Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic
P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror-symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to gener...
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| description | P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror-symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called the parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 051, 15 pages
Dual Invertible Polynomials with Permutation
Symmetries and the Orbifold Euler Characteristic
Wolfgang EBELING † and Sabir M. GUSEIN-ZADE ‡
† Leibniz Universität Hannover, Institut für Algebraische Geometrie,
Postfach 6009, D-30060 Hannover, Germany
E-mail: ebeling@math.uni-hannover.de
‡ Moscow State University, Faculty of Mechanics and Mathematics,
Moscow, GSP-1, 119991, Russia
E-mail: sabir@mccme.ru
Received July 29, 2019, in final form June 01, 2020; Published online June 11, 2020
https://doi.org/10.3842/SIGMA.2020.051
Abstract. P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct
mirror symmetric Calabi–Yau manifolds. They considered a pair consisting of an invertible
polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual
pair. A. Takahashi suggested a method to generalize this construction to symmetry groups
generated by some diagonal symmetries and some permutations of variables. In a previous
paper, we explained that this construction should work only under a special condition on
the permutation group called parity condition (PC). Here we prove that, if the permutation
group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor
fibres of dual pairs coincide up to sign.
Key words: group action; invertible polynomial; orbifold Euler characteristic; mirror sym-
metry; Berglund–Hübsch–Henningson–Takahashi duality
2020 Mathematics Subject Classification: 14J33; 57R18; 32S55
Dedicated to Dmitry Borisovich Fuchs
on the occasion of his 80th birthday
1 Introduction
The idea of mirror symmetry came to mathematics from physics. In the simplest form, it refers
to the observation that there exist pairs of Calabi–Yau manifolds with symmetric sets of Hodge
numbers. It implies, in particular, that their Euler characteristics coincide up to sign. In [2, 3],
P. Berglund, T. Hübsch, and M. Henningson suggested a method to construct mirror symmetric
Calabi–Yau manifolds. They considered pairs (f,G) consisting of a quasihomogeneous polyno-
mial f of a special type (an invertible one) and of a finite (abelian) group G of its diagonal
symmetries. For a pair (f,G) they constructed a dual pair
(
f̃ , G̃
)
. For certain pairs (f,G),
a crepant resolution of the quotient {f = 0}/G of the subvariety defined by the equation f = 0
in the weighted projective space is a Calabi–Yau manifold. Berglund, Hübsch, and Henningson
claimed that the manifolds constructed for the pairs (f,G) and
(
f̃ , G̃
)
are mirror symmetric to
each other. Berglund and Henningson [2] proved a symmetry property for the elliptic genera of
them (see also [12]).
Instead of working with the hypersurface {f = 0} in the weighted projective space one can
consider the Milnor fibre Vf = {f = 1} in the affine space with the action of the group G. In this
This paper is a contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor
of Dmitry Fuchs. The full collection is available at https://www.emis.de/journals/SIGMA/Fuchs.html
mailto:ebeling@math.uni-hannover.de
mailto:sabir@mccme.ru
https://doi.org/10.3842/SIGMA.2020.051
https://www.emis.de/journals/SIGMA/Fuchs.html
2 W. Ebeling and S.M. Gusein-Zade
case one has to compare orbifold Hodge numbers of the Milnor fibres of dual pairs and thus the
reduced orbifold Euler characteristics of them. There were some symmetries found for invariants
of the pairs (Vf , G) and
(
V
f̃
, G̃
)
. In particular, in [4], it was shown that the reduced orbifold Euler
characteristics χ(Vf , G) and χ
(
V
f̃
, G̃
)
coincide up to sign. (This statement holds for arbitrary
pairs (f,G), not only for those giving Calabi–Yau manifolds.) Besides that, in [5], another
special sort of symmetry (called Saito duality) was found between the reduced equivariant Euler
characteristics χG(Vf ) and χG̃
(
V
f̃
)
(with values in the Burnside rings of the groups) of the
Milnor fibres. Initially one could not see a relation of this symmetry with the mirror one. Later
it was understood that the statement for the orbifold Euler characteristics can be deduced from
the Saito duality (in the case of abelian (!) groups; see a discussion below, and see [7] for a proof
of this fact).
Based on an idea of A. Takahashi, in [6], the notion of dual pair was generalized to the
following situation. Let f be an invertible polynomial in n variables, let S be a subgroup of the
group Sn of permutations of the variables preserving the polynomial f , and let G be a group
of diagonal symmetries of f invariant with respect to S. In this case, the semidirect product
Go S is defined and f is Go S-invariant. (The group Go S is, in general, not abelian.) One
can see that the polynomial f̃ participating in the BHH-dual pair
(
f̃ , G̃
)
is preserved by the
group S and that the dual subgroup G̃ is S-invariant. Therefore, f̃ is invariant with respect to
the semidirect product G̃o S. The Berglund–Hübsch–Henningson–Takahashi (BHHT) dual to
the pair (f,Go S) is the pair
(
f̃ , G̃o S
)
.
In [6], a special property of a subgroup S of the permutation group Sn was introduced
which was called parity condition (PC). It was shown that a non-abelian analogue of the Saito
duality between the reduced equivariant Euler characteristics of the Milnor fibres may hold for
BHHT-dual pairs only if the permutation group S satisfies PC. This led to the conjecture that
BHHT-dual pairs correspond to mirror symmetric varieties only if the condition PC is satisfied.
This conjecture found a support in data about Calabi–Yau threefolds presented in [19].
One invariant which has to be the same up to sign for mirror symmetric orbifolds is the
reduced orbifold Euler characteristic. One can conjecture that the reduced orbifold Euler char-
acteristics of BHHT-dual pairs satisfying the PC condition coincide up to sign. In [7], this
conjecture was proved for a very particular case, namely when the polynomial f is atomic of
loop type (see the definition in Section 3).
Here we prove the conjecture for BHHT-dual pairs with a cyclic permutation group, i.e., S is
a cyclic group.
