FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 21 definitions, 5 theorems, 3 lemmas, 1 corollary, 14 examples, 3 constructions and 2 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
In practice \(L\) is often the lattice associated to a generalized Dynkin diagram \(\Gamma\) . However, in general the roots corresponding to the vertices of \(\Gamma\) do not define a vanishing lattice. Theorem 18.1.7 provides a large class of Dynkin diagrams that do not suffer from this defect. As demonstrated in Section 18.2, these can be used to determine the monodromy group of many singularities.
Definition / construction / notation
In this section \(( L , q )\) is a quadratic integral lattice with polarization \(^ { b }\) . By ”root” we shall mean ” \(\left( - 2 \right)\) -root” and accordingly we set \(\epsilon = -\) . With appropriate changes all results are valid for 2-roots and \(\epsilon = +\) . The set of roots in \(L\) is denoted \(\Delta ( L )\) , the lattice generated by these \(L _ { \mathrm { r o o t } }\) , and the Weyl group \({ \mathsf { W } } ^ { - } ( L )\) .
Definition / construction / notation
The lattices we consider in this chapter are spanned by roots which form the nodes of a graph. If \(r , r ^ { \prime }\) are such roots and \(b ( \boldsymbol { r } , \boldsymbol { r } ^ { \prime } ) = 1\) , as usual, we connect the corresponding nodes by an unlabelled edge. In the lattices also pairs \(\{ r , r ^ { \prime } \}\) occur for which \(b ( r , r ^ { \prime } ) = - 2\) . We connect the corresponding nodes by a doubly dashed edge. The resulting graphs \(\Gamma\) are called (generalized) Dynkin diagrams and the corresponding even integral lattice \(L _ { \Gamma }\) they give rise to the associated rootlattice. Identifying the vertices of \(\Gamma\) with roots in \(L _ { \Gamma }\) , the Weyl group \({ \mathsf { W } } ^ { - } ( { \Gamma } )\) of \(\Gamma\) is the group generated by the reflections determined by \(\Gamma\) .
Definition / construction
The distinguished root set is \(1\)-connected when any two relevant roots can be joined through roots with successive pairing \(1\).
Definition / construction / notation
We also need a suitable notation to deal with orbits of a possibly infinite set of roots, say \(\Lambda \subset L\) . If \({ \mathsf { W } } ^ { - } ( \Lambda )\) is the group generated by the reflections \(\sigma _ { \lambda }\) , \(\lambda \in \Lambda\) , we shall denote the orbit of \(\Lambda\) by the corresponding bold letter: \[ \Lambda = \mathsf { W } ^ { - } ( \Lambda ) \Lambda \subset L _ { \mathrm { r o o t } } . \]
Lemma 18.1.1
Lemma 18.1.1. Two roots in a 1-connected component of a set of roots \(\Delta \subset L = L _ { \Gamma }\) belong to the same \({ \mathsf { W } } ^ { - } ( \Delta )\) -orbit.
Definition 18.1.2
Definition 18.1.2. 1. A vanishing lattice consists of a pair \(( L , \Delta )\) of a lattice \(L\) and a set of roots \(\Delta\) spanning \(L\) and forming a single orbit under \(W ^ { - } ( \Delta ) =\) \(\langle \sigma _ { r } \rangle _ { r \in \Delta }\) , that is \(\Delta = \Delta\) . 2. A vanishing lattice \(( L , \Delta )\) contains the vanishing lattice \(( L ^ { \prime } , \Delta ^ { \prime } )\) if \(L ^ { \prime }\) is a primitive sublattice of \(L\) and \(\Delta ^ { \prime } \subset \Delta\) .
