The incidence class and the hierarchy of orbits

THE INCIDEN CE CLASS AND THE HIE RARCHY OF ORBITS L. M. FEH ´ ER AND ZS. P A T AK F AL VI Abstract. R. Rim´ anyi defined the incidence cla ss o f tw o s ing ularities η and ξ as [ η ] | ξ , the restriction of the T ho m p o lynomial of η to ξ . He conjectured that (un der mild conditions) [ η ] | ξ 6 = 0 ⇐ ⇒ ξ ⊂ η . Gener alizing this no tion we define the incidence c lass of tw o orbits η and ξ o f a representation. W e give a sufficient condition (p ositivity) for ξ to have the prop erty that [ η ] | ξ 6 = 0 ⇐ ⇒ ξ ⊂ η for any other orbit η . W e show that for many in teresting cases , e.g. the quiver representations of Dynkin t yp e p o sitivity holds for a ll orbits. In other w ords in these case s the incidence classes completely determine the hier arch y of the orbits. W e also study the case o f singularities where pos itiv ity do esn’t hold for all o rbits. 1. Intro duction Supp ose that a complex algebraic represen tation ρ : G → GL( V ) of a complex a lgebraic Lie group G is giv en and w e w an t to understand the hierarc h y of orbits: whic h orbit is in the closure of another one. The idea is simple: if η and ξ are t w o orbits then w e calculate [ η ]—the G - equiv ar ian t P oincar ´ e dual of η and restrict it to ξ . Since [ η ] is supported on η , if ξ is disjoin t fro m η , then the incidence class [ η ] | ξ is zero. If the G -action is ric h enough then we ha v e a c hance fo r the o pp osite implication. In this pap er • w e giv e a sufficien t condition (p ositivit y) for the Incidence Prop erty of an orbit ξ : fo r an y other G -in v arian t sub v a riet y X ⊂ V w e hav e that [ X ] | ξ 6 = 0 ⇐ ⇒ ξ ⊂ [ X ]. • W e sho w that for man y in teresting cases, e.g. the quiv er represen tations of Dynkin t yp e p ositivit y holds for all orbits. • W e also study the case o f singularities where p ositivit y do esn’t hold f or all orbits. Our w ork w as inspired by a conjecture of R. Rim´ an yi in [R im01], that for singularities of holomorphic maps the incidence classes detect the hierarc hy of contact singularit y classes. T o study the conjecture we generalized the notion of incidence class to the general g roup represen tatio n setting. Let ρ : G → GL( V ) b e an algebraic represen tation of the complex algebraic Lie g roup G on the v ector space V . If X ⊂ V is a G - in v arian t sub v ariety of co dimension d then w e can assign a G -equiv ariant cohomology class [ X ] ∈ H 2 d G ( V ) ∼ = H 2 d ( B G ). In different areas of mat hematics this class has differen t names e.g. equiv ariant Poincar ´ e dual, m ultidegree, Joseph p olynomial and—in the case of singularities—Thom p olynomial. Equiv ar ian t cohomolog y shares some pro p erties of ordinary cohomology . F or example there is a restriction map: if Y ⊂ V is another G -in v arian t subset and α ∈ H ∗ G ( V ) then α | Y ∈ H ∗ G ( Y ). So we can define the incidence class [ X ] | Y measuring the “closeness” of X and Y . Suppo rted b y OTKA T04636 5MA T Keywords: Thom p oly no mial, equiv aria nt maps, equiv ar iant Poincar´ e dual, multidegree, Joseph poly nomial, incidence class AMS Sub ject cla ssification 32S20. 1 2 L. M. FEH ´ ER A ND ZS. P A T AKF AL VI The crucial observ ation is t hat if ξ is an orbit then [ X ] | ξ is an equiv arian t P oincar´ e dual itself: Let x ∈ ξ and let us denote b y G x the maximal compact subgroup of the stabilizer group of x . Then [ X ] | ξ ∈ H ∗ G ( ξ ) ∼ = H ∗ G x ( pt ). Choosing a G x -in v ariant nor mal space N x to ξ a t x w e will sho w that Prop osition 3.1. [ X ] | ξ = [ X ∩ N x ] G x . This observ ation reduces the pro blem to finding a sufficien t condition for a represen ta tion ha ving the prop ert y that all non- empt y G -in v a rian t sub v a riet y has