2 Invertible polynomials and non-abelian duality
A polynomial f in n variables is called invertible if it is quasihomogeneous, consists of n mono-
mials, that is
f(x1, . . . , xn) =
n∑
i=1
ai
n∏
j=1
x
Eij
j ,
where ai are non-zero complex numbers and the matrix E = (Eij) has non-negative integer
entries, and detE 6= 0. This does not imply that f has an isolated critical point at the origin,
e.g., f(x1, x2) = x41 + x21x
2
2 is an invertible polynomial with a non-isolated critical point at
the origin. If f has an isolated critical point at the origin, then the invertible polynomial is
called non-degenerate. Here we will only consider non-degenerate invertible polynomials and we
drop the adjective non-degenerate. Without loss of generality one may assume that ai = 1 for
i = 1, . . . , n.
Dual Invertible Polynomials with Permutation Symmetries 3
The group of (diagonal) symmetries of f is
Gf =
{
λ = (λ1, . . . , λn) ∈ (C∗)n : f(λ1x1, . . . , λnxn) = f(x1, . . . , xn)
}
.
One can see that Gf is an abelian group of order |Gf | = | detE|.
The group Sn of permutations on n elements acts on Cn by permuting the coordinates.
Suppose that the polynomial f is invariant with respect to the action of a subgroup S of Sn. In
this case, S acts on the group Gf by conjugation. The group of transformations of Cn generated
by Gf and S is the semidirect product GfoS and the polynomial f is GfoS-invariant. Because
of that, the group Gf o S acts on the Milnor fibre Vf =
{
x ∈ Cn : f(x) = 1
}
.
Remark 2.1. Elements of GfoS can be represented as pairs (λ, σ) with λ = (λ1, . . . , λn) ∈ Gf ,
σ ∈ S. The multiplication in Gf o S is given by
(λ, σ) · (µ, τ) := (λσ(µ), στ),
where, for µ = (µ1, . . . , µn),
σ(µ) :=
(
µσ−1(1), . . . , µσ−1(n)
)
.
The action of the group Gf o S on Cn is defined by
(λ, σ)x =
(
λ1xσ−1(1), . . . , λnxσ−1(n)
)
(
x = (x1, . . . , xn) ∈ Cn
)
.
The Berglund–Hübsch (BH) transpose of f is
f̃(x1, . . . , xn) =
n∑
i=1
n∏
j=1
x
Eji
j
(see [3]). One can show that the group G
f̃
of diagonal symmetries of f̃ is in a natural way
isomorphic to the group G∗f = Hom(Gf ,C∗) of characters of Gf (see, e.g., [5, Proposition 2]).
Let G be a subgroup of Gf . The (Berglund–Henningson) dual subgroup G̃ in G
f̃
is the set of
characters α : Gf → C∗ vanishing (i.e., being equal to 1) on the subgroup G ([2], see also [14]
or [13]). One has |G̃| = |Gf |/|G|. The pair
(
f̃ , G̃
)
is called the Berglund–Hübsch–Henningson
(BHH) dual of the pair (f,G).
Let S be a subgroup of Sn preserving f and let G be a subgroup of Gf invariant with respect
to S, i.e., σ(G) = G for any σ ∈ S. In this case, the semidirect product G o S is defined
and the polynomial f is G o S-invariant. The BH-transpose f̃ is also preserved by S and the
dual subgroup G̃ is S-invariant. Therefore the group G̃ o S preserves the polynomial f̃ . The
pair
(
f̃ , G̃o S
)
is called the Berglund–Hübsch–Henningson–Takahashi (BHHT) dual to the pair
(f,Go S) (see [6]).
One says that the subgroup S of Sn satisfies the parity condition (PC) if, for any subgroup
T ⊂ S, one has
dim
(
Cn
)T ≡ n mod 2,
where
(
Cn
)T
:=
{
x ∈ Cn : σx = x for σ ∈ T
}
is the fixed point set of T (see [6]).
One can show that, if S satisfies PC, then S ⊂ An. Moreover, if S is a cyclic group (say,
generated by s), then S satisfies PC if and only if s ∈ An.
4 W. Ebeling and S.M. Gusein-Zade
3 Orbifold Euler characteristic and fixed point sets
of symmetries
For a topological space X with an action of a finite group H, its orbifold Euler characteristic is
defined by (see, e.g., [1, 11])
χorb(X,H) =
1
|H|
∑
(g,h)∈H2:
gh=hg
χ
(
X〈g,h〉
)
. (3.1)
Here X〈g,h〉 is the fixed point set of the subgroup of H generated by g and h, i.e., X〈g,h〉 = {x ∈
X : gx = hx = x}, χ(·) is the “additive” Euler characteristic defined as the alternating sum of
the ranks of the cohomology groups with compact support. (One can show that χorb(X,H) is
an integer.) The reduced orbifold Euler characteristic is
χorb(X,H) = χorb(X,H)− χorb(pt, H),
where pt is the one point set with the unique action of H. (If the group H is abelian,
χorb(pt, H) = |H|.)
The orbifold Euler characteristic χorb(X,H) is an additive invariant of H-spaces, i.e., spaces
with an action of the group H. The universal additive invariant of H-spaces is the equiv-
ariant Euler characteristic with values in the Burnside ring A(H) of the group H (see, e.g., [9,
Section 3]). Therefore the orbifold Euler characteristic of H-spaces is the reduction of the equiv-
ariant one under a group homomorphism A(H) → Z. One can speculate that the symmetry
property (coincidence up to sign) for the reduced orbifold Euler characteristics of the Milnor
fibres of BHHT-dual pairs can be deduced from the non-abelian Saito duality. The results of [6]
imply that this is really the case if
χorb
(
Gf o S/H o T,Go S
)
= χorb
(
G
f̃
o S/H̃ o T, G̃o S
)
for a subgroup T of S and for subgroups H and G of Gf (with special properties). Unfortunately,
it is unclear how to prove this equation. In [7] it was proved for H = {e} (and thus H̃ = G
f̃
).
If H is a subgroup of a finite group K, one has the induction operation IndKH which converts H-
spaces to K-spaces. For anH-space X, the space IndKH X is the quotient of the Cartesian product
K ×X by the (right) action of the group H defined by (g, x) ∗ h =
(
gh, h−1x
)
(g ∈ K, x ∈ X,
h ∈ H). The action of the group K on IndKH X is defined in the natural way: g′∗(g, x) = (g′g, x).
One has the following important property of the orbifold Euler characteristic:
χorb
(
IndKH X,K
)
= χorb(X,H)
(see [10, Theorem 1]).