Remark 18.1.3
Remark 18.1.3. The vertices of a generalized Dynkin diagram \(\Gamma\) form a finite set of roots, say \(\Delta\) , spanning the lattice \(L _ { \Gamma } = \mathbb { Z } \cdot \Delta\) . If \(\Delta\) is 1-connected, by Lemma 18.1.1 all roots in \(\Delta\) are in the same orbit under the reflection group \({ \mathsf { W } } ^ { - } ( \Delta )\) , but \(\Delta\) need not be an entire orbit. Taking \(\Delta = \mathsf { W } ^ { - } ( \Delta )\) instead of \(\Delta\) then yields a vanishing lattice. In general, if the subset \(\Delta\) of roots of \(L\) belongs to a single \({ \mathsf { W } } ^ { - } ( \Delta )\) -orbit, this procedure yields a vanishing lattice. Clearly, this also shows that such a vanishing lattice is determined by the lattice itself.
Definition / construction
The basic complete-vanishing-lattice model uses \(L_{\min}=U\oplus U\prime\oplus A_2(-1)\) together with the displayed six-root spanning set \(T_{\min}\); these roots form a basis and define the associated Dynkin graph.
Example / comparison
The minimal vanishing lattice admits a second displayed root basis obtained by replacing two roots; the transition matrix is unimodular, so both displayed graphs describe vanishing-lattice bases of the same lattice.
Definition 18.1.4
Definition 18.1.4. A vanishing lattice \(( L , \Delta )\) is called complete if it contains a copy of \(( L _ { \mathrm { m i n } } , \Gamma _ { \mathrm { m i n } } )\) .
Definition / construction / notation
This concept is obviously preserved under containment of vanishing lattices; since this plays a central role later on, we set this apart.
Lemma 18.1.5
Lemma 18.1.5. A vanishing lattice containing a complete vanishing lattice is itself complete.
Definition 18.1.6
Definition 18.1.6. A (possibly infinite) subset \(\Lambda\) of roots in \(L\) is called a special subset of roots of \(L\) if it contains three roots \(r _ { 1 } , r _ { 2 } , r _ { 3 }\) , by definition a socle of \(\Lambda\) , which have the following two properties: 1. The intersection behaviour of roots in the socle and with other roots \(r \in \Lambda\) is as in the following graph: 
In other words, \(b ( r _ { 1 } , r _ { 2 } ) = 1\) , \(b ( r _ { 2 } , r _ { 3 } ) = - 2\) , \(b ( r _ { 1 } , r ) = 0\) for \(r \neq r _ { 1 } , r _ { 2 }\) and, finally, \(b ( r , r _ { 2 } ) = b ( r , r _ { 3 } )\) for \(r \neq r _ { 1 } , r _ { 2 } , r _ { 3 }\) . In particular it follows that \(r _ { 1 } , r _ { 2 } , r _ { 3 }\) are independent (consider the Gram matrix). 2. The set \(\Lambda ^ { \prime } = \Lambda - \{ r _ { 1 } , r _ { 2 } \}\) is 1-connected.
Definition / construction / notation
We make some observations and introduce related terminology.
Definition / construction
The hyperbolic plane generated by the indicated roots is the socle plane; the associated splitting \(L=U\oplus L^\prime\) is the socle decomposition.
Theorem 18.1.7
Theorem 18.1.7 ([60, Thm 3]). Let \(( L , q )\) be a quadratic lattice of rank at least 4 containing a special subset Λ of roots which span \(L\) . Then \(( L , \pmb { \Lambda } )\) is a vanishing lattice. If, moreover, \(L\) splits off an extra hyperbolic plane besides the socle plane, then \(\mathsf { W } ^ { - } ( L ) = \mathsf { W } ^ { - } ( \Lambda )\) . More precisely, we have: 1. • Let \(U\) be the socle plane and \(L = U \mathbb { \textcircled { 1 } L ^ { \prime } }\) the corresponding socle decomposition. Then \(L ^ { \prime }\) is spanned by \(\Lambda ^ { \prime } = \Lambda - \{ r _ { 1 } , r _ { 2 } \}\) . • Conversely, let \(L\) be a quadratic lattice of rank at least 4 with a decomposition \(L = U \oplus L ^ { \prime }\) , where \(L ^ { \prime }\) is spanned by a 1-connected set of roots \(\Lambda ^ { \prime }\) . Let \(\{ e , f \}\) be the standard basis of \(U\) and \(r _ { 3 } \in \Lambda ^ { \prime }\) . Then the set of roots \(r _ { 1 } = e - f , r _ { 2 } = - e + r _ { 3 } , r _ { 3 } , r \in \Lambda ^ { \prime }\) forms a special subset of roots. 2. With \(\Lambda ^ { \prime }\) as in \(\mathit { 1 }\) , the roots \(r ^ { \prime } \in \Lambda ^ { \prime }\) belong to the same \(\mathsf { W } ^ { - } ( L ^ { \prime } )\) -orbit. 3. \(\psi _ { U } L ^ { \prime } \subset \mathsf { W } ^ { - } ( \Lambda )\) 4. If, moreover, \(L ^ { \prime } = U ^ { \prime } \oplus L ^ { \prime \prime }\) for some hyperbolic plane \(U ^ { \prime }\) , then \(W ^ { - } ( \Lambda ) \ =\) \({ \mathsf { W } } ^ { - } ( L )\) . In other words, the Weyl group of \(L\) is generated by the reflections \(\sigma _ { r }\) , where the roots \(r\) belong to \(\Lambda\) .
