Strange duality between the quadrangle complete intersection singularities

There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and C.T.C.Wall. We derive this duality from the mirror symmetry, the Berglund-H\"ubsch transposition of invertible polynomials, and our previous results about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.


Introduction
V. I. Arnold [A2] observed a strange duality between the 14 exceptional unimodal singularities. It is well known that this duality is a special case of the Berglund-Hübsch duality of invertible polynomials, see e.g. [ET1]. C. T. C. Wall and the first author [EW] discovered an extension of this duality embracing on one hand series of bimodal hypersurface singularities and on the other hand, isolated complete intersection singularities (ICIS) in C 4 . The duals of the ICIS are not themselves singularities but are virtual (k = −1) cases of series (e.g. J 3,k , k ≥ 0) of bimodal singularities. In [EW], the k = −1 cases of the series were called virtual singularities and Milnor lattices were associated to them, but they do not coincide with the Milnor lattices of the germs at the origin by setting k = −1 in Arnold's equations of the series, which are exceptional unimodal singularities with a smaller Milnor number. In [ET3], we showed that the virtual singularities exist in the sense that the equations have to be considered as global polynomials and we derived this extension from the mirror symmetry and the Berglund-Hübsch duality of invertible polynomials.
Arnold's 14 exceptional unimodal singularities are triangle hypersurface singularities, i.e., they are weighted homogeneous singularities obtained from triangles in the hyperbolic plane. More precisely, they are determined by triangles with angles π b 1 , π b 2 , π b 3 , where b 1 , b 2 , b 3 are positive integers called the Dolgachev numbers of the singularity. The k = 0 elements of the bimodal series are quadrangle hypersurface singularities, i.e., they are related in a similar way to quadrangles in the hyperbolic plane. They are determined 2010 Mathematics Subject Classification. 32S20, 32S30, 14J33. by four positive integers b 1 , b 2 , b 3 , b 4 . For 6 quadruples (b 1 , b 2 , b 3 , b 4 ), the corresponding quadrangle singularities are hypersurface singularities. The dual ICIS are triangle complete intersection singularities in C 4 . There are 8 of them determined by 8 triples (b 1 , b 2 , b 3 ). For another 5 quadruples (b 1 , b 2 , b 3 , b 4 ), the quadrangle singularities are ICIS.
These singularities will be considered in this paper. They are again the k = 0 elements of certain series of singularities. These series are the 8 series of K-unimodal ICIS in Wall's classification [W1]. Wall and the first author also observed a duality between the k = −1 cases of these series (see also [E1,Sect. 3.6]). They were called virtual singularites as well. The objective of this paper is to show that these singularities exist as well and to derive this duality from the Berglund-Hübsch duality, too. The quadrangle complete intersection singularities together with their Dolgachev numbers are listed in Table 1. The virtual singularities obtained by setting k = −1 in the equations are listed in Table 2.
We derive this duality from our paper [ET3]. An important tool are matrix factorizations of size two. In [ET4], we showed that such a matrix factorization can be considered as an inverse to Wall's reduction of complete intersection singularities to hypersurface singularities [W2]. We proceed as follows. In [ET3], we classified certain 4 × 3-matrices which provided the duality to complete intersection singularities. Here we consider the polynomials determined by these matrices for the bimodal series. We determine the matrix factorizations of size two of the corank 3 polynomials. It turns out that we get exactly 8 possibilities which correspond to the 8 series of ICIS. We show that one can associate 4 × 4-matrices to these equations such that the duality is given by the Berglund-Hübsch transposition of these matrices. Using the definition of the virtual singularities in [ET3] and matrix factorizations again, we define virtual complete intersection singularities.
Similarly as in [ET3], we associate Dolgachev numbers to the virtual singularities.
These are two pairs of numbers corresponding to a decomposition of the equations into two parts. We also associate Gabrielov numbers to the virtual singularities by considering deformations to cusp singularities. We consider the second function on the zero set of the first function. Again this function has to be considered as a global function. It turns out that these functions have, besides an isolated critical point at the origin, additional critical points outside the origin. We consider Coxeter-Dynkin diagrams of distinguished bases of thimbles corresponding to these functions taking the additional critical points into account.