The computation of the orbifold Euler characteristic χorb(Vf , GoS) of the Milnor fibre of an
invertible polynomial f (in n variables) with an action of a group Go S (G ⊂ Gf , S ⊂ Sn) will
be based on a decomposition of Vf into its intersections with certain unions of the coordinate
tori. For a subset I ⊂ I0 = {1, 2, . . . , n}, let
CI :=
{
x = (x1, . . . , xn) ∈ Cn : xi = 0 for i /∈ I
}
and let
(C∗)I :=
{
x = (x1, . . . , xn) ∈ Cn : xi 6= 0 for i ∈ I, xi = 0 for i /∈ I
}
.
Dual Invertible Polynomials with Permutation Symmetries 5
One has
Cn =
∐
I⊂I0
(C∗)I .
Let f I be the restriction of the polynomial f to CI , and let V I
f = Vf ∩(C∗)I . Each torus (C∗)I is
invariant with respect to the action of the group Gf . Let GIf :=
{
λ ∈ Gf : λx = x for x ∈ (C∗)I
}
be the isotropy group of the action of Gf on (C∗)I .
The group S acts on the set 2I0 of subsets of I0. One can represent the space Cn as the
disjoint unions
Cn =
∐
J∈2I0/S
∐
J∈J
(C∗)J .
The union of tori
∐
J∈J (C∗)J is invariant with respect to the action of the group GoS. Therefore
χorb(Vf , Go S) =
∑
J∈2I0/S
χorb
( ∐
J∈J
V J
f , Go S
)
.
For a subset I ⊂ I0, let SI ⊂ S be the isotropy subgroup of I for the S-action on 2I0 . Let
I := I0 \ I be the complement of I. One has SI = SI . One can see that, for a representative I
of an S-orbit J , one has∐
J∈J
V J
f = IndGoS
GoSI V
I
f
(as a Go S-set). Therefore
χorb
( ∐
J∈J
V J
f , Go S
)
= χorb
(
IndGoS
GoSI V
I
f , Go S
)
= χorb
(
V I
f , Go SI
)
.
A polynomial f is invertible if and only if it is the (Sebastiani–Thom) sum of “atomic”
polynomials in different (non-intersecting) sets of variables of one of the forms:
1) xp11 x2 + xp22 x3 + · · ·+ x
pm−1
m−1 xm + xpmm , m ≥ 1 (chain type);
2) xp11 x2 + xp22 x3 + · · ·+ x
pm−1
m−1 xm + xpmm x1, m ≥ 2 (loop type).
This classification appeared first in [15] with a reference to proofs in [16]. Sometimes (for
example in [14]) one also distinguishes the so-called Fermat type: xp11 . Here we consider it as
a special case of the chain type with m = 1. (There are some reasons to consider it as a special
case of the loop type with m = 1 as well, writing it as xp1−11 x1.)
Let f be an invertible polynomial and let S be a permutation group preserving f . An
element σ of S respects the decomposition of f into atomic polynomials and sends each of them
into an isomorphic one. For an atomic summand fα of f , let N be the minimal power of σ which
sends fα to itself. One may have the following two (somewhat different) situations. First, the
action of σN on the set of variables of fα may be trivial. This always happens if fα is of chain
type. If fα is of loop type, the action of σN on the set of its variables may be non-trivial. A non-
trivial automorphism of a loop can be a rotation. This means the following. The length m of the
loop fα = xp11 x2 + · · ·+xpmm x1 is divisible by `, 0 < ` < m, m = k`, the sequence p1, p2, . . . , pk` is
`-periodic, that is, pi+` = pi, where the index i is considered modulo k`, and the automorphism
sends the variable xi to the variable xi+s` with 0 < s < k. Another option for a non-trivial
automorphism is a flip. This means that there exists an index q such that the automorphism
6 W. Ebeling and S.M. Gusein-Zade
sends the variable xi to the variable xq−i. Such an automorphism exists if and only if all the
exponents pi are equal to 1. In this case, the polynomial fα has either a non-isolated critical
point at the origin, or a non-degenerate one (i.e., its Hessian is different from zero), depending
on the parity of the length m. We exclude flips from consideration, i.e., assume that σN is
a rotation.
For a computation of the orbifold Euler characteristic χorb(Vf , Go S) with the use of equa-
tion (3.1), one has to consider mutual fixed point sets V
〈g,h〉
f of pairs of commuting elements
g, h ∈ GoS. Here we shall consider the fixed point set (V I
f )〈g〉 (I ⊂ I0) of an element g = (λ, σ),
σ ∈ SI , and give a condition for it to be non-empty. First we consider the case I = I0.
For an element σ ∈ S, let η = (i1, . . . , ir) be a cycle in σ. For an element λ ∈ Gf , the cycle
product of λ corresponding to η is λi1 · · ·λir . Let x = (x1, x2, . . . , xn) ∈ (C∗)n be a fixed point
of g = (λ, σ) ∈ Go S. This means that
(λ, σ)x = λσ(x) = x, (3.2)
where σ(x) = (xσ−1(1), xσ−1(2), . . . , xσ−1(n)). For a cycle η = (i1, . . . , ir) in σ, equation (3.2)
means that
(λi1xir , λi2xi1 , . . . , λirxir−1) = (xi1 , xi2 , . . . , xir). (3.3)
Here xij 6= 0, j = 1, . . . , r, and therefore a solution of (3.3) exists if and only if λi1 · · ·λir = 1,
i.e., if the corresponding cycle product is equal to 1. One can take, e.g.,
(xi1 , . . . , xir) = (λi1 , λi1λi2 , . . . , λi1 · · ·λir−1 , 1).
This implies that the fixed point set
(
(C∗)n
)〈(λ,σ)〉
is non-empty if and only if, for all cycles of σ,
the cycle products of λ are equal to 1.
For an element σ ∈ S, let ` be the number of cycles of the permutation σ.
Definition 3.1. The cycle homomorphism Cσ is the map from Gf to (C∗)` which sends an
element λ ∈ Gf to the collection of the cycle products of λ.
The discussion above means that the fixed point set
(
(C∗)n
)〈(λ,σ)〉
is non-empty if and only
if λ ∈ KerCσ. In this case one has dim
(
Cn
)〈(λ,σ)〉
= dim
(
Cn
)〈σ〉
= `.