a nonzero equiv ariant P oincar´ e dual. This condition can b e given in terms of the w eights of the represen tation. Supp ose that T ⊂ G is a maximal torus. Then V splits in to 1-dimensional represen tations of T with w eigh ts w i ∈ ˇ T = Hom( T , U (1)). If T has rank r then ˇ T ∼ = Z r . Definition. The represen tation ρ : G → GL( V ) is p ositive if the conv ex hull of its w eigh ts do esn’t con ta in zero. Theorem 4.3. I f the r epr esentation ρ : G → GL( V ) is p ositive then al l non-empty G -invariant subvariety has a nonzer o e quivariant Poinc a r´ e dual. F rom this w e immediately get: Theorem 4.5. A l l orbits of the r epr esentation ρ : G → GL( V ) have the Incidenc e Pr op erty if for al l x ∈ V the normal r ep r esentation of G x is p ositive (normal p ositivity). In section 5. w e sho w a simple condition whic h implies normal p ositivity and sho w that e.g. ev ery quiv er represen tation of Dynkin t yp e satisfy this condition. After this generalization w e faced the problem, familiar to man y mathematicians, that our theory do esn’t apply to the original conjecture. Mos t orbit s of the represen tations corresp onding to the case o f singularities are not p ositiv e. M an y times it has a trivial reason: the stabilizer is trivial. Eve n if w e restrict ourselv es to the orbits ha ving at least a U (1) symmetry , there are plen t y of non p ositiv e examples. Nonetheles s, calculations sh ow that in man y cases incidence still detects the hierarc h y of or bits. W e list some examples, coun terexamples and conjectures. The author s thank M. Ka zar ia n and R. R im´ an yi for v a luable discussions. 2. The e quiv ariant Poincar ´ e dual and the Incidence class W e define equiv ariant cohomology o f a G -space X via the Borel construction and use cohomol- ogy with ratio nal co efficien ts: H ∗ G ( X ) := H ∗ ( E G × G X ; Q ), where E G → B G is the univ ersal principal G -bundle o ver the classifying space B G of G . W e will frequen tly use the following simple prop erties o f equiv ariant coho mo lo gy: (1) H ∗ G ( V ) ∼ = H ∗ G ( pt ) if V is con tractible. (2) H ∗ G ( X ) ∼ = H ∗ K ( X ), where K is a maximal compact subgroup of G . More generally it is true if K is a defor mation retract of G . (3) W e can restrict to a n in v arian t subspace, or to a subgroup. W e can also com bine them. If the subspace Y ⊂ X is in v ariant for a subgroup S < G , then w e hav e a restriction map H ∗ G ( X ) → H ∗ S ( Y ). If X is a smoot h (alwa ys complex in this pap er) alg ebraic v ariet y (typic ally a complex v ector space in this pa p er) a nd Y is a G - in v arian t sub v ariety then w e can assign a class [ Y ] ∈ H ∗ G ( X ). F or this class not only the names are nume rous (some listed in the Introduction) but the definitions, to o. These definitions are equiv alen t fo r the algebraic setting when the acting group is an algebraic tor us. F or an accoun t of these results see [P at06]. Our definition is ba sed on the follo wing fact in ordinary cohomology (see e. g.[F ul97], page 219): THE INCIDENCE CLASS AND THE HI ERARCHY OF ORBITS 3 Prop osition 2.1 ( Definition). I f X is a smo oth algebr aic varie ty and Y is an i rr e ducible subvariety of c omplex c o dimension d then ther e is a unique ele ment [ Y ] ∈ H ∗ ( X ) such that (1) [ Y ] is supp o rte d on Y , i . e. [ Y ] r estricte d to X \ Y is zer o, (2) [ Y ] | X \ Sing Y = [ Y o ⊂ ( X \ Sing Y )] . Her e Sing Y deno tes the singular subvariety of Y and Y o = Y \ Sing Y . The c ohomolo gy class [ Y o ⊂ ( X \ Sing Y )] is de fi ne d by extending the Thom c l a ss of a tu bular neighb ourho o d of the pr op er submanifold Y o ⊂ ( X \ Sing Y ) via e x cision. If Y has many c om p onents (of the sam e c o dim ension) then we take the sum. This definition