The Gerstenhaber product $\HH^2(A)\times \HH^2(A)\to \HH^3(A)$ of affine toric varieties
aa r X i v : . [ m a t h . AG ] D ec THE GERSTENHABER PRODUCT HH ( A ) × HH ( A ) → HH ( A ) OFAFFINE TORIC VARIETIES
MATEJ FILIP
Abstract.
For an affine toric variety
Spec( A ) , we give a convex geometricinterpretation of the Gerstenhaber product HH ( A ) × HH ( A ) → HH ( A ) between the Hochschild cohomology groups. In the case of Gorenstein toricsurfaces we prove that the Gerstenhaber product is the zero map. As an appli-cation in commutative deformation theory we find the equations of the versalbase space (in special lattice degrees) up to second order for not necessarily iso-lated toric Gorenstein singularities. Our construction reproves and generalizesresults obtained in [1] and [13]. Introduction
It is well known that non-commutative deformations of an affine variety X =Spec( A ) are controlled by the Hochschild differential graded Lie algebra (dgla forshort). There are two important A -modules: the second Hochschild cohomologygroup HH ( A ) that describes the first order deformations and the third Hochschildcohomology group HH ( A ) that contains the obstructions for extending deforma-tions of X to larger base spaces. HH ( A ) can be decomposed as H ( A ) ⊕ H ( A ) , where H ( A ) describes thefirst order commutative deformations. Moreover, there exists the Harrison dgla,that is a sub-dgla of the Hochschild dgla, controlling the commutative deformationsof X .Focusing on commutative deformations, computing the versal deformation ofaffine varieties with isolated singularities is a challenging problem. For toric sur-faces Kollár and Shepherd-Barron [9] showed that there is a correspondence betweencertain partial resolutions (P-resolutions) and the reduced versal base space com-ponents. Furthermore, in [5] and [15] Christophersen and Stevens gave a set ofequations for each reduced component of the versal base space. For higher dimen-sional toric varieties the versal base space was computed by Altmann [2] in the caseof isolated toric Gorenstein singularities.In order to better understand the deformation theory of X = Spec( A ) , we needto understand the cup product T ( A ) × T ( A ) → T ( A ) between Andre-Quillencohomology groups. The associated quadratic form describes the equations of theversal base space (if exists) up to second order. A formula for computing the cupproduct for toric varieties that are smooth in codimension 2 was obtained in [1].Since this formula is especially simple in the case of three-dimensional isolatedtoric Gorenstein singularities, it helped Altmann to construct the versal base space Mathematics Subject Classification.
Key words and phrases.
Toric varieties, Hochschild cohomology, Harrison cohomology, Ger-stenhaber product. in [2]. The cup product of toric varieties was also analysed by Sletsjøe [13] butunfortunately there is a mistake in the paper (see Section 3).The cup product is coming from the differential graded Lie algebra (dgla forshort) arising from the cotangent complex. This dgla is isomorphic to the Harrisondgla. The Lie bracket induces the product H ( A ) × H ( A ) → H ( A ) betweenthe Harrison cohomology groups, which is isomorphic to the cup product T ( A ) × T ( A ) → T ( A ) .In this paper we give a convex geometric description of the Harrison productfor an affine toric variety Spec( A ) . This gives us a general cup product formula T ( A ) × T ( A ) → T ( A ) (without the assumption of smoothness in codimension2) that agrees in the case of Gorenstein isolated singularities with Altmann’s cupproduct formula. We obtain a nice expression of the cup product especially forGorenstein not necessarily isolated singularities. This gives us an idea how theversal base space in special lattice degrees could look like. Note that since we aredealing with the non-isolated case, the T ( A ) is non-zero in infinitely many latticedegrees.We also generalize the above description to the product HH ( A ) × HH ( A ) → HH ( A ) , induced by the Lie bracket (also called the Gerstenhaber bracket) of theHochschild dgla. This product is also known as the Gerstenhaber product. As anapplication we obtain that this product is zero for Gorenstein toric surfaces. Thisis interesting since it might lead to the formality theorem for (singular) Gorensteintoric surfaces. The formality theorem has been proved for smooth affine varieties(see [10], [7]).The paper is organized as follows: in Section 2 we recall deformation theory oftoric varieties. In Section 3 we give a convex geometric description of the product H ( A ) × H ( A ) → H ( A ) for toric varieties. The cup product in the specialcase of toric Gorenstein singularities is computed in Section 4 (see Theorem 4.4and Subsection 4.2), where we also show that our product agrees with Altmann’scup product formula for isolated toric Gorenstein singularities (see Corollary 4.5).We describe the quadratic equations of the versal base space in the Gorensteindegree − R ∗ in Corollary 4.6. In Section 5 we analyse the Gerstenhaber product HH ( A ) × HH ( A ) → HH ( A ) for toric varieties. The proof that this product isthe zero map for Gorenstein toric surfaces is done in Proposition 5.3.2. Preliminaries
Toric geometry.
Let k be a field of characteristic 0. Let M, N be mutuallydual, finitely generated, free Abelian groups. We denote by M R , N R the associatedreal vector spaces obtained via base change with R . Let σ = h a , ..., a N i ⊂ N R be arational, polyhedral cone with apex in and let a , ..., a N ∈ N denote its primitivefundamental generators (i.e. none of the a i is a proper multiple of an element of N ). We define the dual cone σ ∨ := { r ∈ M R | h σ, r i ≥ } ⊂ M R and denote by Λ := σ ∨ ∩ M the resulting semi-group of lattice points. Its spectrum Spec ( k [Λ]) iscalled an affine toric variety .2.2. The Hochschild dgla.
For any finitely generated k -algebra, we can definethe cotangent complex L A | k and its derived exterior powers ∧ i L A | k (see e.g. [11]). HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 3
The n -th cohomology group of Hom A ( ∧ i L A | k , A ) is called the n -th (higher) André-Quillen cohomology group and denoted by T n ( i ) ( A ) . We will also use the followingnotation T n ( A ) := T n (1) ( A ) for n ≥ .Using the notation from [8] we denote by C • ( A ) the Hochschild cochain complexand by C n ( A ) = C n (1) ( A ) ⊕· · ·⊕ C n ( n ) ( A ) the Hodge decomposition which induces thedecomposition in cohomology HH n ( A ) ∼ = H n (1) ( A ) ⊕ · · · ⊕ H n ( n ) ( A ) , where HH n ( A ) is the n -th Hochschild cohomology group and H n ( i ) ( A ) is the n -th cohomology of C • ( i ) ( A ) . It is well known (see [11, Proposition 4.5.13]) that T n − i ( i ) ( A ) ∼ = H n ( i ) ( A ) forall i ≥ .In order to get a dgla structure on the Hochschild cochain complex we need toshift it by . The Lie bracket [ · , · ] : C n ( A ) × C m ( A ) → C m + n − ( A ) , which is alsocalled the Gerstenhaber bracket , is well known so we skip the definition of it (seee.g. [8, Section 2]). In particular, the Gerstenhaber bracket induces the product(1) [ · , · ] : HH ( A ) × HH ( A ) → HH ( A ) , between the important A -modules mentioned in Introduction. The product in (1)is called the Gerstenhaber product .We denote the projectors of HH n ( A ) to H n ( i ) ( A ) by e n ( i ) . Lemma 2.1.
For an element p ∈ H ( A ) and an element q ∈ H ( A ) we havethe following: • the equation e (3)[ p, p ] = 0 is the Jacobi identity, e (2)[ p, p ] = 0 • [ p, q ] = e (2)[ p, q ] and [ q, q ] = e (1)[ q, q ] .Proof. An easy computation, see also [12]. (cid:3)
Using Lemma 2.1, we see that the Gerstenhaber product consists of the products H i ) ( A ) × H ( A ) → H i ) ( A ) , for i = 1 , and H ( A ) × H ( A ) → HH ( A ) .In [8] we showed that every Poisson structure p ∈ H ( A ) on an affine toricvariety X σ = Spec( A ) can be quantized, which implies that [ p, p ] = 0 ∈ HH ( A ) .Note also that for an element p ∈ H ( A ) that is not a Poisson structure, Lemma 2.1implies that [ p, p ] = 0 . In this paper we will focus in understanding the remainingtwo products H i ) ( A ) × H ( A ) → H i ) ( A ) , for i = 1 , and A an affine toric variety.2.3. The Hochschild dgla of toric varieties.