Definition 3.2. For an element σ ∈ S, the shift homomorphism Aσ is the map from Gf to itself
defined by
Aσ(λ) = λ(σ(λ))−1.
Remark 3.3. Two elements (λ, σ) and (λ′, σ′) commute if and only if σσ′ = σ′σ and λσ(λ′) =
λ′σ′(λ). The latter condition is equivalent to Aσ′(λ) = Aσ(λ′).
Proposition 3.4. One has
KerCσ = ImAσ.
Proof. It is easy to see that ImAσ ⊂ KerCσ. Indeed, for µ = Aσ(λ) and for a cycle (i1, . . . , ir)
of σ, one has µij = λij (λij−1)−1, where the index j − 1 is considered modulo r, and therefore
the cycle product of Aσ(λ) is equal to
µi1 · · ·µir = λi1(λir)
−1λi2(λi1)−1 · · ·λir(λir−1)−1 = 1.
We shall show that the order | ImAσ| of the subgroup ImAσ is equal to the order |KerCσ|.
Dual Invertible Polynomials with Permutation Symmetries 7
Let f =
⊕
α∈A fα be the representation of the invertible polynomial f as the Sebastiani–
Thom sum of atomic polynomials fα. The permutation σ sends each fα to an isomorphic
one. Let us regard f as
⊕
ω∈A/〈σ〉
⊕
α∈ω fα where the first sum is over all orbits of the action
of the group 〈σ〉 (generated by σ) on the set A of indices. Since Gf =
⊕
ω∈A/〈σ〉G
⊕
α∈ω fα
and σ preserves each summand
⊕
α∈ω fα, it is sufficient to prove the statement for one block⊕
α∈ω fα with ω ∈ A/〈σ〉. Thus we may assume that f =
⊕N
i=1 fi where fi are isomorphic
atomic polynomials and σ sends fi to fi+1 (the indices are considered modulo N). The proof is
somewhat different for the cases when the fi are of chain type and when the fi are of loop type.
1) Let
fi = xp1i,1xi,2 + xp2i,2xi,3 + · · ·+ x
pm−1
i,m−1xi,m + xpmi,m
be of chain type, i = 1, . . . , N . The permutation σ sends the variable xi,j to the variable xi+1,j .
The order |Gf | of the group Gf is equal to |Gf1 |N (in the case under consideration |Gf1 | =
p1 · · · pm). Let λi = (λi,1, . . . , λi,m), i = 1, . . . , N . For λi ∈ Gfi , one has
λi,j = λ
(−1)j−1p1···pj−1
i,1 (3.4)
for j = 1, . . . ,m. The kernel KerAσ consists of the elements (λ1, . . . , λN ) ∈ GNf1 such that
λ1 = λ2 = · · · = λN . Therefore |KerAσ| = |Gf1 | and | ImAσ| = |Gf1 |N−1. Because of (3.4) the
cycle relation for λi,j follows from the cycle relation for λi,1. Therefore the kernel KerCσ consists
of (λ1, . . . , λN ) ∈ GNf1 such that λ1,1 · λ2,1 · · ·λN,1 = 1. In the elements of Gf , the components
λ1,1, λ2,1,. . . , λN,1 are arbitrary roots of degree |Gf1 | of 1. Therefore |KerCσ| = |Gf |/|Gf1 | =
|Gf1 |N−1 = | ImAσ|.
2) Let
fi = xp1i,1xi,2 + xp2i,2xi,3 + · · ·+ x
pm−1
i,m−1xi,m + xpmi,mxi,1
be of loop type, i = 1, . . . , N . The permutation σ sends the variable xi,j to the variable xi+1,j
for 1 ≤ i ≤ N − 1 and sends the variable xN,j to the variable x1,j+L, where 0 ≤ L ≤ m− 1 (the
index j is considered modulo m). In this case pj+L = pj . If L = 0, the proof literally coincides
with the one for chains. Otherwise let ` = gcd (L,m). The sequence p1, . . . , pm of the exponents
is `-periodic, i.e., pj+` = pj . Let k := m
` , P := p1 · · · p`. One has |Gfi | = P k − (−1)k`, |Gf | =
|Gfi |N . Let λi = (λi,1, . . . , λi,m), i = 1, . . . , N . For λi ∈ Gfi , one has λi,j = λ
(−1)j−1p1···pj−1
i,1 for
j = 1, . . . ,m; cf. (3.4). Because of this the cycle relation for λi,j follows from the cycle relation
for λi,1. The kernel KerAσ consists of the elements (λ1, . . . , λN ) ∈ GNf1 such that
λ1,1 = λ2,1 = · · · = λN,1 = λ1,`+1 = · · · = λN,`+1 = λ1,2`+1 = · · · = λN,(k−1)`+1,
i.e., λi = λ1 for 1 ≤ i ≤ N and, in addition,
λ1,1 = λ1,`+1 = λ1,2`+1 = · · · = λ1,(k−1)`+1. (3.5)
Since λ1,`+1 = (λ1,1)
(−1)`P , equation (3.5) means that (λ1,1)
(−1)`P = λ1,1, so (λ1,1)
P−(−1)` = 1,
i.e., λ1,1 is an arbitrary root of degree P − (−1)` of 1. Therefore
|KerAσ| = P − (−1)`
and
| ImAσ| =
P k − (−1)k`
P − (−1)`
.
8 W. Ebeling and S.M. Gusein-Zade
The cycle relation for λ1,1 is
λ1,1 · · ·λN,1λ1,`+1 · · ·λN,`+1λ1,`(k−1)+1 · · ·λN,`(k−1)+1 = 1,
i.e.,
(λ1,1 · · ·λN,1)1+(−1)`P+(−1)2`P 2+···+(−1)(k−1)`Pk−1
= 1.
This means that the product λ1,1 · · ·λN,1 is an arbitrary root of degree
P k−1 + (−1)`P k−2 + · · ·+ (−1)(k−1)` =
P k − (−1)k`
P − (−1)`
of 1. Since λ1,1, . . . , λN,1 are arbitrary roots of degree P k − (−1)k` of 1, this implies that
|KerCσ| =
(
P k − (−1)k`
)N
P − (−1)`
= | ImAσ|. �
Let I be a non-empty subset of I0 = {1, . . . , n} and let σ ∈ SI . The discussion above about
a condition for (λ, σ) to have a non-empty fixed point set in (C∗)n gives the following analogue
for (C∗)I : the fixed point set
(
(C∗)I
)〈(λ,σ)〉
is non-empty if and only if, for all cycles of σ contained
in I, the cycle products of λ are equal to 1. In this case, the dimension of
(
CI
)〈(λ,σ)〉
is equal to
the dimension of
(
CI
)〈σ〉
and is equal to the number of cycles contained in I.