Invertible polynomials
We recall some general definitions.
A weighted homogeneous polynomial f (x 1 , . . . , x n ) is called invertible if it can be written where a i ∈ C * , E ij are non-negative integers, and the n×n-matrix E := (E ij ) is invertible over Q.
An invertible polynomial is called non-degenerate if it has an isolated singularity at the origin.
Let f be an invertible polynomial given as above. By rescaling of the variables, one can assume that a i = 1 for i = 1, . . . , n. Moreover, we can assume that det E > 0.
The Berglund-Hübsch transpose [BH] f of f is defined by the transpose matrix E T of E, i.e.
f (x 1 , . . . , Let f (x 1 , . . . , x n ) be an invertible polynomial. The canonical system of weights W f of f is the system of weights (w 1 , . . . , w n ; d) given by the unique solution of the equation We define The maximal group of diagonal symmetries of f is the group It always contains the exponential grading operator g 0 := (e 2πiq 1 , . . . , e 2πiqn ).
Denote by G 0 the subgroup of G f generated by g 0 .
By [BHe] (see also [EG2,Proposition 2]), Hom(G f , C * ) is isomorphic to G f . For a subgroup G ⊂ G f , Berglund and Henningson [BHe] defined its dual group G by

Wall's reduction
Let (X, 0) be an ICIS in C 4 given by an equation where a(z, w) and c(x, z, w) are polynomials of degree ≥ 2, b(z, w) is a polynomial of degree ≥ 1, and x, b(z, w) form a regular sequence in C[x, z, w]. Then we can consider the reduction L y F (x, z, w) = xc(x, z, w) + a(z, w)b(z, w) of [W2] corresponding to the variable y. This means that we eliminate the variable y to get the equation of a hypersurface singularity in C 3 . Geometrically, this elimination corresponds to the projection along the y-axis on the coordinate space of the remaining variables x, z, w. It is proved in [W2,Theorem 7.9], for the case b(z, w) = z, that the Milnor number increases by one.
In [ET4], we considered certain polynomials of the form with the conditions on a(z, w), b(z, w), and c(x, z, w) as above and associated a complete intersection singularity to a graded matrix factorization of size two of f . We showed that, in this way, we get an inverse to Wall's reduction. More precisely, a matrix factorization of f is given by two matrices such that We associate to this the complete intersection singularity (X Q , 0) given by F Q (x, y, z, w) = (F Q,1 (x, y, z, w), F Q,2 (x, y, z, w)) := (a(z, w) − xy, c(x, z, w) + yb(z, w)).

An extension of the Berglund-Hübsch duality
We shall now show that the duality between the quadrangle complete intersection singularities can be derived from the Berglund-Hübsch transposition of invertible polynomials in 4 variables. We use the procedure in [ET3] to associate a weighted homogeneous non-invertible polynomial with 4 terms in 4 variables to each of the quadrangle complete intersection singularities. We consider the complete intersection singularities associated to the matrix factorizations in Table 4 defined by equations (F Q,1 , F Q,2 , where we set a i = 1, i = 1, . . . , 4, and where we take a suitable order of the terms. Moreover, in the equation for I 1,0 we replace XZ + Y Z by X 2 + Y 2 . We also substitute temporarily the variables x, y, z, w by capital letters X, Y, Z, W . We have the following 4 cases: We make the following coordinate substitutions in F Q,2 (X, Y, Z, W ): for a 4 × 4-matrix E of exponents. The corresponding polynomials are listed in Table 5.
This procedure can be explained as follows. We observe that the kernel of the matrix E is generated by one of the following vectors: There exists a Z-graded structure on R given by the respective Let R = i∈Z R i be the decomposition of R according to one of these Z-gradings. The new coordinates X, Y, Z, W are some invariant polynomials with respect to these actions and they satisfy the relation given by the corresponding first equation.
x 12 w 6 + y 12 w 6 + z 2 + x 6 y 6 w 6 I 1,0 An inspection of Table 5 shows that the Berglund-Hübsch transpose of the polynomial f is either the polynomial f itself or another polynomial appearing in the table.
This leads to the indicated duality.