From [8] we recall the following.In the group ring of the permutation group S n one defines the shuffle s i,n − i tobe P (sgn π ) π , where the sum is taken over those permutations π ∈ S n such that π (1) < π (2) < · · · < π ( i ) and π ( i + 1) < π ( i + 2) < · · · < π ( n ) . Let s n = P n − i =1 s i,n − i . Definition 1. L ⊂ Λ is said to be monoid-like if for all elements λ , λ ∈ L therelation λ − λ ∈ Λ implies λ − λ ∈ L . Moreover, a subset L ⊂ L of a monoid-likeset is called full if ( L + Λ) ∩ L = L . For any subset P ⊂ Λ and n ≥ we introduce S n ( P ) := { ( λ , ..., λ n ) ∈ P n | P nv =1 λ v ∈ P } . If L ⊂ L are as in the previous definition, then this givesrise to the following vector spaces ( ≤ i ≤ n ): C n ( i ) ( L, L \ L ; k ) := { ϕ : S n ( L ) → k | ϕ ◦ s n = (2 i − ϕ, ϕ vanishes on S n ( L \ L ) } , MATEJ FILIP which turn into a complex with the differentials(2) d : C n − i ) ( L, L \ L ; k ) → C n ( i ) ( L, L \ L ; k ) , ( dϕ )( λ , ..., λ n ) := ϕ ( λ , ..., λ n ) + n − X i =1 ( − i ϕ ( λ , ..., λ i + λ i +1 , ..., λ n ) + ( − n ϕ ( λ , ..., λ n − ) . Definition 2. By H n ( i ) ( L, L \ L ; k ) we denote the cohomology groups of the abovecomplex C • ( i ) ( L, L \ L ; k ) . We denote H n ( i ) ( L, ∅ ; k ) shortly by H n ( i ) ( L ; k ) .It is a trivial check that for A = k [Λ] the Hochschild differentials respect thegrading given by the degrees R ∈ M . Thus we get the Hochschild subcomplex C • ,R ( i ) and we denote the corresponding cohomology groups by H n,R ( i ) ( A ) ∼ = T n − i,R ( i ) ( A ) .When an algebra A will be clear from the context, we will also write H n ( i ) ( R ) . Itholds that HH n ( A ) = L R ∈ M HH n,R ( A ) and HH n,R ( A ) ∼ = ⊕ i H n,R ( i ) ( A ) . Thus wecan analyse the Hochschild cohomology groups by analysing them in every degree R ∈ M .For an element R ∈ M we denote Λ( R ) := Λ + R . Proposition 2.2.
Let A = k [Λ] and R ∈ M . It holds that (3) H n, − R ( i ) ( A ) ∼ = T n − i, − R ( i ) ( A ) ∼ = H n ( i ) (Λ , Λ \ Λ( R ); k ) . Proof.
See [8, Proposition 4.2]. (cid:3)
Proposition 2.2 tells us how to compute the Hochschild comology groups indegree − R .The next lemma describes the Gerstenhaber product in the toric setting. Lemma 2.3.
For each i ∈ { , } the Gerstenhaber bracket induces the product [ · , · ] : C i ) (Λ , Λ \ Λ( R ); k ) × C (Λ , Λ \ Λ( S ); k ) → C i ) (Λ , Λ \ Λ( R + S ); k ) , [ f, g ] = f ( − S + λ + λ , λ ) g ( λ , λ ) − f ( λ , − S + λ + λ ) g ( λ , λ )+ g ( − R + λ + λ , λ ) f ( λ , λ ) − g ( λ , − R + λ + λ ) f ( λ , λ ) . This product induces the Gerstenhaber product in cohomology (4) H i ) (Λ , Λ \ Λ( R ); k ) × H (Λ , Λ \ Λ( S ); k ) → H i ) (Λ , Λ \ Λ( R + S ); k ) . Proof.
See [8, Lemma 5.4]. (cid:3)
Remark 1.
Note that in [8, Lemma 5.4] we obtained a complete description ofthe Gerstenhaber product (also for other parts of the Hodge decomposition). Ingeneral the Gerstenhaber product does not respect the Hodge decomposition like inLemma 2.3. For us only the two products in Lemma 2.3 will be important. In (4) wedescribe the Gerstenhaber product H , − R ( i ) ( k [Λ]) × H , − S (1) ( k [Λ]) → H , − R − S ( i ) ( k [Λ]) .The Gerstenhaber product always respects the toric grading (i.e. two elements ofdegree R and S are mapped to an element of degree R + S ). Thus we can analysethe Gerstenhaber product by analysing it in toric degrees. HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 5
In the following we will recall from [8] how the groups appearing in (4) can benicely interpreted. This will later lead to a nice interpretation of the Gerstenhaberproduct.For a face τ in σ (denoted τ ≤ σ ) we define the convex sets introduced in [4]:(5) K Rτ := Λ ∩ ( R − int τ ∨ ) . The above convex sets admit the following properties ( σ = h a , ..., a N i ): • K R = Λ and K Rj := K Ra j = { r ∈ Λ | h a j , r i < h a j , R i} for j = 1 , ..., N . • For τ = 0 the equality K Rτ = ∩ a j ∈ τ K Ra j holds. • Λ \ ( R + Λ) = ∪ Nj =1 K Ra j . Example 1.
Let a = ( − , and a = (1 , . Let σ = h a , a i and thus theHilbert basis of Λ = σ ∨ ∩ M is H = { ( − , , ( − , , (0 , , (1 , , (2 , } . Let R = (0 , . We have Λ \ Λ( R ) = K Ra ∪ K Ra , where K Ra = { ( r , r ) ∈ Λ | − r + 2 r < } = { (0 , , (2 , , (1 , , (4 , , (3 , , ... } ,K Ra = { ( r , r ) ∈ Λ | r + 2 r < } = { (0 , , ( − , , ( − , , ( − , , ( − , , ... } . For each i ≥ we have the following important double complexes (see [8, Section4.2]). We define C q ( i ) ( K Rτ ; k ) := C q ( i ) ( K Rτ , ∅ ; k ) and C q ( i ) ( K Rp ; k ) := ⊕ τ ≤ σ, dim τ = p C q ( i ) ( K Rτ ; k ) (0 ≤ p ≤ dim σ, q ≥ i ) . The differentials(6) δ : C q ( i ) ( K Rp ; k ) → C q ( i ) ( K Rp +1 ; k ) are defined in the following way: we are summing (up to a sign) the images of therestriction map C q ( i ) ( K Rτ ; k ) → C q ( i ) ( K Rτ ′ ; k ) , for any pair τ ≤ τ ′ of p and ( p + 1) -dimensional faces, respectively. The sign arises from the comparison of the (pre-fixed) orientations of τ and τ ′ (see also [6, pg. 580] for more details). For each i ≥ this construction gives us the double complexes that we shortly denote by(7) C • ( i ) ( K R • ; k ) . Proposition 2.4. T n − i, − R ( i ) ( A ) = H n (cid:0) tot • ( C • ( i ) ( K R • ; k )) (cid:1) for ≤ i ≤ n .Proof. See [8, Proposition 4.4]. (cid:3)
Proposition 2.5. If τ ≤ σ is a smooth face, then H q ( i ) ( K Rτ ; k ) = 0 for q ≥ i + 1 .Proof. See [8, Proposition 4.6]. (cid:3)
Theorem 2.6.