Let the subset I be such that the number of monomials of the polynomial f I is equal to |I|.
Let I = I0 \ I and |I| = k. By renumbering the coordinates, we can assume without loss of
generality that I = {1, . . . , k}. Then the matrix E = (Eij) is of the form
E =
(
EI 0
∗ EI
)
,
where EI and EI are square matrices of sizes k× k and (n− k)× (n− k) respectively, and EI is
the matrix corresponding to f I . Since detE 6= 0, it follows that detEI 6= 0. Moreover, f I also
has an isolated critical point at the origin, see, e.g., [8, Proposition 5]. This implies that f I is
an invertible polynomial.
Proposition 3.5. In the situation described above, the fixed point set
(
(C∗)I
)〈(λ,σ)〉
is non-empty
if and only if λ ∈ KerCσ +GIf .
Proof. Let f =
⊕
ω∈A/〈σ〉
⊕
α∈ω fα be the decomposition of f into the Sebastiani–Thom sum of
polynomials fα of atomic type. One has Gf =
⊕
ω∈A/〈σ〉G
⊕
α∈ω fα
. The subset I is the disjoint
union of the subsets Iω where Iω is the intersection of I with the set of the indices corresponding
to the coordinates in
⊕
α∈ω fα. Since, for ω1 6= ω2, the subsets Iω1 and Iω2 are disjoint, one
has G⊕
α∈ω1
fα ⊂ G
Iω2
f . Therefore it is sufficient to prove the statement for f =
⊕N
i=1 fi and σ
sending fi to fi+1. If the polynomials fi, i = 1, . . . , N , are of loop type, then I consists of all
the coordinates and the statement says that λ has a non-empty fixed point set in the maximal
torus if and only if λ ∈ KerCσ, i.e., λ satisfies the cycle relation(s), and GIf = {1}.
If
fi = xp1i,1xi,2 + xp2i,2xi,3 + · · ·+ x
pm−1
i,m−1xi,m + xpmi,m, i = 1, . . . , N,
are of chain type, then I = Ir =
∐N
i=1{(i, r), . . . , (i,m)} where 1 ≤ r ≤ m. An element
λ = (λi,j) ∈ (C∗)mN belongs to Gf if and only if λi,1 is a root of degree p1 · · · pm of 1 and
Dual Invertible Polynomials with Permutation Symmetries 9
λi,j = (λi,1)
(−1)j−1p1···pj−1 for j = 2, . . . ,m. In particular, λi,j can be an arbitrary root of degree
pj · · · pm of 1. Since, for j > r, λi,j = (λi,r)
(−1)j−rpr···pj−1 , the cycle relation λ1,jλ2,j · · ·λm,j = 1
follows from the cycle relation λ1,rλ2,r · · ·λm,r = 1. In this case, one can write λi,r = exp 2π
√
−1ni
pr···pm
where
∑N
i=1 ni = 0. Let λ̃ be the element of Gf defined by λ̃i,1 = exp 2π
√
−1ni
p1···pm . One has
λ̃ ∈ KerCσ and λλ̃
−1
∈ GIrf . This proves the statement. �
As above, let I be a non-empty subset of I0 such that the number of monomials of the
polynomial f I is equal to |I| and let g = (λ, σ), σ ∈ SI , be such that the fixed point set(
(C∗)I
)〈(λ,σ)〉
is non-empty.
Proposition 3.6. One has
χ
((
V I
f
)〈(λ,σ)〉)
= (−1)dim(CI)
〈σ〉−1 |KerAσ|∣∣KerAσ ∩GIf
∣∣ . (3.6)
Proof. The Euler characteristic under consideration is the Euler characteristic of the Milnor
fibre of the restriction of the function f to
(
CI
)〈(λ,σ)〉
. Let f =
⊕
ω∈A/〈σ〉
⊕
α∈ω fα be the
decomposition of f into atomic polynomials and let Iα be the set of indices of the variables
in fα. The fixed point set
(
CI
)〈(λ,σ)〉
is the direct sum of the spaces
(⊕
α∈ω CIα
)〈(λ,σ)〉
over
all ω such that I ∩
(∐
α∈ω Iα
)
is non-empty. The restriction of f to
(
CI
)〈(λ,σ)〉
is the Sebastiani–
Thom sum of its restrictions to
(⊕
α∈ω CIα
)〈(λ,σ)〉
. Therefore its Milnor fibre is homotopy
equivalent to the join of the Milnor fibres of the restrictions of f to
(⊕
α∈ω CIα
)〈(λ,σ)〉
and its
Euler characteristic is equal up to sign to the product of the corresponding Euler characteristics
for
⊕
α∈ω CIα . The groups whose orders are in the numerator and in the denominator of (3.6)
are direct products of the corresponding groups for I ∩
(∐
α∈ω Iα
)
. Therefore it is sufficient to
prove (3.6) for the polynomial
⊕
α∈ω fα with ω ∈ A/〈σ〉. Thus, as in Proposition 3.5, we may
assume that f =
⊕N
i=1 fi (fi are atomic) and σ sends fi to fi+1. Again we have to distinguish
between two cases.
1) Let
fi = xp1i,1xi,2 + xp2i,2xi,3 + · · ·+ x
pm−1
i,m−1xi,m + xpmi,m, i = 1, . . . , N,
be of chain type. The permutation σ sends the variable xi,j to the variable xi+1,j . The subset I
(invariant with respect to σ) is of the form
N∐
i=1
{(i, r), (i, r + 1), . . . , (i,m)}
with 1 ≤ r ≤ m. The fixed point set
(
(C∗)I
)〈(λ,σ)〉
consists of points of the form
(λ1y, λ1λ2y, . . . , λ1 · · ·λm−1y, y),
where λi ∈ Gfi (see the notations in the proof of Proposition 3.5),
y = (yr, . . . , ym) = (xN,r, . . . , xN,m).