Example / comparison
Example 18.1.8. The roots in the Coxeter–Dynkin graph \(T^1_{p,q,r}\) form a special set and \(T^1_{p,q,r}\simeq\widetilde T_{p,q,r}\oplus U\). For \(p\ge2\), \(q\ge3\), \(r\ge7\), this yields a complete vanishing lattice of the class used in Theorem 18.1.9.
Theorem 18.1.9
Theorem 18.1.9 ([63, Thm. 5.3.4]). Let \(( L , \Delta )\) be a complete vanishing lattice. Then \(\mathsf { W } ^ { - } ( \Delta ) = \mathsf { W } ^ { - } ( L ) = \mathsf { O } ^ { - , \# } ( L )\) .
Lemma 18.1.10
Lemma 18.1.10. For a complete vanishing lattice \(( L , \Delta )\) , the set \(\Delta\) is 1-connected.
Remark: Remark
Remark. \(A _ { 2 } ( - 1 )\) is not 1-connected: the set of roots \(\{ a _ { \mathrm { 1 } } , \alpha _ { \mathrm { 2 } } , - ( \alpha _ { \mathrm { 1 } } + \alpha _ { \mathrm { 2 } } ) \}\) as well as the set \(\left\{ - \alpha _ { 1 } - \alpha _ { 2 } , \alpha _ { 1 } + \alpha _ { 2 } \right\}\) is 1-connected, but these two sets cannot be connected.
Example / comparison
8, its reflection group is maximal. It forms a complete \(p \geq 2 , q \geq 3 , r \geq 7\) since \({ \cal T } _ { p , q , r } ^ { 1 } = { \cal T } _ { p , q , 4 } \oplus { \cal U } \oplus { \cal U } ^ { \prime } \oplus { \cal A } _ { r - 6 } ( - 1 )\) , which contains \(U \oplus U ^ { \prime } \oplus A _ { 2 } ( - 1 )\) (all legs of \(T _ { p , q , 4 }\) have \(\geq 2\) nodes).
Example / comparison
Every vanishing lattice containing \(T _ { p , q , r } ^ { 1 }\) as in the first example is a complete vanishing lattice.
Definition / construction / notation
18.2.A Introduction to isolated hypersurface singularities. We let \(z _ { 1 } , \dots\) \(z _ { n + 1 }\) be the standard coordinates on \(\mathbb { C } ^ { n + 1 }\) and we shall write \(\pmb { z } = ( z _ { 1 } , \dots , z _ { n + 1 } )\) for the vector with coordinates ??1, . . . , ????+1
Definition / construction / notation
which is symmetric and bilinear, but not necessarily non-degenerate. We denote this pairing with a dot. The resulting lattice \[ \Lambda _ { X , x } = ( H _ { n } ( M _ { * } , \mathbb { Z } ) , \cdot ) \]
Definition / construction / notation
is called the Milnor lattice of \(( X , x )\) . It comes equipped with the action of the monodromy operator \(T _ { X , x }\) which is induced by moving the Milnor fiber of \(f\) once about the origin in anti-clockwise direction.