The k -th cohomology group of the complex → H i ( i ) (Λ; k ) → M j H i ( i ) ( K Rj ; k ) → M τ ≤ σ, dim τ =2 H i ( i ) ( K Rτ ; k ) → M τ ≤ σ, dim τ =3 H i ( i ) ( K Rτ ; k ) is isomorphic to T k, − R ( i ) ( A ) , for k = 0 , , ( H i ( i ) (Λ; k ) is the degree term).Proof. Follows from the proof of [8, Theorem 4.6]. (cid:3)
MATEJ FILIP The product H ( k [Λ]) × H ( k [Λ]) → H ( k [Λ]) In this section we give a general formula for the product H ( k [Λ]) × H ( k [Λ]) → H ( k [Λ]) , extending Altmann’s cup product formula on toric varieties that aresmooth in codimension 2. Note that Altmann obtained the cup product formulawith different methods (using Laudal’s description, coming from the cotangent com-plex). For R, S ∈ M and A = k [Λ] we give a convex geometric description of theproduct H , − R (1) ( A ) × H , − S (1) ( A ) → H , − R − S (1) ( A ) , which we call the Harrison cupproduct .As mentioned in Introduction, the Harrison cup product was also analysed bySletsjøe [13] but unfortunately with a mistake that we point out now. We start byrecalling basic constructions from [13].For R ∈ M recall that Λ( R ) := Λ + R . We have an exact sequence of complexes: → C • (1) (Λ , Λ \ Λ( R ); k ) → C • (1) (Λ; k ) → C • (1) (Λ \ Λ( R ); k ) → . Note that H q (1) (Λ; k ) = 0 for q ≥ by Proposition 2.5. Thus we can write thecorresponding long exact sequence in cohomology and we get the following. Corollary 3.1.
The sequence → H (Λ , Λ \ Λ( R ); k ) → H (Λ; k ) → H (Λ \ Λ( R ); k ) d −→ H (Λ , Λ \ Λ( R ); k ) → is exact and H n (1) (Λ \ Λ( R ); k ) ∼ = H n +1(1) (Λ , Λ \ Λ( R ); k ) , for n ≥ . This isomorphism is induced by the map d . Remark 2.
Here with the map d we mean that we first extend a function from Λ \ Λ( R ) to the whole of Λ by and then we apply our differential d . Both mapswe will denote by d and the meaning will be clear from the context. Remark 3.
Let L ⊂ L be as in Definition 1. Elements in C ( L ; k ) = { f : L → k } are functions on L . If we additionally have f ∈ C ( L, L \ L ; k ) , then a function f vanishes on L \ L . Restricting a function f ∈ C ( L ; k ) to some monoid-likesubset K ⊂ L means that we look on f as an element in C ( K ; k ) . Immediatelyfrom definition we obtain that H ( L ; k ) is the space of functions f ∈ C ( L ; k ) such that f ( a + b ) = f ( a ) + f ( b ) if a + b ∈ L . Thus we call elements from H ( L ; k ) additive functions on L .Let ξ be an element from H (Λ \ Λ( R ); k ) . We extend (not additively) ξ to thewhole of Λ by (i.e. ξ ( λ ) = 0 for λ ∈ Λ( R )) . This extended function we denote by ξ ∈ C (Λ; k ) . We have T , − R ( A ) ∼ = H (Λ , Λ \ Λ( R ); k ) by Proposition 2.2 andthe surjective map H (Λ \ Λ( R ); k ) d −→ H (Λ , Λ \ Λ( R ); k ) by Corollary 3.1. Thus we see that every element of T ( − R ) ∼ = H ( − R ) can bewritten as dξ for some ξ ∈ H (Λ \ Λ( R ); k ) . Here dξ denotes the cohomologyclass of dξ ∈ C (Λ , Λ \ Λ( R ); k ) . HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 7
Example 2.
Continuing with Example 1, let ξ ∈ H (Λ \ Λ( R ); k ) , with ξ ( r , r ) = (cid:26) r if ( r , r ) ∈ K Ra if ( r , r ) ∈ K Ra . Since K Ra ∩ K Ra = ∅ , ξ is well defined. Note that R Λ \ Λ( R ) . For ξ ∈ C (Λ; k ) we have ξ ( R ) = 0 .From Lemma 2.3 recall the Gerstenhaber product for i = 1 , called the (Harrison)cup product. Construction 1.
Let
R, S ∈ M and let ξ and µ be elements from H (Λ \ Λ( R ); k ) and H (Λ \ Λ( S ); k ) , respectively. The Harrison cup product [ dξ , dµ ] ∈ H (Λ , Λ \ Λ( R + S ); k ) ∼ = T , − R − S ( A ) can be seen as the cohomology class of [ dξ , dµ ] ∈ C (Λ; k ) in tot • [ C • (1) ( K R + S • ; k )] by Proposition 2.4 (recall that K R + S = Λ ).We can find an element C ∈ C (Λ; k ) such that dC = [ dξ , dµ ] ∈ C (Λ; k ) since H (Λ; k ) = 0 by Proposition 2.5.We inject C into ( C , ..., C N ) ∈ ⊕ j C ( K R + Sa j ; k ) . There exist functions G j ∈ C ( K R + Sa j ; k ) for j = 1 , ..., N , such that dG j = C j . Indeed, H ( K R + Sa j ; k ) = 0 for all j = 1 , ..., N by Proposition 2.5.Let us denote G := ( G , ..., G N ) ∈ ⊕ j C ( K R + Sa j ; k ) . Recall the definition of themap δ from (6). By the above construction δG = [ dξ , dµ ] ∈ tot • [ C • (1) ( K R + S • ; k )] .In the following we try to find the function G explicitly. Then δG ∈ M τ ≤ σ, dim τ =2 H ( K R + Sτ ; k ) . The latter space is easier to work with, which will give us a nice description of thecup product and thus bring us closer to understanding the versal base space.
Proposition 3.2.
We define C ∈ C (Λ; k ) as C ( λ , λ ) := ξ ( λ ) µ ( λ )+ ξ ( λ ) µ ( λ ) − dξ ( λ , λ ) µ ( − R + λ + λ ) − dµ ( λ , λ ) ξ ( − S + λ + λ ) , where µ ( − R + λ + λ ) := 0 (resp. ξ ( − S + λ + λ ) := 0 ) if − R + λ + λ Λ (resp. − S + λ + λ Λ ). It holds that [ dξ , dµ ] = dC. Proof.
See [13, Theorem 4.8]. (cid:3)
Sletsjøe [13] claimed that Proposition 3.2 gives us a nice cup product formula,but unfortunately there is a mistake in his paper: in [13] it was written that onlythe first two terms of C ( λ , λ ) matter for the computations of the cup productformula and that the other two vanish with d . This is not correct since dξ C (Λ , Λ \ Λ( R + S ); k ) , which was wrongly assumed in the paper. We only have dξ ∈ C (Λ , Λ \ Λ( R ); k ) .Thus we need to consider C ( λ , λ ) with all 4 terms and we will try to simplifythis using the double complex C • (1) ( K R • ; k ) (see the equation (7)). MATEJ FILIP
Following Construction 1 we now inject C into ( C , ..., C N ) ∈ ⊕ j C ( K R + Sa j ; k ) .In the following we will define the functions h j which will serve as a first approx-imation of the functions G j from Construction 1, i.e. dh j ( λ , λ ) = C j ( λ , λ ) "almost" holds (we will be more precise later).For each j = 1 , ..., N we choose ˜ ξ j ∈ ( M ⊗ Z k ) ∗ such that ˜ ξ j restricted to K Ra j equals ξ , i.e. ˜ ξ j = ξ as elements in H ( K Ra j ; k ) . Note that this is always possiblesince H ( K Ra j ; k ) is isomorphic to (span k ( K Ra j )) ∗ , which is the space of k -linearfunctions on span k ( K Ra j ) (see [4, Proposition 4.2]; for the description of span k ( K Ra j ) see the discussion after Theorem 4.2). Note also that K Ra j ⊂ Λ \ Λ( R ) so it makessense to consider ξ restricted to K Ra j . If h a j , R i = 0 holds, then a choice of ˜ ξ j is notunique. In the same way we define ˜ µ j .We define ξ j , µ j ∈ C ( M ; k ) as follows: ξ j ( λ ) := (cid:26) ˜ ξ j ( λ ) if λ ∈ K R + Sa j otherwise µ j ( λ ) := (cid:26) ˜ µ j ( λ ) if λ ∈ K R + Sa j otherwise.Note that by construction ξ j and µ j are additive functions on K R + Sa j (i.e. for λ , λ ∈ K R + Sa j we have ξ j ( λ ) + ξ j ( λ ) = ξ j ( λ + λ ) if λ + λ ∈ K R + Sa j andsimilarly for µ j ). Moreover, for each j = 1 , ..., N we define ξ j , µ j ∈ C ( M ; k ) as ξ j ( λ ) := (cid:26) ξ j ( λ ) if λ ∈ K R + Sa j ∩ (cid:0) Λ \ Λ( R ))0 otherwise, µ j ( λ ) := (cid:26) µ j ( λ ) if λ ∈ K R + Sa j ∩ (cid:0) Λ \ Λ( S ))0 otherwise.Note that for λ ∈ Λ we have ξ j ( λ ) = (cid:26) ξ j ( λ ) if λ ∈ (cid:0) K R + Sa j \ K Ra j (cid:1) ∩ (cid:0) ∪ k ; k = j K Ra k (cid:1) ξ ( λ ) otherwiseand similarly for µ j . This description explains the notation. It will be used inExample 3 and Remark 4. Example 3.