Therefore the restriction of f to this set is equal to
N
(
yprr yr+1 + y
pr+1
r+1 yr+2 + · · ·+ y
pm−1
m−1 ym + ypmm
)
.
The Euler characteristic of the intersection of its Milnor fibre with the corresponding torus is
equal up to sign (not depending on λ) to the determinant of the matrix of exponents (see, e.g.,
10 W. Ebeling and S.M. Gusein-Zade
[18, Theorem 7.1]), which in our case is equal to pr · · · pm. The group KerAσ consists of the
elements of the form (λ1, λ1, . . . , λ1), λ1 ∈ Gf1 , and its order |KerAσ| is equal to p1 · · · pm.
The group KerAσ ∩ GIf consists of the elements of the form (λ1, λ1, . . . , λ1) with λ1 ∈ G
I∩I1
f1
.
This means that λ1,1 is an arbitrary root of degree p1 · · · pr−1 of 1 and therefore the order∣∣KerAσ ∩GIf
∣∣ is equal to p1 · · · pr−1.
2) Let
fi = xp1i,1xi,2 + xp2i,2xi,3 + · · ·+ x
pm−1
i,m−1xi,m + xpmi,mxi,1, i = 1, . . . , N,
be of loop type, m = k`, pi = pi+`, the permutation σ sends the variable xi,j to the variable
xi+1,j for 1 ≤ i ≤ N − 1 and sends the variable xN,j to the variable x1,j+s`, gcd(s, k) = 1, and
the set I consists of the indices of all the variables. (One can say that in this case I = I0.) In
[6, Proposition 3], it was shown that the element (λ, σ) ∈ GoS (with non-empty fixed point set(
(C∗)Nk`
)〈(λ,σ)〉
) is conjugate in Gf o S to the element (1, σ) (in fact by an element of the form
(µ, 1), µ ∈ Gf ). This means that the fixed point set
(
CI
)〈(λ,σ)〉
is obtained from
(
CI
)〈σ〉
by the
translation by µ. This translation preserves f . The fixed point set
(
(C∗)Nk`
)〈σ〉
consists of the
points x = (xi,j) with xi,j = x1,j for i = 1, . . . , N , j = 1, . . . , k` and x1,j+` = x1,j (the index j
is considered modulo k`). Therefore, as coordinates on
(
(C∗)Nk`
)〈σ〉
, one can take yj = x1,j for
1 ≤ j ≤ ` and the restriction of the polynomial f to this subspace is equal to
kN
(
yp11 y2 + yp22 y3 + · · ·+ y
p`−1
`−1 y` + yp`` y1
)
. (3.7)
As in 1), the Euler characteristic of the intersection of its Milnor fibre with the maximal torus is
equal up to sign (not depending on λ) to P−(−1)` where P = p1 · · · p`. We have |KerAσ| = P−
(−1)` and GIf = {1} and therefore
∣∣KerAσ∩GIf
∣∣ = 1. Note that, for ` = 1, the polynomial (3.7)
is equal to kNyp11 y1 (i.e., is of Fermat type) and the equation for the Euler characteristic holds
as well. This proves the statement up to sign. However the sign of the Euler characteristic of
the Milnor fibre is determined by the dimension. �
4 Symmetry for cyclic permutation groups
Let f be an invertible polynomial (in n variables) and let S ⊂ Sn be a subgroup of the group of
permutations of the coordinates preserving f . Let G be an S-invariant subgroup of Gf , and let
the pair
(
f̃ , G̃o S
)
be the BHHT-dual to (f,Go S).
Theorem 4.1. If S is a cyclic group satisfying the condition PC, then
χorb(Vf , Go S) = (−1)nχorb
(
V
f̃
, G̃o S
)
. (4.1)
One has Cn =
∐
I⊂I0(C∗)I (I0 = {1, . . . , n}) and therefore
Vf =
∐
I⊂I0, I 6=∅
V I
f
(0 /∈ Vf ). The group S acts on 2I0 . One has the decomposition
Cn \ {0} =
∐
J∈(2I0\{∅})/S
∐
J⊂J
(C∗)J ,
where the (disjoint) unions are over the orbits J of the S-action except the one of the empty
set and over the elements of the orbit. Therefore
Vf =
∐
J∈(2I0\{∅})/S
∐
J⊂J
V J
f .
Dual Invertible Polynomials with Permutation Symmetries 11
For I ⊂ I0, let SI be the isotropy subgroup of I for the action of S on 2I0 . It is easy to see
that ∐
J⊂J
V J
f = IndGoS
GoSI V
I
f
for an element I of J . One has
χorb
(
IndGoS
GoSI V
I
f , Go S
)
= χorb
(
V I
f , Go SI
)
(see [10, Theorem 1]).
Recall that, for a subset I ⊂ I0, f
I denotes the restriction of f to CI . The polynomial f I
has not more than |I| monomials and it has |I| monomials if and only if f̃ I has |I| monomials.
If f I has less than |I| monomials, then
χorb
(
V I
f , Go SI
)
= 0.
This follows from [6, Lemma 1], which says that, in this case, the equivariant Euler characteristic
χGfoS
I(
V I
f
)
with values in the Burnside ring A(GfoSI) is equal to zero, together with the facts
that χorb
(
V I
f , GoSI
)
is a reduction of χGoSI(V I
f
)
and χGoSI(V I
f
)
is a reduction of χGfoS
I(
V I
f
)
.
Therefore, if f I has less than |I| monomials, then both χorb
(
V I
f , Go SI
)
and χorb
(
V I
f̃
, G̃o SI
)
are equal to zero.
One has
χorb
(
V I
f , Go SI
)
=
1
|G| · |SI |
∑
((λ,σ),(λ′,σ′))∈(GoSI )2:
(λ,σ)(λ′,σ′)=(λ′,σ′)(λ,σ)
χ
((
V I
f
)〈(λ,σ),(λ′,σ′)〉)
=
1
|SI |
∑
(σ,σ′)∈(SI )2 :
σσ′=σ′σ
1
|G|
∑
(λ,λ′)∈G2 :
(λ,σ)(λ′,σ′)=(λ′,σ′)(λ,σ)
χ
((
V I
f
)〈(λ,σ),(λ′,σ′)〉)
.