Example / comparison
The function \(f ( z ) = z _ { 1 } ^ { 2 } + \cdot \cdot \cdot + z _ { n + 1 } ^ { 2 }\) has a single critical point at the origin. The corresponding singularity is an ordinary double point. If \(t \neq 0\) , the fiber \(f ^ { - 1 } ( t )\) is the smooth hypersurface \(\begin{array} { r } { \sum _ { j } z _ { j } ^ { 2 } = t } \end{array}\) . The Milnor number is 1 since \(H _ { n } ( M _ { * } , \mathbb { Z } ) = \mathbb { Z }\) is generated by a so-called vanishing cycle \(\delta\) given by \(\{ x _ { 1 } ^ { 2 } + \cdot \cdot \cdot + x _ { n + 1 } ^ { 2 } = t _ { * } \} \cap \{ | z | ^ { 2 } < \eta \}\) , where \(z _ { j } = x _ { j } + i y _ { j }\) , \(j = 1 , \ldots , n + 1\) , and \(t _ { * }\) is the \(t\) -coordinate of the point \(^ *\) . It is well known (cf. e.g. [134, §6]) that \(\delta \cdot \delta = 2 \epsilon\) and that the monodromy operator \(T _ { X , x }\) is the reflection \(s _ { \delta }\) , corresponding to \(\delta\) .
Example / comparison
More generally, consider the du Val singularities of Table 4.5.1. The minimal resolution of these singularities is a connected chain of rational curves of selfintersection \(- 2\) whose dual graph is one of the Dynkin diagrams \(A _ { k } , D _ { k }\) , \(E _ { 6 } , E _ { 7 } , E _ { 8 }\) . The construction on page 102 shows that a tubular neighborhood of such a chain is a disc bundle over a wedge of 2-spheres, and from this it follows that the Milnor fibre has the same homotopy type as the latter wedge of 2-spheres. Du Val-type singularities occur in any dimension and are denoted by the same symbols as for dimension 2. Their Milnor lattices are given in Table 18.2.1. The Milnor number for \(A _ { k } , D _ { k }\) and \(E _ { k }\) equals \(k\) . Table 18.2.1: Milnor lattice of the du Val singularities 
Definition / construction / notation
The multi-index notation we use means this: if \(B _ { j } = ( b _ { 1 , j } , \dotsc , b _ { n + 1 , j } )\) , one sets \[ z ^ { B _ { j } } = z _ { 1 } ^ { b _ { 1 , j } } \cdot \cdot \cdot z _ { n + 1 } ^ { b _ { n + 1 , j } } . \]
Definition / construction / notation
For a semi-universal unfolding \(\mathbf F\), its critical-value set is the discriminant locus \(D_{\mathbf F}\) in the base.
Example / comparison
The ordinary double point \(z _ { 1 } ^ { 2 } + \cdot \cdot \cdot + z _ { n + 1 } ^ { 2 } = 0\) of Example 18.2.1 has the function \(z _ { 1 } ^ { 2 } + \dots + z _ { n + 1 } ^ { 2 } - t\) as its semi-universal unfolding and so \(\tau = \mu = 1\) . The discriminant locus is the origin and the modality is \(0\) .
Example / comparison
The other Du Val singularities have \(\mu = \tau\) as well and the modality is \(0\) . For instance, for \(E _ { 6 }\) the jacobian ring has a basis \(\{ 1 , z _ { 2 } , z _ { 3 } , z _ { 3 } ^ { 2 } , z _ { 2 } z _ { 3 } , z _ { 2 } z _ { 3 } ^ { 2 } \}\) consisting of 6 elements and it also contains the defining polynomial.