Continuing with Example 2 we see that ˜ ξ and ˜ ξ are unique in thiscase and we have ˜ ξ ( λ , λ ) = λ and ˜ ξ ( λ , λ ) = 0 . Moreover, let us choose S = R = (0 , . We see that K Ra ∩ K Ra = (cid:0) K Ra \ K Ra (cid:1) ∩ K Ra = { (1 , } and thuswe have ξ ( λ ) = ξ ( λ ) for all λ ∈ Λ except (1 , , for which ξ (1 ,
1) = ξ (1 ,
1) = 1 and ξ (1 ,
1) = ξ (1 ,
1) = 0 .For each j = 1 , ..., N we define the function h j ∈ C ( K R + Sa j ; k ) as(8) h j ( λ ) := − ξ j ( λ ) · µ j ( λ ) + ξ j ( − S + λ ) µ j ( λ ) + µ j ( − R + λ ) ξ j ( λ ) and C j ∈ C ( K R + Sa j ; k ) as C j ( λ , λ ) := ξ j ( λ ) µ j ( λ )+ ξ j ( λ ) µ j ( λ ) − dξ j ( λ , λ ) µ j ( − R + λ + λ ) − dµ j ( λ , λ ) ξ j ( − S + λ + λ ) . The following proposition is very surprising and it is crucial for later constructionof the cup product.We consider the lattice M as a partially ordered set where positive elementslie in Λ . Thus if for λ ∈ Λ we write λ ≥ R , it means that λ ∈ Λ( R ) = Λ + R .Equivalently, if for λ ∈ Λ we write λ R , it means that λ ∈ Λ \ Λ( R ) . HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 9
Proposition 3.3.
It holds that (9) d ( h j ) = C j ∈ C ( K R + Sa j ; k ) . Proof.
From the definition of the differential d in (2) we have d ( h j )( λ , λ ) = − ξ j ( λ ) µ j ( λ ) + ξ j ( − S + λ ) µ j ( λ ) + µ j ( − R + λ ) ξ j ( λ ) − (cid:0) − ξ j ( λ + λ ) µ j ( λ + λ ) + ξ j ( − S + λ + λ ) µ j ( λ + λ ) + µ j ( − R + λ + λ ) ξ j ( λ + λ ) (cid:1) − ξ j ( λ ) µ j ( λ ) + ξ j ( − S + λ ) µ j ( λ ) + µ j ( − R + λ ) ξ j ( λ ) . Recall that by the definition of C ( K R + Sa j ; k ) we need to verify that d ( h j )( λ , λ ) = C j ( λ , λ ) holds for those ( λ , λ ) ∈ K R + Sa j × K R + Sa j such that λ + λ ∈ K R + Sa j .(1) λ R, S and λ R, S :Note that in this case λ , λ ∈ Λ \ Λ( R ) , Λ \ Λ( S ) . Thus by definition ξ j ( λ k ) = ξ j ( λ k ) and µ j ( λ k ) = µ j ( λ k ) hold for k = 1 , . We consider nowthe following subcases. • λ + λ ≥ R, S :In this subcase we have dh j ( λ , λ ) = ξ j ( λ ) µ j ( λ ) + ξ j ( λ ) µ j ( λ ) − ξ j ( − S + λ + λ ) (cid:0) µ j ( λ )+ µ j ( λ ) (cid:1) − µ j ( − R + λ + λ ) (cid:0) ξ j ( λ )+ ξ j ( λ ) (cid:1) .Moreover, dξ j ( λ , λ ) = ξ j ( λ ) + ξ j ( λ ) since ξ j ( λ + λ ) = 0 . Simi-larly, dµ j ( λ , λ ) = µ j ( λ ) + µ j ( λ ) and thus the equality dh j = C j follows. • λ + λ ≥ R , λ + λ S : dh j ( λ , λ ) = ξ j ( λ ) µ j ( λ )+ ξ j ( λ ) µ j ( λ ) − µ j ( − R + λ + λ ) (cid:0) ξ j ( λ )+ ξ j ( λ ) (cid:1) . Moreover, dξ j ( λ , λ ) = ξ j ( λ ) + ξ j ( λ ) and dµ j ( λ , λ ) = 0 fromwhich the equality dh j = C j follows. • λ + λ R , λ + λ ≥ S : dh j ( λ , λ ) = ξ j ( λ ) µ j ( λ ) + ξ j ( λ ) µ j ( λ ) − ξ j ( − S + λ + λ ) (cid:0) µ j ( λ ) + µ j ( λ ) (cid:1) . • λ + λ R, S : dh j ( λ , λ ) = ξ j ( λ ) µ j ( λ ) + ξ j ( λ ) µ j ( λ ) . In the last two cases we conclude that dh j = C j holds in a similar way.(2) λ R, S and λ ≥ R, S : We have ξ j ( λ ) = ξ j ( λ ) and µ j ( λ ) = µ j ( λ ) . Note that these equalitiesdoes not necessarily hold for λ . We also know that λ + λ ≥ R, S and thuswe can easily check that dh j ( λ , λ ) = − µ j ( λ ) (cid:0) ξ j ( λ ) + ξ j ( − S + λ ) (cid:1) − ξ j ( λ ) (cid:0) µ j ( − R + λ ) + µ j ( λ ) (cid:1) . On the other hand we have C j ( λ , λ ) = − ξ j ( λ ) (cid:0) µ j ( − R + λ ) + µ j ( λ ) (cid:1) − µ j ( λ ) (cid:0) ξ j ( − S + λ ) + ξ j ( λ ) (cid:1) . Since λ ∈ K R + Sa j we have − R + λ S and − S + λ R and thus µ j ( − R + λ ) = µ j ( − R + λ ) and ξ j ( − S + λ ) = ξ j ( − S + λ ) . It follows that the equality dh j = C j is also satisfied in this case.(3) Similarly as above we can check that the equality (9) is satisfied also in theremaining cases. (cid:3) Remark 4.