Let
χIf,G(σ, σ′) :=
1
|G|
∑
(λ,λ′)∈G2:
(λ,σ)(λ′,σ′)=(λ′,σ′)(λ,σ)
χ
((
V I
f
)〈(λ,σ),(λ′,σ′)〉)
. (4.2)
One has
χorb
(
pt, Go SI
)
=
1
|SI |
∑
(σ,σ′)∈(SI)2
1
|G|
∣∣{(λ, λ′) ∈ G2 : (λ, σ)(λ′, σ′) = (λ′, σ′)(λ, σ)}
∣∣.
Let
χ∅
f,G(σ, σ′) := − 1
|G|
∣∣{(λ, λ′) ∈ G2 : (λ, σ)(λ′, σ′) = (λ′, σ′)(λ, σ)}
∣∣.
In these terms, one has
χorb
(
Vf , Go SI
)
=
∑
J=[I]∈2I0/S
1
|SI |
∑
(σ,σ′)∈(SI )2 :
σσ′=σ′σ
χIf,G(σ, σ′), (4.3)
where the first sum runs over all the S-orbits in 2I0 including the orbit of the empty set,
[I] denotes the orbit of the subset I.
Let SI be a cyclic group (∼= Zq) and let s be a generator of SI .
12 W. Ebeling and S.M. Gusein-Zade
Proposition 4.2. Let σ = sm, σ′ = sm
′
. For m∗ = gcd(m,m′, q), one has χIf,G(σ, σ′) =
χIf,G(sm
∗
, 1).
Proof. For (λ, σ) and (λ′, σ′) from GoS, let
(
ν, σ(σ′)−1
)
:= (λ, σ)(λ′, σ′)−1, where ν = ν(λ, λ′).
Then (λ, σ) commutes with (λ′, σ′) if and only if (λ, σ) commutes with (ν, σ(σ′)−1). Moreover one
has
(
V I
f
)〈(λ,σ),(λ′,σ′)〉
=
(
V I
f
)〈(λ,σ),(ν,σ(σ′)−1)〉
. Therefore the correspondence
(
(λ, σ), (λ′, σ′)
)
←→(
(λ, σ),
(
ν(λ, λ′), σ(σ′)−1
))
preserves the summands in (4.2) and therefore
χIf,G(σ, σ′) = χIf,G
(
σ, σ(σ′)−1
)
.
Then the Euclidian algorithm implies the statement. The arguments are valid for I = ∅ as
well. �
Now we shall compute χIf,G(σ, 1) for I such that f I has |I| monomials. (This includes I = ∅.)
Proposition 4.3. In this case,
χIf,G(σ, 1) = (−1)d
I
σ−1 1
|G|
· |KerAσ|
|KerAσ ∩GIf |
×
∣∣G ∩ ( ImAσ +GIf
)∣∣ · ∣∣G ∩ (KerAσ ∩GIf
)∣∣, (4.4)
where dIσ = dim
(
CI
)〈σ〉
.
Proof. First, let I be non-empty. The fixed point set of (λ, σ) in (C∗)I is not empty if and
only if λ ∈ KerCσ + GIf (Proposition 3.5). Thus the number of the elements λ ∈ G with a
non-empty fixed point set
(
(C∗)I
)〈(λ,σ)〉
is equal to
∣∣G ∩ (KerCσ +GIf
)∣∣ =
∣∣G ∩ ( ImAσ +GIf
)∣∣
(see Proposition 3.4).
The fixed point set of (λ′, 1) in (C∗)I is empty for λ′ /∈ GIf and is equal to (C∗)I for λ′ ∈ GIf .
The element (λ′, 1) ∈ G o S commutes with (λ, σ) if and only if Aσ(λ′) = A1(λ) = 1, i.e., if
λ′ ∈ KerAσ. Thus, for a fixed λ ∈ G with a non-empty fixed point set
(
(C∗)I
)〈(λ,σ)〉
, the number
of elements (λ′, 1) ∈ GoS commuting with (λ, σ) and having a non-empty fixed point set in (C∗)I
(coinciding with (C∗)I) is equal to
∣∣G∩(KerAσ∩GIf
)∣∣. In this case
(
V I
f
)〈(λ,σ),(λ′,1)〉
=
(
V I
f
)〈(λ,σ)〉
(since λ′ ∈ GIf ) and according to Proposition 3.6 one has
χ
((
V I
f
)〈(λ,σ),(λ′,1)〉)
= (−1)d
I
σ−1 |KerAσ|∣∣KerAσ ∩GIf
∣∣ .
This proves the statement for a non-empty I.
Let us show that equation (4.4) holds for I = ∅ as well. In this case GIf = Gf and the right
hand side of (4.4) degenerates to −|G∩KerAσ| (since KerAσ∩GIf = KerAσ, G∩
(
ImAσ+GIf
)
=
G, G ∩
(
KerAσ ∩ GIf
)
= G ∩ KerAσ). For an arbitrary element λ ∈ G (their number being
equal to |G|), (λ, σ) commutes with (λ′, 1) if and only if Aσ(λ′) = A1(λ) = 1 (see Remark 3.3),
i.e., if λ′ ∈ G ∩KerAσ. Therefore∣∣{(λ, λ′) ∈ G2 : (λ, σ) and (λ′, 1) commute
}∣∣ = |G| · |G ∩KerAσ|
and χIf,G(σ, 1) = −|G ∩KerAσ|. �
Proof of Theorem 4.1. From (4.3) together with Proposition 4.2, it follows that it is sufficient
to show that, for I such that f I has |I| monomials, one has
χIf,G(σ, 1) = (−1)nχI
f̃ ,G̃
(σ, 1), (4.5)
Dual Invertible Polynomials with Permutation Symmetries 13
σ ∈ SI . According to Proposition 3.6, the signs in all non-zero summands on the left hand side
and on the right hand side of (4.5) (see Definition (4.2)) are
(−1)dim(CI)
〈σ〉−1 and (−1)n(−1)dim
(
CI
)〈σ〉
−1,
respectively. The condition PC gives
dim
(
Cn
)〈σ〉
= dim
(
CI
)〈σ〉
+ dim
(
CI
)〈σ〉 ≡ n mod 2.
Therefore the signs on the left hand side and on the right hand side coincide and to prove the
statement it is sufficient to show that in this case∣∣χIf,G(σ, 1)
∣∣ =
∣∣χI
f̃ ,G̃
(σ, 1)
∣∣.