Example / comparison
The hyperbolic singularities \(T _ { p , q , r }\) . In dimension 2 these singularities are given by: \[ f = 0 , \quad f = x ^ { p } + y ^ { q } + z ^ { r } + a x y z , \quad a \in \mathbb { C } ^ { \times } , \quad { \frac { 1 } { p } } + { \frac { 1 } { q } } + { \frac { 1 } { r } } < 1 . \] With \(J _ { f }\) the jacobian ideal, we see that the vector space \(\mathbb { C } [ \left\lfloor x , y , z \right\rfloor ] / ( J _ { f } , f )\) is generated by 1 together with the monomials \(x ^ { k }\) , \(k = 1 , \ldots , p - 1\) , \(y ^ { \ell }\) , \(\ell = 1 , \ldots , q - 1\) , \(z ^ { m }\) , \(m = 1 , \ldots , r - 1\) , and so \(\tau ( f ) = p + q + r - 2\) . Because of the relation \[ p ^ { - 1 } x f _ { x } + q ^ { - 1 } y f _ { y } + r ^ { - 1 } z f _ { z } - f = ( p ^ { - 1 } + q ^ { - 1 } + r ^ { - 1 } - 1 ) a x y z , \] we find \(\mu = p + q + r - 2\) . For all non-zero \(a\) the singularities have the same Milnor number. It turns out (see [5, Ch. I.2.3]) that the isomorphism class of the singularity varies with \(a\) , and so it is a unimodal singularity.
Example / comparison
Milnor lattices can have a non-zero null-space. The simplest are the so-called parabolic singularities. These are also unimodal, again by [5, Ch. I.2.3]. Table 18.2.2: Milnor lattice of the parabolic singularities 
Definition / construction
The local monodromy group is the image of the fundamental group of the discriminant complement acting on the homology of a reference Milnor fibre.
Definition / construction / notation
and such that the fiber \(\boldsymbol { F } _ { * , 0 }\) of \(\pmb { F }\) passing through the point \(( * , 0 )\) intersected with the ball \(| z | < \eta\) is the Milnor fiber \(M _ { * }\) of \(( X , x )\) . By definition \(H _ { n } ( M _ { * } , \mathbb { Z } ) = \Lambda _ { X , x }\) is free of rank \(\mu\) . It is generated by vanishing cycles as follows. A general line \(\ell\) intersects the discriminant hypersurface \(\mathrm { D } _ { \mathbf { F } }\) transversally. It turns out that \(\ell \cap\) \(U \cap \mathsf { D } _ { \mathbf { F } }\) consists of precisely \(\mu\) points, say \(t _ { 1 } , \ldots , t _ { \mu }\) , and the fiber of \(\pmb { F }\) over each of these points has exactly one ordinary double point, say \(s _ { j }\) , and no other singular points. In Example 18.2.1 we found a vanishing cycle \(\delta _ { j } ^ { \prime }\) in the fibre near \(s _ { j }\) . We want to transport it to \(( * , 0 ) \in U\) along a well-chosen path.
Definition / construction / notation
A distinguished system of generators of the local discriminant-complement fundamental group is formed by the standard loops \(\gamma_j\) around the critical values.
Theorem 18.2.3
Theorem 18.2.3. The set \(\Delta _ { X , x } = \{ \delta _ { 1 } , . . . , \delta _ { \mu } \}\) of vanishing cycles is a basis for the Milnor lattice \(\Lambda _ { X , x }\) . All of these vanishing cycles are in the same orbit under the action of the monodromy group. In particular \(( \Lambda _ { X , x } , \Delta _ { X , x } )\) is a vanishing lattice.
Example / comparison
A, but are obtained from \(T _ { p q r } ^ { \mathrm { 1 } }\) by deleting the root \(r _ { 1 }\) . Since then \(r _ { 2 } - r\) is orthogonal to itself and all other roots, the lattice is isometric to \(\widetilde { T } _ { p , q , r } \oplus 0\) .
Example / comparison
For the \(14\) exceptional unimodal singularity families, the vanishing lattice is \(T^1_{p,q,r}\); the resulting vanishing lattices are complete and their monodromy group equals the reflection/stable orthogonal group.
Example / comparison
For Du Val singularities the monodromy group is maximal in the stated sense; the quotient of the full isometry group by monodromy is the symmetry group of the Dynkin diagram.
Example / comparison
Adjacency of isolated singularities yields an inclusion of vanishing lattices; completeness therefore descends from an adjacent singularity with complete vanishing lattice.