By definition ξ j ( λ ) and ξ ( λ ) can be for λ ∈ K R + Sa j different only for λ ∈ (cid:0) K R + Sa j \ K Ra j (cid:1) ∩ ( ∪ k ; k = j K Ra k ) . The latter space is "not large" (see Example 3where it is just a point for j = 2 ; later we will also see examples when it is empty).Similarly, µ j ( λ ) and µ ( λ ) can be on K R + Sa j different only for λ ∈ (cid:0) K R + Sa j \ K Sa j (cid:1) ∩ ( ∪ k ; k = j K Sa k ) . Using the notation C j from Construction 1 we see by the definitionthat C j ( λ , λ ) = C j ( λ , λ ) if ξ j ( λ ) = ξ ( λ ) and µ j ( λ ) = µ ( λ ) for λ ∈ K R + Sa j . InProposition 3.3 we verified that dh j = C j holds, which gives us that C j − dh j hasmany zeros (since (cid:0) K R + Sa j \ K Ra j (cid:1) ∩ ( ∪ k ; k = j K Ra k ) and (cid:0) K R + Sa j \ K Sa j (cid:1) ∩ ( ∪ k ; k = j K Sa k ) are"not large"). Thus it is easier to find F j ∈ C ( K R + Sa j ; k ) such that dF j = C j − dh j (note that such F j exists since H ( K R + Sa j ; k ) = 0 by Proposition 2.5). We thendefine G j := F j + h j and proceed with Construction 1 to obtain a nice descriptionof the cup product.In the next section we will explicitly find the functions F j (and thus also thefunctions G j from Construction 1) in the special case of Gorenstein toric varieties.4. The cup product of affine Gorenstein toric varieties
Recall that toric Gorenstein varieties are obtained by putting a lattice polytope P ⊂ A into the affine hyperplane A ×{ } ⊂ A × R =: N R and defining σ := Cone ( P ) ,the cone over P . Then the canonical degree R ∗ ∈ M equals (0 , . Definition 3.
We define the vector space V ⊂ R N by(10) V := V ( P ) := { ( t , ..., t N ) | X j t j ǫ j d j = 0 | for every -face ǫ ≤ P } , where ǫ = ( ǫ , ..., ǫ N ) ∈ { , ± } N is the sign vector of ǫ (see [2, Definition 2.1]). Proposition 4.1.
For Gorenstein toric varieties it holds that T ( − R ∗ ) ∼ = V /k · .Proof. See [3]. (cid:3)
For simplicity we will assume that X σ = Spec( A ) is a three-dimensional Goren-stein singularity given by a cone σ = h a , ..., a N i , where a , ..., a N are arranged in acycle ( n -dimensional case for n > can be then treated with collecting informationabout all -faces of P ).We define a N +1 := a . Let us denote d j := a j +1 − a j and let(11) V = { t = ( t , ..., t N ) ∈ k N | N X j =1 t j d j = 0 } , which is the special case of (10) in the -dimensional case. With ℓ ( d j ) we willdenote the lattice length of d j . Remark 5.
Note that if X σ is isolated, we have T ( − R ∗ ) = T . In general T isnon-zero also in other degrees (see [3, Theorem 4.4]).In the following we recall some results from [4], which even simplify the sequenceappearing in Theorem 2.6.The complex H ( K R • ; k ) in Theorem 2.6 (for i = 1 ) has (span k K R • ) ∗ as asubcomplex. Here for n ∈ N we define (span k K Rn ) ∗ := L τ ≤ σ, dim τ = n (span k K Rτ ) ∗ and (span k K Rτ ) ∗ denotes the space of linear functions on span k K Rτ . HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 11
Theorem 4.2.
Assume that X σ = Spec( A ) is Gorenstein. Then T k, − R ( A ) = H k (cid:0) (span k K R • ) ∗ (cid:1) for k = 0 , , .Proof. See [4, Proposition 5.4]. (cid:3)
Note that for every j = 1 , ..., N it holds that Span k K Ra j = if h a j , R i ≤ a j ) ⊥ if h a j , R i = 1 M ⊗ Z k if h a j , R i ≥ . In the following we compute T ( − R m ) for R m = mR ∗ with m ≥ using Theorem4.2. For all j = 1 , ..., N it holds that span k K R m a j = M ⊗ Z k and span k ( K R m a j ∩ K R m a j +1 ) = ( δ j d j ) ⊥ , where δ j := (cid:26) if ℓ ( d j ) < m if ℓ ( d j ) ≥ m. Thus the complex (span k K R m • ) ∗ for R m = mR ∗ with m ≥ becomes → N k ψ −→ N Nk δ −→ ⊕ j ( N k /δ j d j ) η −→ (Span k ( ∩ j K R m a j )) ∗ , where ψ ( x ) = ( x, ..., x ) , δ ( b , ..., b N ) = ( b − b , b − b , ..., b N − b ) , η ( q , ..., q N ) = P Nj =1 q j . Proposition 4.3.
It holds that T ( − R m ) ∼ = ker η/ im δ . Moreover, if m = 2 and X σ is isolated, then ker η/ im δ ∼ = ( M k /R ∗ ) ∗ holds.Proof. The first statement follows from the above calculation and Theorem 4.2.The second statement follows from the exactness of the complex N Nk δ −→ N Nk η −→ N k . (cid:3) The cup product T ( − R ∗ ) × T ( − R ∗ ) → T ( − R ∗ ) . In the case of isolatedthree-dimensional toric Gorenstein singularities Altmann [1] obtained the followingcup product(12)
V / ( k · × V / ( k · ( M k /R ∗ ) ∗ ( t, s ) N X j =1 s j t j d j . Recall that it holds T ( − R ∗ ) ∼ = H ( − R ∗ ) and T ( − R ∗ ) ∼ = H ( − R ∗ ) . Wewill now generalize Altmann’s cup product formula to the case of not necessar-ily isolated toric Gorenstein singularities. Note that Altmann was using differentmethods (Laudal’s cup product) in his proof.We will first recall the isomorphism map V / ( k · f −→ H (cid:0) (span k K R ∗• ) ∗ (cid:1) from[3, Section 2.7]. Note that both spaces are isomorphic to T , − R ∗ ( k [Λ]) by Theorem4.2 and Proposition 4.1. By the definition it holds that (span k K R ∗ ) ∗ = ⊕ j ( a ⊥ j ) ∗ . We will define u j ∈ ( a ⊥ j ) ∗ such that f ( t ) = ( u , ..., u N ) . There exist b j ∈ R ⊥ for j = 1 , ..., N such that ∀ j it holds that(13) b j +1 − b j = t j ( a j +1 − a j ) . Since P Nj =1 t j d j = 0 we have a one-parameter solution of this system of equations,namely b = b + t d , b = b + t d + t d ,..., b N = b + P N − i =1 t i d i . Our function u j ∈ ( a ⊥ j ) ∗ is defined by u j ( x ) = h b j , x i . Note that for different choices of b westill obtain the same element in H (cid:0) (span k K R ∗• ) ∗ (cid:1) and thus f is well defined. Notealso that we indeed have u j − u j +1 = 0 on a ⊥ j ∩ a ⊥ j +1 . Theorem 4.4.
The cup product T ( − R ∗ ) × T ( − R ∗ ) → T ( − R ∗ ) equals thebilinear map V / ( k · × V / ( k · ker η/ im δ ( t, s ) ( s t d , ..., s N t N d N ) . Proof.