Virtual isolated complete intersection singularities
We now derive the equations for the virtual singularities.
In [ET3, Section 4], we associated a polynomial h to f, which defines the corresponding virtual bimodal hypersurface singularity. This is done as follows. We consider the polynomial f(x, z, w) from Table 3 with the choice of coefficients a 1 , a 2 , a 3 , a 4 given in the last column. Then the corresponding equation defines a non-isolated singularity. We consider the cusp singularity and perform the coordinate change indicated in Table 6. Then this polynomial is trans- Table 6. Functions h of 4 of the quadrangle hypersurface singularities formed to where the polynomial h(x, z, w) is indicated in Table 6. The polynomial h(x, z, w) has an isolated singularity at the origin, but also an additional critical point of type A 1 outside the origin. Moreover, if we consider the 1-parameter family h(x, z, w)−t·xzw for t ∈ [0, 1], then, for t = 0, 1, the polynomial h(x, z, w) − t · xzw has two additional critical points of type A 1 outside the origin. One of them merges with the singularity of h(x, z, w) at the origin for t = 0 and the other one merges with the singularity of f(x, z, w) − xzw at the origin for t = 1.
Example 1. Consider the case Q 2,0 . Then The polynomial h(x.z, w) has a singularity of Arnold type Q 12 at the origin. On the other hand, for t = 0, Using the proof of [ET1, Theorem 10], one can show that, for t = 1, this is a cusp singularity of type T 3,3,6 . For t = 1, it is a cusp singularity of type T 3,3,7 . Using this, one can check the above statements. Now we are looking at possible matrix factorizations of the polynomials h of Table 6.
They are listed together with the corresponding isolated complete intersection singularities in Table 7. The corresponding isolated complete intersection singularity is denoted by . The resulting singularities defined by H = (h 1 , h 2 ) are called the virtual singularities and they are denoted by replacing the index 0 by −1. There is another Table 7. Virtual singularities matrix factorization in the case U 1,−1 , namely It is equivalent to the matrix factorization corresponding to I 1,−1 .

Let
H(x, y, z, w) = (h 1 (x, y, z, w), h 2 (x, y, z, w)) be the equations defining a virtual singularity and let Supp( be the Newton polygon of h 2 at infinity [Ko], i.e. Γ ∞ (h 2 ) is the convex closure in R 4 of Supp(h 2 ) ∪ {0}. The Newton polygon Γ ∞ (h 2 ) has two faces which do not contain the origin. Call these faces Σ 1 and Σ 2 . Let Then (h 1 , h 2,k ) defines a non-isolated weighted homogeneous complete intersection singularity. The polynomials h 1 and h 2 and their systems of weights are listed in Table 8.

Dolgachev numbers
We shall now define Dolgachev numbers for our virtual singularities.
(C) V i is not of the form of (A) or (B).
In case (A) we consider those exceptional orbits which are not contained in L. In case (B) we consider those exceptional orbits which are not contained in U. In case (C) we consider those exceptional orbits which do not coincide with the singular locus of V i . We call these the principal orbits. It turns out that in all cases we have exactly two principal orbits.
The Dolgachev numbers of the virtual singularities are computed as follows. The two pairs of polynomials (h 1 , h 2,i ), i = 1, 2, of Table 8 define non-isolated weighted homogeneous complete intersection singularities of certain types. The systems of weights correspond to the five quadrangle ICIS and three elliptic complete intersection singularities as considered by Wall [W3]. We indicate the notation of Wall [W3] in Table 9. The corresponding orbifold curves have genus zero. We list the orders of the isotropy groups of the exceptional orbits of these ICIS in this table (see also [E3]). Some of them correspond to the orders of the isotropy groups of the exceptional orbits for the non-isolated singularities given by the pairs (h 1 , h 2,i ), i = 1, 2. Those ones which do not occur are stroken out. The orders of the isotropy groups of the principal orbits are indicated in bold face. We also indicate for each pair which of the corresponding cases (A), (B), or (C) applies. An exceptional orbit which coincides with the singular locus is marked by * .
Remark 3. Using the primary decomposition algorithm of the computer algebra software Singular [DGPS], one can show that, for each pair (h 1 , h 2,i ) where we have case (A), the subspace L is an irreducible component of V i . If one removes this component L in case (A), the component U in case (B), and the point corresponding to the singular line in case (C), one gets P 1 α 2i−1 ,α 2i with one point removed. Here P 1 α 2i−1 ,α 2i denotes the complex projective line with two orbifold points with singularities Z/α j Z, j = 2i − 1, 2i.
Proposition 4. The Gabrielov numbers of the virtual quadrangle complete intersection singularities are given by Table 10.
In the case I 1,−1 , we indicate the claimed K-equivalence. We add the first polynomial h 1 (x, y, z, w) to the second one h 2 (x; y, z, w) − zw and obtain (xy − w 3 , xy − w 3 − xw + z 2 + yz + xz − zw). (6.1) By the transformation w → w + y, this is transformed to where p 1 (y, w) and q 1 (y, w) are certain polynomials of degree 3 in the variables y and w. Using [A1,Lemma 7.3] and the fact that y divides p 1 (y, w), one can get rid of the polynomial p 1 (y, w) with the help of the term xy. Similarly, one can get rid of the polynomial q 1 (y, w) in the second equation with the help of the term zw. Now we apply the transformation w → w + x. Then the pair (6.2) gets where p 2 (x, w) and q 2 (x, w) are again polynomials which can be removed. Applying the again with certain removable polynomials p 3 (x, z) and q 3 (x, z). Finally, by the transformation z → 1 2 (z − x) followed by w → w − x and rescaling, we obtain (xy − z 3 − w 3 + p 4 (x, w), x 3 + y 3 − zw), (6.5) again with a removable polynomial p 4 (x, w).
By [E1], one can compute Coxeter-Dynkin diagrams of the (global) singularities defined by H = (h 1 , h 2 ). Let X (1) := {(x, y, z, w) ∈ C 4 | h 1 (x, y, z, w) = 0} and consider the function h 2 : X (1) → C. It has besides the origin one or two additional critical points which are of type A 1 . The singularity at the origin is indicated in Table 10. We define the Milnor number of the virtual singularity by the sum of the Milnor numbers of the singular points. It is equal to 12 in all cases.