can b e extended to stratified spaces (stratified b y complex submanifolds). F or a v ar iety w e ta k e the singular stratification: Y \ Sing Y , Sing Y \ Sing( Sing Y ) and so on. Now w e define the the equiv ariant Poincar ´ e dual as a G c haracteristic class (see [Kaz97]): Theorem 2.2 ( Definition). L et ρ : G → GL( V ) b e an algebr aic r epr es entation and Y ⊂ V is a G -invariant irr e ducible subvariety o f c omplex c o dimension d , then ther e is a unique ele m ent [ Y ] ∈ H ∗ G ( V ) such that fo r al l algebr aic ρ -bund le π : E → M over a smo oth alge b r aic variety M with class i f ying map k : M → B G [ Y ( E ) ⊂ E ] = π ∗ k ∗ [ Y ] , wher e Y ( E ) = P × G Y for P denoting the princip al G - b und le of E . If the group action is not clear from the con text w e will use the notation [ Y ] G for the G - equiv ar ian t P oincar ´ e dual. Notice that the class [ Y ] is lo calized a t 0 in the sense that it dep ends only on the germ of Y at 0. It fo llo ws fro m the fact that the restriction map H ∗ G ( V ) → H ∗ K ( U ) is an isomorphism for any K - in v arian t con tractible neigh b ourho o d U of 0. Here we use the maximal compact subgroup K to ensure the existenc e of small in v arian t con tractible neigh b ourho o ds. Supp ose no w that ξ is an orbit of V . Then w e ha v e the restriction map | ξ : H ∗ G ( V ) → H ∗ G ( ξ ). Observ e that H ∗ G ( ξ ) ∼ = H ∗ G x ( pt ), where x ∈ ξ and G x is the (maximal compact subgroup of the) stabilizer group of x . W e can mak e this more explic it if w e restrict to the p oin t x , whic h is naturally a G x -space. It is not difficult to calculate the map H ∗ G ( pt ) → H ∗ G x ( pt ). Let T < G b e a maximal torus of rank r . Then by the Borel injectivit y theorem t he restriction map H ∗ G ( pt ) → H ∗ T ( pt ) is injective . (This is v alid only with rational co efficien ts. That is the reason w e use cohomology with rational co efficien t s.) So w e can iden tify a class in H ∗ G ( pt ) with a p olynomial p ∈ Q [ α 1 , . . . , α r ] ∼ = H ∗ T ( pt ). W e can c ho ose x ∈ ξ and a maximal tor us T x < G x suc h tha t T x < T . Then restriction of p is a linear substitution in the v ariables β 1 , . . . , β s , where H ∗ T x ( pt ) ∼ = Q [ β 1 , . . . , β s ]. The substitution is determined b y the inclusion T x ֒ → T . 3. The Incidence class as an equiv ariant Poincar ´ e dual Since G x —the maximal compact subgroup of the stabilizer group of x —is a compact group, w e can c ho ose a G x -in v ariant normal space N x < T x V suc h that N x ⊕ T x ξ = T x V . W e ha ve the exp o nen tial map e x : N x → V whic h is transv ersal to ξ . The map e x is G x -in v ariant so it induces a homomor phism H ∗ G ( V ) → H ∗ G x ( N x ). Prop osition 3.1. L et Y ⊂ V b e a G -invariant subva riety, then [ e − 1 x ( Y )] = e ∗ x [ Y ] , or, with som e abuse of n otation [ Y ∩ N x ⊂ N x ] = [ Y ] | ξ . 4 L. M. FEH ´ ER A ND ZS. P A T AKF AL VI Pr o of. It is enough to sho w that e x is t ransv ersal to Y in a neigh b ourho o d of x . On transv ersality w e mean that e x is tr a nsv ersal to every stratum of Y (see e.g. [P at0 6, § 2.2]). The action o f G defines a bundle map ϕ : Lie( G ) × V → T V . A t x = e x (0) the map e x is transv ersal to Im( ϕ x ), so tra nsve rsalit y also holds in an op en neigh b ourho o d. But Im( ϕ y ) is the tangen t spac e of the orbit of y , so in this neigh b ourho o d e x is transv ersal to an y orbit o f G , therefore to an y G -inv ariant submanifold, in par ticular to the strata of Y .  