We write for short R = R ∗ . Let ξ = (0 , t d , ..., P n − j =1 t j d j ) ∈ ⊕ j H ( K Ra j ; k ) and µ = (0 , s d , ..., P n − j =1 s j d j ) ∈ ⊕ j H ( K Ra j ; k ) . By the above description of theisomorphism map f (using b = 0 ) and by Construction 1 it is enough to prove δG = ( t s d , ..., t N s N d N ) ∈ M τ ≤ σ, dim τ =2 H ( K R + Sτ ; k ) . Here G is constructed with ξ and µ as in Construction 1.By the description of ξ and µ we define ξ j , µ j ∈ C ( M ; k ) as ξ = µ = 0 and ξ j ( λ ) := (cid:26) h P j − k =1 t k d k , λ i if λ ∈ K Ra j otherwise , µ j ( λ ) := (cid:26) h P j − k =1 s k d k , λ i if λ ∈ K Ra j otherwisefor ≤ j ≤ N . Recall from the equation (8) that the functions h j ∈ C ( K Ra j ; k ) are h j ( λ ) = − ξ j ( λ ) µ j ( λ ) + ξ j ( − R + λ ) µ j ( λ ) + µ j ( − R + λ ) ξ j ( λ ) . Recall that dh j = C j by Proposition 3.3. From Remark 4 we know that in orderto determine when ( dh j − C j )( λ , λ ) ∈ C ( K Ra j ; k ) is zero, we need to considerthe space P j := ( K Ra j \ K Ra j ) ∩ ( ∪ k ; k = j K Ra k ) . Denoting P j := ( K Ra j \ K Ra j ) ∩ K Ra j +1 ,P j := ( K Ra j \ K Ra j ) ∩ K Ra j − , we see that P j = P j ∪ P j for each j .If ℓ ( d j ) > (or equivalently h a j , a j +1 i is not smooth), then P j = P j = ∅ andthus dh j = C j ∈ C ( K Ra j ; k ) for all j = 1 , ..., N by Proposition 3.3 and Remark 4.If ℓ ( d j ) = 1 (or equivalently if h a j , a j +1 i is smooth), then P j ⊂ Λ and P j ⊂ Λ areinfinite sets contained in the lines parallel to a ⊥ j ∩ a ⊥ j +1 and a ⊥ j − ∩ a ⊥ j , respectively.If λ ∈ P j ∪ P j , then h λ, a j i = 1 . We want to find the functions F j ∈ C ( K Ra j ; k ) for which dh j + dF j = C j ∈ C ( K Ra j ; k ) holds. Let F j ( c ) := − ξ ( c ) s j d j ( c ) − µ ( c ) t j d j ( c ) = ξ ( c ) s j + µ ( c ) t j if c ∈ P j ξ ( c ) s j − d j − ( c ) + µ ( c ) t j − d j − ( c ) = − ξ ( c ) s j − − µ ( c ) t j − if c ∈ P j otherwise . HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 13
Using that dh j = C j (see Proposition 3.3) it is straightforward to verify that dh j + dF j = C j indeed holds. Let us just verify it for ( λ j , λ j ) ∈ (Λ ∩ a ⊥ j ∩ a ⊥ j +1 , P j ) .In this case we have dF j ( λ j , λ j ) = − ξ ( λ j ) s j − µ ( λ j ) t j , ( C j − dh j )( λ , λ j ) = ξ ( λ j ) µ ( λ j ) + µ ( λ j ) ξ ( λ j ) − ξ j ( λ j ) µ j ( λ j ) − µ j ( λ j ) ξ j ( λ j ) . Since ξ ( λ j ) = ξ j ( λ j ) , µ ( λ j ) = µ j ( λ j ) and ( ξ − ξ j )( λ j ) = t j d j ( λ j ) = − t j , ( µ − µ j )( λ j ) = s j d j ( λ j ) = − s j , we conclude the claim.To conclude the proof of Theorem 4.4, we need to show that(14) δG = δ ( F , ..., F N ) + δ ( h , ..., h N ) = ( t s d , ..., t N s N d N ) ∈ H ( K Rj ∩ K Rj +1 ; k ) . Since K Rj ∩ K Rj +1 = P j ∪ P j +12 ∪ (cid:0) kR + Λ ∩ a ⊥ j ∩ a ⊥ j +1 (cid:1) for k ∈ { , } , we needto distinguish the following cases.(1) c ∈ P j : it holds that h a j , c i = 1 , h a j +1 , c i = 0 and thus we have F j +1 ( c ) =0 , F j ( c ) = ξ ( c ) s j + µ ( c ) t j . In this case we have ξ j +1 ( c ) = ξ ( c ) . Using ξ j +1 ( c ) = ξ j ( c ) + t j d j ( c ) , ( h j − h j +1 )( c ) = − ξ j ( c ) µ j ( c ) + ξ j +1 ( c ) µ j +1 ( c ) and the fact that d j ( c ) = − we obtain that ( F j − F j +1 )( c ) + ( h j − h j +1 )( c ) = − s j t j . (2) c ∈ P j +12 : it holds that h a j , c i = 0 , h a j +1 , c i = 1 and thus we have F j ( c ) = 0 , F j +1 ( c ) = − ξ ( c ) s j − µ ( c ) t j . In this case we have ξ j ( c ) = ξ ( c ) and similarlyas in the case (1) we obtain that ( F j − F j +1 )( c ) + ( h j − h j +1 )( c ) = t j s j . (3) c ∈ kR + Λ ∩ a ⊥ j ∩ a ⊥ j +1 : it holds that ( F j − F j +1 )( c ) + ( h j − h j +1 )( c ) = 0 .Since d j ( c ) = − for c ∈ P j and d j ( c ) = 1 for c ∈ P j +12 we see that the equation (14)indeed holds. Note that if ℓ ( d j ) > , then we already mentioned that P j = P j = ∅ for all j , which implies F j = F j +1 = 0 and as in the case (3) above we have δh = 0 which agrees with our cup product formula since t j s j d j = 0 on N k /d j . Thus weconclude the proof. (cid:3) Corollary 4.5. If X σ is an isolated Gorenstein singularity, then Theorem 4.4 givesus Altmann’s cup product (12) .Proof. Follows from Proposition 4.3 since the isomorphism ker µ/ im δ ∼ = ( M k /R ∗ ) is explicitly given by the summation of components in ker µ ∈ N Nk . (cid:3) We will denote the cup product from Theorem 4.4 by t ∪ s ∈ ker µ/ im δ . Corollary 4.6. (1) If all edges of P have lattice length 1 (i.e. X σ is an isolatedsingularity), then we have t ∪ s = 0 if and only if d t s + · · · + d N t N s N = 0 .(2) P has one edge d j with ℓ ( d j ) > and for all other edges d k holds either ℓ ( d k ) = 1 for all k = j or there exists d i parallel to d j with ℓ ( d i ) > . Inthis case we have t ∪ s = 0 if and only if d t s + · · · + d N t N s N = 0 on a ⊥ j ∩ a ⊥ j +1 = a ⊥ i ∩ a ⊥ i +1 .(3) P has at least two non parallel edges that have lattice length ≥ . In thiscase t ∪ s = 0 always holds. Proof.
From the definition of the map δ we see that t ∪ s = 0 if and only if for all j such that ℓ ( d j ) > there exist functions f j ( t, s ) such that (cid:0) X j ; ℓ ( d j )=1 t j s j d j (cid:1) + (cid:0) X j ; ℓ ( d j ) > f j ( t, s ) d j (cid:1) = 0 . From this the proof easily follows. (cid:3)
Remark 6.
By standard deformation theory arguments (see e.g. [14, pp. 64]) weknow that the quadratic equations corresponding to t ∪ t = 0 give us the quadraticequations of the versal base space in degree − R ∗ .4.2. The cup product between nonnegative degrees.
Let X σ be a non iso-lated three-dimensional toric Gorenstein singularity. In this section we compute thecup product T ( − R ) × T ( − S ) → T ( − R − S ) for R, S , i.e. for R, S Λ . If R and S have the last component equal to , then the computations in this sectionhave implications in deformation theory of projective toric varieties.Let s , ..., s N be the fundamental generators of the dual cone σ ∨ , labelled so that σ ∩ ( s j ) ⊥ equals the face spanned by a j , a j +1 ∈ σ .Let R p,qj := qR ∗ − ps j with ≤ q ≤ ℓ ( d j ) and p ∈ Z sufficiently large such that R p,qj int ( σ ∨ ) . In this case we already know that T ( − R p,qj ) is one dimensional by[3, Theorem 4.4]. Lemma 4.7. If { a j | h a j , R i > } ≤ it holds that T ( − R ) = 0 .Proof. If { a j | h a j , R i > } ≤ , the statement is trivial. Without loss of general-ity we assume h a i , R i > for i = 1 , and h a j , R i ≤ for other j . Now the statementfollows from the fact that T = 0 for a Gorenstein surface h a , a i ⊂ N R ∼ = R . (cid:3) Proposition 4.8.