Strange duality
We now consider the duality defined in Section 1. We summarise the results on the Dolgachev and Gabrielov numbers of the virtual singularities in Table 11. From this table, we get the following result:  For another feature of this duality, we have to introduce some notions.
We shall consider the Poincaré series P X (t) = ∞ s=0 dim A s ·t s of this graded algebra. One has For a map ϕ : Z → Z of a topological space Z, the zeta function is defined to be If, in the definition, we use the actions of the operators ϕ * on the reduced homology groups H p (Z; Z), we get the reduced zeta function For 0 ≤ j ≤ k, let X (j) be the complete intersection given by the equations f 1 = .
One can show that (ϕ (j) * ) d j = id and therefore ζ X,j (t) can be written in the form Following K. Saito [S1, S2], we define the Saito dual to ζ X,j (t) to be the rational function Let Y (k) = (X (k) \ {0})/C * be the space of orbits of the C * -action on X (k) \ {0} and Y (k) m be the set of orbits for which the isotropy group is the cyclic group of order m. For a topological space Z, denote by χ(Z) its Euler characteristic. Define Now let X be an ICIS in C 4 defined by two polynomial equations f 1 = f 2 = 0 and assume that both X (1) = f −1 1 (0) and X (2) = f −1 2 (0) have isolated singularities at the origin. Moreover, assume that X (1) has a singularity of type A 1 . Consider the mapping F := (f 1 , f 2 ) : C 4 → C 2 . Let C F be the critical locus of F and D F = F (C F ). The Assume that (1, 1) ∈ D F . Let V (1) = f −1 1 (1) and V 2 = f −1 2 (1) ∩ V (1) . (Note that V 2 = V (2) but V 2 and V (2) are homeomorphic to each other.) Then V 2 ⊂ V (1) and the monodromy transformation ϕ (1) : V (1) → V (1) induces a relative monodromy operator be the characteristic polynomial of this operator.
A k = 0 element of one of the series can again be given as the zero set of two quasihomogeneous functions of weights w 1 , w 2 , w 3 , w 4 and degrees d 1 , d 2 . We are now ready to state the following analogue of [ET3, Theorem 6]: Theorem 7. Let X be a virtual ICIS and X 0 be the k = 0 element of the dual series.
On the other hand, we can compute the Poincaré series from the weights and degrees of the dual ICIS given in Table 5. The polynomial Or X (t) is given by Or X (t) = (1 − t) −2 where γ 1 , γ 2 ; γ 3 , γ 4 are the Gabrielov numbers of X which are the Dolgachev numbers of X. Comparing these polynomials, we obtain the first equality of Equation (7.2).
In each case, the polynomial 2 j=1 ζ X 0 ,j (t) has already been indicated in [E2, Table 7] under the heading π * .
Remark 8. The spectrum of an ICIS was defined in [ESt]. In a similar way one can define the spectrum of a virtual ICIS X. Spectra for the series of ICIS above have been calculated by Steenbrink [St2]. The spectrum of a virtual ICIS agrees with the spectrum defined by setting k = −1 in the corresponding formulas of [St2]. The spectral numbers coincide with the exponents of the roots of 2 j=1 ζ X,j (t).