4. The positivity condition In this section w e giv e a condition whic h implies that ev ery non-empt y G -inv ariant sub v ariet y Y has a non-zero equiv arian t P o incar´ e dual. A simple condition is if the represen tation ρ : G → GL( V ) “con tains the scalars” , i.e. the scalars of GL( V ) ar e in the image of ρ . In this case Y is automatically a cone and its pro jectiv e degree can b e calculated from [ Y ] by a substitution ( see e.g. [FNR0 7]) so it is nev er 0. But the stabilizer group of x ∈ V t ypically do esn’t contain the scalars, so w e nee d a more general condition. By the Borel injectivit y theorem w e can assume that G = T is a complex torus. This is the p oint where w e use that G is a complex group, so its maximal torus is con tained in a complex torus of the same rank. The real to rus is to o small, it alwa ys has orbits with zero equiv ar ian t P oincar ´ e dual. Let us denote the set of w eights by W ρ . Theorem 4.1. F or a r epr esentation ρ : T → GL( V ) of the c ompl e x torus T ∼ = GL(1) r the fol lowing c ondi tion s ar e e quivalent (1) for al l no n-empty T -invariant subvarie ty Y ⊂ V the class [ Y ] ∈ H ∗ T ( V ) is non-zer o. (2) The c onve x hul l of W ρ do esn ’t c ontain 0 . This theorem w as first pro v ed in [Pat06, Thm 5.2.1 .]. Indep enden t ly it w as noticed in [KS06] that (2) = ⇒ ( 1) follow s from [KM05, Thm D.]. Here w e pro vide a short direct pro o f . W e refer to condition (ii) a s p ositivity since it means that there is a linear functional λ o n Z r suc h that λ ( w ) > 0 fo r all w ∈ W ρ . another p ossible name would b e instabilit y as p ositivit y is equiv alent to the condition that all p oints of V are unstable in the GIT sense. W e need o ne more prop ert y of the equiv ariant P oincar´ e dual (see [P at06, prop 4 .1]) to pro v e Theorem 4.1. Theorem 4.2. If Y ⊂ V is a T -invariant subvariety then the class [ Y ] ∈ H ∗ T ( V ) c an b e exp r esse d as a n o n-zer o p olynomial of the weights of ρ with non-n e gative inte ger c o efficie n ts. Theorem 4.2 immediately follo ws from the fact—w ell kno wn and widely used by the sp ecialists— that for complex torus a ctions the no tions of equiv aria nt P oincar ´ e dual and multide gr e e coincide. The reason w e refer to [P a t06] is that we ha v en’t found an o lder pro of in the literat ure. The pro of is based on the iden tification of the m ultidegree to the equiv aria n t in tersection class [EG98], and then the later to the equiv ariant P oincar ´ e dual. See [KMS06], page 2 55, for the first equiv alence. As fo r the second one, there is a cycle map from the equiv ariant Chow group to the equiv arian t cohomology ring of V ([EG98], page 605). Scrutinizing t his map, it turns out that it is in fact the ordinary cycle map of the pro duct of some pro jectiv e spaces, for whic h it is easy to pro v e tha t it is an isomorphism. A basic fact a b out m ultidegree is that the multidegree of a T -in v ariant sub v ariet y is equal to the sum of m ultidegrees of T -inv aria n t sub(v ector)spaces (with p ossible p ositiv e in teger m ultiplicities). No t icing that the equiv aria n t P oincar ´ e dual of a subspace is the THE INCIDENCE CLASS AND THE HI ERARCHY OF ORBITS 5 pro duct o f some w eights of ρ w e prov ed Theorem 4.2 . Here w e use the identific ation of a w eight w : T → GL (1) with the first Chern class c 1 ( L w ) of the line bundle L w = E T × w C . Pr o of of The or em 4.1. W e can t hink of the w eights as linear functionals on R r —the real part of the Lie algebra of the torus T . If the con vex h ull of W ρ do esn’t con tain 0 then there is a substitution α i 7→ z i ∈ R suc h that for all w ( α 1 , . . . , α r ) ∈ W ρ w e ha v e that w ( z 1 , . . . , z r ) > 0. By Theorem 4 .2 the class [ Y ] is a non- trivial linear com binat ion of monomials of the w eights of ρ with non- negativ e co efficien ts, so if w e apply the same substitution w e get a p o sitiv e n um b er, whic h implies that [ Y ] = p ( α 1 , . . . , α r ) 6 = 0. On the other hand, if 0 is in the con vex h ull, then there is a non-t r ivial linear comb ination P n i w i = 0 with n i ≥ 0 in tegers. It implies that p = Q x n i i is a non-zero inv aria n t p olynomial (the co ordinate x i corresp onds to the o ne- dimensional eigenspace with w eigh t w i ). Then Y := { x ∈ V : p ( x ) = 1 } is a non-empt y inv ariant sub v ariety . W e ha v e that 0 6∈ Y so b y the fact that [ Y ] is lo calized to 0 w e get that [ Y ] = 0.  