Let R := R p ,q j and R := R p ,q k , where j and k are chosensuch that either j = k or it does not exist a -face F of σ ∨ with s j , s k ∈ F . Thecup product T ( − R ) × T ( − R ) → T ( − R − R ) is the zero map.Proof. We will use computations obtained in Section 3 (see Construction 1 and Re-mark 4). Let ξ ∈ H (Λ \ Λ( R ); k ) and µ ∈ H (Λ \ Λ( R ); k ) represent basis ele-ments for T ( − R ) and T ( − R ) , respectively. Note that h a j , R i , h a j +1 , R i > and h a i , R i ≤ for i = j, j + 1 . Similarly, h a k , R i , h a k +1 , R i > and h a i , R i ≤ for i = k, k + 1 . If j = k , then the statement follows from Lemma 4.7. Other-wise, we know by the assumption that it holds h a j , R + R i ≤ h a j , R i and thus K R + R a j ⊂ K R a j holds. By the assumptions we also have K R + R a j +1 ⊂ K R a j +1 , K R + R a k ⊂ K R a k , K R + R a k +1 ⊂ K R a k +1 . This implies that ξ j ( λ ) = ξ ( λ ) for λ ∈ K R + R a j , ξ j +1 ( λ ) = ξ ( λ ) for λ ∈ K R + R a j +1 , µ k ( λ ) = µ ( λ ) for λ ∈ K R + R a k and µ k +1 ( λ ) = µ ( λ ) for λ ∈ K R + R a k +1 . By Construc-tion 1 and Remark 4 then follows that the cup product is equal to δ ( h , ..., h N ) ∈⊕ Nj =1 H ( K R + R j,j +1 ; k ) , which is clearly equal to zero (since h j = 0 for all j ). (cid:3) The following example shows that we can also compute the cup product betweenthe elements of degrees R := R p ,q j and R := R p ,q j +1 . HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 15
Example 4.
A typical example of a non-isolated, three dimensional toric Goren-stein singularity is the affine cone X σ over the weighted projective space P (1 , , .The cone σ is given by σ = h a , a , a i , where a = ( − , − , , a = (2 , − , , a = ( − , , . We obtain σ ∨ = h s , s , s i with s = (0 , , , s = ( − , − , , s = (1 , , .H is nonzero in degrees R α := 2 R ∗ − αs , R β := 2 R ∗ − βs and R γ := 2 R ∗ − γs with α ≥ , β ≥ and γ ≥ . Let us denote the corresponding basis element of R α , R β and R γ by z α , z β and z γ , respectively.We have h a , R α i = h a , R α i = 2 , h a , R α i = 2 − α, h a , R β i = h a , R β i = 2 , h a , R β i = 2 − β, h a , R γ i = h a , R γ i = 3 , h a , R γ i = 3 − γ. From Lemma 4.7 we know that the only possible nonzero cup products can be z ∪ z and z ∪ z , since in other cases we have T ( R ij + R kl ) = 0 . Using computationsin Section 3 (more precisely using Construction 1 and Remark 4) we can easilyverify that it indeed holds z ∪ z = 0 and z ∪ z = 0 . In this case the equations z · z = z · z = 0 are already the generalized (infinite dimensional) versal basespace. J. Stevens checked this using the computer algebra system Macaulay, see [3,Section 5.2].5. The Gerstenhaber product HH ( k [Λ]) × HH ( k [Λ]) → HH ( k [Λ]) Recall from the discussion after Lemma 2.3 that the only remaining missing partfor understanding the Gerstenhaber product HH ( k [Λ]) × HH ( k [Λ]) → HH ( k [Λ]) is the product H ( k [Λ]) × H ( k [Λ]) → H ( k [Λ]) . We will analyse it in thissection.As before let R, S ∈ M . Every element from H (Λ , Λ \ Λ( S )) can be writtenas dµ for some µ ∈ H (Λ \ Λ( S ); k ) . By [8, Proposition 4.7] we know that H (Λ , Λ \ Λ( R )) is isomorphic to the spaceof multi-additive, skew-symmetric functions f : Λ × Λ → Λ , such that f ( λ , λ ) = 0 if λ + λ ∈ Λ \ Λ( R ) . Remark 7.
Multi-additivity means that f ( a + b, c ) = f ( a, c ) + f ( b, c ) and f ( a, b + c ) = f ( a, b ) + f ( a, c ) hold for all a, b, c ∈ Λ .From Lemma 2.3 recall the description of the Gerstenhaber product. Proposition 5.1.
Let µ ∈ H (Λ \ Λ( S ); k ) and ξ ∈ H (Λ , Λ \ Λ( R ); k ) . Let B ( λ , λ ) := B ( λ , λ ) − B ( λ , λ ) ∈ C (Λ; k ) , where B ( λ , λ ) := ξ ( − S + λ + λ , λ ) µ ( λ ) + ξ ( λ , − S + λ + λ ) µ ( λ ) ,B ( λ , λ ) := ξ ( λ , λ ) µ ( λ + λ − R ) . Let λ := λ + λ + λ .(1) If λ + λ ≥ S , λ + λ ≥ S we have dB ( λ , λ , λ ) = [ ξ, dµ ]( λ , λ , λ ) . (2) If λ + λ S , λ + λ ≥ S we have (cid:0) dB − [ ξ, dµ ] (cid:1) ( λ , λ , λ ) = µ ( λ ) (cid:0) ξ ( − S + λ , λ ) + ξ ( λ , λ ) (cid:1) + µ ( λ ) (cid:0) ξ ( λ , − S + λ ) − ξ ( λ , λ ) (cid:1) . (3) If λ + λ ≥ S , λ + λ S we have (cid:0) dB − [ ξ, dµ ] (cid:1) ( λ , λ , λ ) = µ ( λ ) (cid:0) ξ ( λ , λ ) − ξ ( − S + λ , λ ) (cid:1) + µ ( λ ) (cid:0) ξ ( − S + λ , λ ) − ξ ( λ , λ ) (cid:1) . (4) If λ + λ S , λ + λ S we have (cid:0) dB − [ ξ, dµ ] (cid:1) ( λ , λ , λ ) = µ ( λ ) (cid:0) ξ ( − S + λ , λ ) + ξ ( λ , λ ) (cid:1) ++ µ ( λ ) (cid:0) ξ ( λ , − S + λ ) − ξ ( − S + λ , λ ) (cid:1) ++ µ ( λ ) (cid:0) ξ ( − S + λ , λ ) − ξ ( λ , λ ) (cid:1) . Proof.
It holds that [ ξ, dµ ]( λ , λ , λ ) == µ ( λ ) (cid:0) ξ ( − S + λ + λ , λ ) − ξ ( λ , λ ) (cid:1) + µ ( λ ) (cid:0) ξ ( − S + λ + λ , λ ) − ξ ( λ , − S + λ + λ ) (cid:1) + µ ( λ ) (cid:0) − ξ ( λ , − S + λ + λ ) + ξ ( λ , λ ) (cid:1) − µ ( λ + λ ) ξ ( − S + λ + λ , λ )+ µ ( λ + λ ) ξ ( λ , − S + λ + λ ) − dB ( λ , λ , λ ) , where we used the fact that ξ is multi-additive. We will conclude the proof case bycase.(1) λ + λ ≥ S , λ + λ ≥ S We have µ ( λ + λ ) = µ ( λ + λ ) = 0 . Thus we compute dB ( λ , λ , λ ) == µ ( λ ) (cid:0) ξ ( − S + λ + λ , λ ) + ξ ( λ , λ ) (cid:1) ++ µ ( λ ) (cid:0) ξ ( − S + λ + λ , λ ) − ξ ( λ , − S + λ + λ ) (cid:1) ++ µ ( λ ) (cid:0) − ξ ( λ , − S + λ + λ ) − ξ ( λ , λ ) (cid:1) . It holds that ξ ( − S + λ + λ , λ ) − ξ ( λ , − S + λ + λ ) == ξ ( − S + λ + λ + λ , λ ) − ξ ( λ , λ ) − ξ ( λ , − S + λ + λ + λ ) + ξ ( λ , λ ) == ξ ( − S + λ + λ , λ ) − ξ ( λ , − S + λ + λ ) and thus we see that in this case dB ( λ , λ , λ ) = [ ξ, dµ ]( λ , λ , λ ) holds.(2) λ + λ S , λ + λ ≥ S : HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 17
We have µ ( λ + λ ) = 0 and µ ( λ + λ ) = µ ( λ ) + µ ( λ ) . It holdsthat dB ( λ , λ , λ ) == µ ( λ ) ξ ( − S + λ + λ + λ , λ ) + µ ( λ ) (cid:0) ξ ( − S + λ + λ , λ ) − ξ ( − S + λ + λ + λ , λ ) (cid:1) ++ µ ( λ ) (cid:0) ξ ( λ , − S + λ + λ ) − ξ ( λ + λ , − S + λ + λ + λ ) (cid:1) , [ ξ, dµ ]( λ , λ , λ ) == µ ( λ )( − ξ ( λ , λ )) + µ ( λ )( − ξ ( λ , − S + λ + λ ))++ µ ( λ ) (cid:0) − ξ ( λ , − S + λ + λ ) + ξ ( λ , λ ) (cid:1) . If we compute (cid:0) dB − [ ξ, dµ ] (cid:1) ( λ , λ , λ ) we see that the term before µ ( λ ) vanishes because ξ ( λ , − S + λ + λ ) − ξ ( λ + λ , − S + λ + λ + λ ) == ξ ( λ , − S + λ + λ + λ ) − ξ ( λ , λ ) − ξ ( λ + λ , − S + λ + λ + λ ) == − ξ ( λ , − S + λ + λ ) + ξ ( λ , λ ) . This gives us the desired result.Similarly we can consider the remaining cases. (cid:3)
Corollary 5.2.