Definition / construction / notation
parametrizes the smooth hypersurfaces. A generic line \(\ell \subset \mathbb { P } _ { n , d }\) meets \(\mathsf { D } _ { d }\) transversally in points \(P\) corresponding to hypersurfaces with exactly one ordinary double point. Every such point \(P\) gives rise to a vanishing cycle \(\delta _ { P }\) in a nearby smooth fiber. This situation resembles the local situation we just treated. The present situation is the one Lefschetz had in mind in the classical monograph [138] and this is why the restriction \(F _ { n , d }\) to the line \(\ell\) is called a Lefschetz pencil. The global theory mirrors the local theory exactly. This leads to an associated vanishing lattice, but this time in cohomology, as we shall now explain.
Theorem 18.3.1
Theorem 18.3.1 ([134, §7]). Let \(F _ { n , d }\) be the universal family of degree ?? hypersurfaces in \(\mathbb { P } ^ { n + 1 }\) with discriminant \(\mathsf { D } _ { d }\) , \(\ell \subset \mathbb { P } _ { n , d }\) a line transversally intersecting \(\mathsf { D } _ { d }\) so that \(F _ { n , d } | _ { \ell }\) is a Lefschetz pencil. Choose a base point \(* \in \mathcal { E } \cap U _ { n , d }\) . Then 1. Under Poincar´e duality the set \(\Delta _ { d , n }\) of vanishing cocycles obtained by taking the Poincar´e duals of the vanishing cycles \(\delta _ { P }\) , \(P \in \mathsf { D } _ { d } \cap \ell\) (transported to ∗), generate the primitive homology \(H _ { \mathrm { p r i m } } ^ { n } ( X _ { * } )\) . 2. All vanishing cocycles are in the same orbit under the monodromy group, and \(( H _ { \mathrm { p r i m } } ^ { n } ( X _ { * } ) , \Delta _ { d , n } )\) is a vanishing lattice.
Construction / comparison
For a general line in the hypersurface parameter space, restriction induces a surjection from the fundamental group of the line-section discriminant complement onto the fundamental group of the full smooth locus.
Construction / comparison
When the Milnor intersection form is non-degenerate, inclusion of a Milnor fibre into a nearby smooth hypersurface embeds the local vanishing lattice into the global primitive-cohomology vanishing lattice.
Construction / comparison
The global monodromy action fixes the hyperplane class and extends across the full middle cohomology, giving an embedding of the primitive monodromy group into the stabilizer of that class in the full orthogonal group.
Theorem 18.3.2
Theorem 18.3.2 ([18]). The monodromy group \(\mathsf { M o n } ( \Lambda _ { n , d } )\) of the universal family of degree ?? smooth hypersurfaces in \(\mathbb { P } ^ { n + 1 }\) , \(n\) even, equals \(0 ^ { \in \mathcal { H } } ( \Lambda _ { d , n } )\) . Under the embedding (18.5) the group \({ \mathsf { M o n } } ( \Lambda _ { n , d } )\) has index 2 in \(\mathsf { O } _ { h } ( H ^ { n } ( X _ { * } , \mathbb { Z } ) )\) if \(d \ge 4\) or \(d = 3 , n \neq 2\) . In the remaining cases these groups are equal.
Corollary 18.3.3
Corollary 18.3.3. Let \(n\) be even and \(\geq 4\) and let \(\ell \in H ^ { n } ( X _ { * } , \mathbb { Z } )\) be the class of \(a\) linear section. Then \[ \mathrm { I m } \left[ \mathsf { D i f f } ^ { + } ( X _ { * } ) \longrightarrow \mathsf { O } ( H ^ { n } ( X _ { * } , \mathbb { Z } ) ) \right] = \left\{ \begin{array} { l l } { \mathsf { O } _ { \ell } ( H ^ { n } ( X _ { * } , \mathbb { Z } ) ) \quad } & { i f n \equiv 0 \bmod 4 . } \\ { \mathsf { O } _ { \ell } ( H ^ { n } ( X _ { * } , \mathbb { Z } ) ) \times \{ \pm 1 \} \quad } & { i f n \equiv 2 \bmod 4 . } \end{array} \right. \]