of \(X\) will not in general be \({\mathbb Q}\)-Cartier. In particular it is not, in general, defined by an automorphic form. Therefore one should choose a compactification adapted to the pair \((X,D)\), i.e. to \(X^\circ=X\setminus D\), in as natural a way as possible. Such a compactification \(\widehat{X^\circ}\) of \(X^\circ\) is what is constructed here.

To construct \(\widehat{X^\circ}\) one first blows up \(X^{\text{bb}}\) along the boundary so as to make each irreducible component of the closure of \(D\) be a Cartier divisor (or at least \({\mathbb Q}\)-Cartier). To do so both economically and constructively one makes use of combinatorial data supplied by \(D\). This part of the procedure is a special case of semitoric partial compactifications, described originally in the Nijmegen preprint and here in §§3–6. These compactifications include, at one extreme, the Baily-Borel compactification, and at the other, toroidal compactifications in the sense of Mumford. An arithmetic hyperplane arrangement \({\mathcal H}\) induces such a compactification \(X^{\Sigma({\mathcal H})}\), given by some cones \(\Sigma({\mathcal H})\) satisfying compatibility conditions weaker than those of the cones of a fan. Trivial data (a single cone, \(\{0\}\)) recover the Baily-Borel compactification. The construction is described for \(1\)-dimensional boundary components in §3 and for \(0\)-dimensional ones in §4: there are no other cases to be considered in the type IV situation.

The proofs that \(X^\Sigma\) has the required properties are not given in full. Instead, for reasons of space and clarity, the author gives an outline and refers to the unpublished Nijmegen preprint for details. The outline will be sufficient for experts, however, and the details of the Nijmegen preprint have been checked independently by several people, the reviewer among them.

The other part of the construction of \(\widehat{X^\circ}\) follows in §5. It takes place within \(X\), not at the cusps. One makes some birational modifications on \(D\), in a way not very different to the ball quotient (unitary) case of the first part. The reason for wanting to do this is that one then has an ample line bundle \(\widehat{\mathcal L}\) on \(\widehat{X^\circ}\) that agrees with the bundle \({\mathcal L}\) of automorphic forms in \(X^\circ\). Some consequences of this are described below.

Even before the construction is given the paper contains much of interest. The first two sections are an introduction to boundary components of locally symmetric varieties and the Baily-Borel compactification. In §3, beyond what has already been mentioned, a simple criterion is given (Proposition 3.4) for the closure of \(D\) to be a Cartier divisor in \(X^{\text{bb}}\) (similar results can be found elsewhere, for instance J. H. Bruinier and E. Freitag [Ann. Inst. Fourier 51, No. 1, 1–27 (2001; Zbl 0966.11021)]). Using this the author gives a proof that the moduli space of amply polarised \(K3\) surfaces of genus \(g>2\) without double points cannot be affine, in contrast to the \(g=2\) case described by R. E. Borcherds et al. [J. Algebr. Geom. 7, No. 1, 183–193 (1998; Zbl 0946.14021)]. In other words, the discriminant locus in the moduli space of \(K3\) surfaces of genus \(g\) is not definable by an automorphic form. This was proved by Nikulin for \(g\gg 0\). However, the author asserts that the discriminant locus is irreducible for \(g>2\), although he does not use this nor attempt to prove it. This is a slip: it is false if \(g\equiv 0\pmod 4\).

The analysis of arrangements in tube domains in §5 includes a subsection on arrangements defined by product expansions. In this case there are strong restrictions on the Weyl group, and this leads to a proof of the first part of the mirror symmetry conjecture of V. A. Gritsenko and V. V. Nikulin [Int. J. Math. 9, No. 2, 153–199 (1998; Zbl 0935.11015)]. In the case of the even unimodular lattice of signature \((2,26)\) (two hyperbolic planes plus the Leech lattice) one verifies that a product formula defining the hyperplanes orthogonal to roots should exist: of course it does, being Borcherds’ famous denominator formula.

The main results about arithmetic arrangements are assembled and stated in §7. In particular, under rather weak conditions, one has that the ring \(R=\bigoplus H^0({\mathbb D}^\circ, {\mathcal O}({\mathbb L}^k))^\Gamma\) is finitely generated (in positive degree), and \(X^\circ=\text{Proj}R\). Often this Proj arises as a GIT quotient. Some examples of this, all related to \(K3\) surfaces, are listed in the introduction.

The paper concludes with three cases described in more detail, in reverse order to their historical development: \(K3\) surfaces, Enriques surfaces and the moduli of triangle singularities. The results on Enriques surfaces are mostly due to H. Sterk [Math. Z. 220, No. 3, 427–444 (1995; Zbl 0841.14031); ibid. 207, No. 1, 1–36 (1991; Zbl 0736.14017)]. The work on triangle singularities gave the first motivation for the author to consider this circle of ideas, as long ago as 1980.

Reviewer: G. K. Sankaran (Bath)

### MSC:

14J15 | Moduli, classification: analytic theory; relations with modular forms |

32N15 | Automorphic functions in symmetric domains |

14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |

14J28 | \(K3\) surfaces and Enriques surfaces |

32S22 | Relations with arrangements of hyperplanes |

### Citations:

Zbl 1052.14036; Zbl 0966.11021; Zbl 0946.14021; Zbl 0935.11015; Zbl 0841.14031; Zbl 0736.14017
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\textit{E. Looijenga}, Duke Math. J. 119, No. 3, 527--588 (2003; Zbl 1079.14045)

### References:

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