The product [ dµ , ξ ] ∈ H ( − R − S ) is equal to the cohomologicalclass of the element ( δ ( B ) , d ( B ) − [ dµ , ξ ]) ∈ C ( K R + S ; k ) ⊕ C (Λ; k ) in the total complex of the complex C • (2) ( K R • ; k ) (see the equation (7) ). Note thatthe map d ( B ) − [ dµ , ξ ] has many zeros by Proposition 5.1 and thus it is easier tocompute it. In the following we will compute it in a special case of toric surfaces and foundout that the Gerstenhaber product is the zero map.Let X σ n = Spec( A n ) be the Gorenstein toric surface given by g ( x, y, z ) = xy − z n +1 . Λ n := σ ∨ n ∩ M is generated by S := (0 , , S := (1 , and S := ( n + 1 , n ) ,with the relation S + S = ( n + 1) S . We recall now from [8, Example 3] that wehave(15) dim k H ( − R ) = dim k H ( − R ) = (cid:26) if R = kS for ≤ k ≤ n + 10 otherwise, H ( A n ) = H ( A n ) = 0 . Let { dµ k ∈ H , − kS (1) ( A n ) | ≤ k ≤ n + 1 } be a basis of T ( A n ) ∼ = H ( A n ) ,such that µ k ∈ C (Λ \ Λ( kS ); k ) is defined by µ k ( λ ) = (cid:26) a if λ = aS , for a ∈ N otherwisefor all ≤ k ≤ n + 1 . Proposition 5.3.
For all toric Gorenstein surfaces
Spec( A n ) it holds that theGerstenhaber product HH ( A n ) × HH ( A n ) → HH ( A n ) is equal to the zero map. Proof.
We choose some ξ ∈ H (Λ , Λ \ Λ( R )) for arbitrary R ∈ M . Using Propo-sition 5.1 we will show that dB = [ dµ k , ξ ] holds for all k . Indeed, considering thecases (2), (3) and (4) of Proposition 5.1 we see that in the case (2) we need to provethat for λ + λ S , λ + λ ≥ S it holds that(16) µ ( λ ) (cid:0) ξ ( − S + λ , λ )+ ξ ( λ , λ ) (cid:1) + µ ( λ ) (cid:0) ξ ( λ , − S + λ ) − ξ ( λ , λ ) (cid:1) = 0 . Let a , a be arbitrary natural numbers. We choose λ = a S , λ = a S and λ arbitrary such that λ + λ ≥ S (note that in all other cases the equation (16)holds trivially). Now that the equation (16) holds follows from multi-additivity andskew-symmetry of ξ : µ ( λ ) (cid:0) ξ ( − S + λ , λ ) + ξ ( λ , λ ) (cid:1) + µ ( λ ) (cid:0) ξ ( λ , − S + λ ) − ξ ( λ , λ ) (cid:1) = a ( ξ ( − S + λ + λ , λ )+ ξ ( λ , λ )+ ξ ( λ , λ ))+ a ( ξ ( λ , − S + λ + λ ) − ξ ( λ , λ )) = a ( a ξ ( − S + λ + λ , S )+ a ξ ( S , λ ))+ a ( a ξ ( S , − S + λ + λ ) − a ξ ( S , λ )) = 0 , where in the second equality we used that ξ ( λ , λ ) = a a ξ ( S , S ) = 0 and in thelast equality we used that ξ ( − S + λ + λ , S ) = − ξ ( S , − S + λ + λ ) .Similarly we can check the cases (3) and (4) and we obtain the claim that dB =[ dµ k , ξ ] for all k .From the description of H ( A n ) from the equation (15), we see that we onlyneed to consider the cases, where R is chosen such that R + S ∈ { kS | for ≤ k ≤ n + 1 } , since in all other cases HH , − R − S ( A n ) = H , − R − S (2) ( A n ) = 0 so theGerstenahber bracket is automatically zero. If R + S ∈ { kS | for ≤ k ≤ n + 1 } ,then we easily see that B ( λ , λ ) = B ( λ , λ ) = B ( λ , λ ) = 0 for all ( λ , λ ) ∈ Λ × Λ such that λ + λ R + S . By Corollary 5.2 we conclude the proof. (cid:3) Acknowledgements
I would like to thank to my PhD advisor Klaus Altmann, for his constant supportand for providing clear answers to my many questions. I am also grateful to ArneB. Sletsjøe for useful discussions.
References [1] K. Altmann:
Infinitesimal deformations and obstructions for toric singularities.
J. PureAppl. Alg. (1997), 211–235.[2] K. Altmann:
The versal deformation of an isolated, toric Gorenstein singularity.
Invent.Math. (1997), 443–479.[3] K. Altmann:
One parameter families containing three-dimensional toric Gorenstein singu-larities , Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. , Cambridge Univ. Press, Cambridge (2000), 21–50.[4] K. Altmann, A. B. Sletsjøe:
André-Quillen cohomology of monoid algebras
J. Alg. (1998), 1899–1911.[5] J.A. Christophersen:
On the components and discriminant of the versal base space of cyclicquotient singularities . In Singularity theory and its applications, Part I (Coventry 1988/1989),volume 1462 of Lecture Notes in Math., Springer, Berlin (1991), 81–92.[6] D. A. Cox, J. B. Little, H.K. Schenk:
Toric varieties , Graduate Studies in Mathematics ,AMS (2011).[7] V. Dolgushev, D. Tamarkin, B. Tsygan:
The homotopy Gerstenhaber algebra of Hochschildcochains of a regular algebra is formal , J. Noncommut. Geom. (2007), iss. 1, 1–25.[8] M. Filip: Hochschild cohomology and deformation quantization of affine toric varieties , J.Alg. (2018), 188–214.[9] J. Kollár, N.I. Shepherd-Barron:
Threefolds and deformations of surface singularities , Invent.Math. , (1988), 299-338. HE GERSTENHABER PRODUCT OF AFFINE TORIC VARIETIES 19 [10] M. Kontsevich:
Deformation quantization of algebraic varieties , Lett. Math. Phys. (2001),iss. 3, 271–294.[11] J.-L. Loday: Cyclic homology , Grundlehren der mathematischen Wissenschaften ,Springer-Verlag, (1992).[12] V.P. Palamodov:
Infinitesimal deformation quantization of complex analytic spaces , Lett.Math. Phys. (2007), iss. 2, 131–142.[13] A. B. Sletjøe: Cohomology of monoid algebras , J. Alg. (1993), 102–1911.[14] J. Stevens:
Deformations of singularities , Lecture notes in Mathematics , Springer,Berlin (2003).[15] J. Stevens:
On the versal deformation of cyclic quotient singularities , Singularity theoryand its applications, Part I (Coventry 1988/1989), volume 1462 of Lecture Notes in Math.,Springer, Berlin (1991), 302-319.
JGU Mainz, Institut fur Mathematik, Staudingerweg 9, 55099 Mainz, Germany
E-mail address ::