One has
∣∣G ∩ ( ImAσ +GIf
)∣∣ =
|G| ·
∣∣ ImAσ +GIf
∣∣∣∣G+
(
ImAσ +GIf
)∣∣ ,
∣∣G ∩ (KerAσ ∩GIf
)∣∣ =
|G| ·
∣∣KerAσ ∩GIf
∣∣∣∣G+
(
KerAσ ∩GIf
)∣∣ .
Therefore Proposition 4.3 gives
∣∣χIf,G(σ, 1)
∣∣ = |KerAσ| · |G| ·
∣∣ ImAσ +GIf
∣∣∣∣G+
(
ImAσ +GIf
)∣∣ · ∣∣G+
(
KerAσ ∩GIf
)∣∣ .
The subgroup of G
f̃
dual to GIf is GI
f̃
(see [5, Lemma 1]). The subgroups KerAσ and ImAσ
of Gf are dual to the subgroups ImA∗σ and KerA∗σ of G
f̃
, respectively. (The homomorphism
A∗σ : G
f̃
→ G
f̃
is in fact the corresponding homomorphism Aσ on this group. We keep the
notation A∗σ to recall what it is acting on.) In particular, |KerAσ| =
|Gf |
| ImA∗σ |
. Therefore the
subgroups dual to ImAσ + GIf , G +
(
ImAσ + GIf
)
, and G +
(
KerAσ ∩ GIf
)
are KerA∗σ ∩ GIf̃ ,
G̃ ∩
(
KerA∗σ ∩GIf̃
)
, and G̃ ∩
(
ImA∗σ +GI
f̃
)
, respectively. Hence one gets
∣∣χIf,G(σ, 1)
∣∣ =
|G|
|Gf |
· |KerAσ|∣∣KerA∗σ ∩GIf̃
∣∣ · ∣∣G̃ ∩ (KerA∗σ ∩GIf̃
)∣∣ · ∣∣G̃ ∩ ( ImA∗σ +GI
f̃
)∣∣. (4.6)
One has |KerAσ| =
|Gf |
| ImAσ | . Since the subgroup dual to ImAσ is KerA∗σ, one gets |KerAσ| =
|KerA∗σ|. Therefore the right hand side of (4.6) coincides with
∣∣χI
f̃ ,G̃
(σ, 1)
∣∣.
This proves Theorem 4.1. �
Remark 4.4. The result in [7, Theorem 4] is a particular case of Theorem 4.1.
Remark 4.5. One can show that
χorb(Vf , Go S) = −χorb
(
Cn, Vf ;Go S
)
,
where χorb
(
Cn, Vf ;GoS
)
is the orbifold Euler characteristic of the pair (Cn, Vf ) (which is equal
to χorb
(
Cn/Vf , Go S
)
). Therefore equation (4.1) is equivalent to
χorb
(
Cn, Vf ;Go S
)
= (−1)nχorb
(
Cn, V
f̃
; G̃o S
)
.
14 W. Ebeling and S.M. Gusein-Zade
5 Examples
Theorem 4.1 gives the orbifold Euler characteristics of the Milnor fibres of pairs BHHT-dual to
a number of examples from [17, Table 2] which are not present in the table. For instance, in
Examples 33 and 34 from [17] one has the pairs (f,Go S), where
f = x41x2 + x42x1 + x43x4 + x44x3 + x55, G =
〈
1
3(1, 2, 0, 0, 0), 13(0, 0, 1, 2, 0), J
〉
,
S = 〈(13)(24)〉 in Example 33 and S = 〈(12)(34)〉 in Example 34. Here 1
m(a1, . . . , a5) means
the operator diag
(
exp 2πa1i
m , . . . , exp 2πa5i
m
)
and J = 1
5(1, 1, 1, 1, 1) is the exponential grading
operator. For the BHHT-dual pairs
(
f̃ , G̃o S
)
one has f̃ = f and G̃ =
〈
1
5(1, 1, 4, 4, 0), J
〉
. The
data from [17, Table 2] (compiled with the use of a computer) together with Theorem 4.1 give
that in the cases dual to Examples 33 and 34 the orbifold Euler characteristics of the Milnor
fibres are equal to 8 and −16, respectively.
In the same way, one gets the following table of orbifold Euler characteristics of the Milnor
fibres of the BHHT-dual pairs for some other examples. Here the first line indicates the number
of the example from [17, Table 2] and the second one gives the orbifold Euler characteristic for
the dual pair. (Let us recall that in all these cases the dual pairs are not present in [17, Table 2].)
36 45 46 48 50 53 60 65 68 69 71 72 77 79 81 87
8 40 −16 16 8 16 −16 8 112 112 112 88 112 88 88 40
Acknowledgements
This work was partially supported by DFG. The work of the second author (Sections 2 and 4)
was supported by the grant 16-11-10018 of the Russian Foundation for Basic Research. We are
very grateful to the referees of the paper for their useful comments.
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Dual Invertible Polynomials with Permutation Symmetries 15
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1 Introduction
2 Invertible polynomials and non-abelian duality
3 Orbifold Euler characteristic and fixed point sets of symmetries
4 Symmetry for cyclic permutation groups
5 Examples
References
|
| id | nasplib_isofts_kiev_ua-123456789-210699 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:31Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ebeling, Wolfgang Gusein-Zade, Sabir M. 2025-12-15T15:24:07Z 2020 Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic. Wolfgang Ebeling and Sabir M. Gusein-Zade. SIGMA 16 (2020), 051, 15 pages 1815-0659 2020 Mathematics Subject Classification: 14J33; 57R18; 32S55 arXiv:1907.11421 https://nasplib.isofts.kiev.ua/handle/123456789/210699 https://doi.org/10.3842/SIGMA.2020.051 P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror-symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called the parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign. This work was partially supported by DFG. The work of the second author (Sections 2 and 4) was supported by the grant 16-11-10018 of the Russian Foundation for Basic Research. We are very grateful to the referees of the paper for their useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic Article published earlier |
| spellingShingle | Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic Ebeling, Wolfgang Gusein-Zade, Sabir M. |
| title | Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic |
| title_full | Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic |
| title_fullStr | Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic |
| title_full_unstemmed | Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic |
| title_short | Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic |
| title_sort | dual invertible polynomials with permutation symmetries and the orbifold euler characteristic |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210699 |
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