Theorem 4.1 immediately implies Theorem 4.3. If the r epr ese ntation ρ : G → GL( V ) is p ositive then al l non-empty G -invariant subvariety has a nonzer o e quivariant Poinc a r´ e dual. Let us recall no w the definition of Incidence Prop erty from the Intro duction. Definition 4.4. Given a r epr esentation ρ : G → G L( V ) we say that an orbit ξ has the Incid e n c e Pr op erty if for any other G -invarian t subvariety X ⊂ V w e have that [ X ] | ξ 6 = 0 ⇐ ⇒ ξ ⊂ [ X ] . Prop osition 3.1 a nd Theorem 4.3 no w implies that Theorem 4.5. A l l orbits of the r epr esen tation ρ : G → GL( V ) have the I ncidenc e Pr op erty if for al l x ∈ V the normal r ep r esentation of G x is p ositive (normal p ositivity). 5. Examples of represent a tions with the Incidence Proper ty First w e give a stronger condition than p ositivity , whic h is easier to chec k. The examples w e ha v e in mind are sub-represen tat ions o f the left-righ t action of a subgroup of GL( W ) × GL( W ) on the ve ctor space Hom( W, W ). In these cases the we igh ts are of the form t − s where t is a “target” w eight a nd s is a “source” weigh t. Prop osition 5.1. Supp ose that ρ : G → GL( V ) is a c omplex r epr esentation, T is a maximal torus of G a nd ther e ar e line arly indep endent elements e 1 , . . . , e s ∈ ˇ T such that al l weights of ρ ar e of the fo rm e i − e j for some i < j . The n ρ is p ositive, i.e. 0 is not in the c onv e x hul l of the weights of ρ . Pr o of. It is enough to ch ec k tha t 0 is not in the con v ex h ull of { e i − e j : i < j } . Assume that P n ij ( e i − e j ) = 0 for some no n-negativ e in tegers n ij . But P n ij ( e i − e j ) = P m k e k for some in tegers m k with m 1 = P n 1 j . It implies that n 1 j = 0 for all j . An induction o n s finishes the pro of.  6 L. M. FEH ´ ER A ND ZS. P A T AKF AL VI 5.1. Sch ub ert -v arieties of flag manifolds. Now we sho w the Incidence Prop ert y to the rep- resen tat io n discusse d in [FR03] and [KM05]. Let the group B + × B − act on Hom( C n , C n ) ∼ = { ( m ij ) n i,j =1 } , where B + and B − are t he groups of n × n upp er and low er triang ular matrices, resp ectiv ely , and ( R, L ) · M = LM R − 1 for R ∈ B + , L ∈ B − and M ∈ Hom( C n , C n ). Ev ery orbit contains a unique 0-1 matrix A , ev ery column and row of whic h contains a t most o ne 1. (Note that the full rank matrices corresp ond to the Sc h ub ert v arieties o f the n - dimensional flag-manifold.) W e can expand suc h a matrix uniquely to the righ t a nd do wn to a 2 n × 2 n p ermutation matrix. This expansion is obtained by adding 1’s t o row s not con taining 1’s starting fr o m the top, putting the 1 in the left-most p osition, where it is p ossible, outside of the upp er-left n × n matr ix. This w a y w e can asso ciate a p erm utation π ∈ S 2 n to an orbit, by setting π ( i ) = j , if there is a 1 in the ( i, j ) p o sition of the p ermu tation matr ix obtained. According to [FR03, § 4.] the maximal torus of the stabilizer group of the matrix A with expanded permutation π is T π = {  diag ( α 1 , . . . , α n ) , diag( α π (1) , . . . , α π ( n ) )  | α i ∈ U (1) } and an in v ariant normal space to the orbit o f A is N A = { ( m ij ) n i,j =1 | m ij = 0 if π ( i ) ≤ j or π − 1 ( j ) ≤ i } of Hom( C n , C n ) inv ariant under G π action. Using the basis e 1 , . . . , e n for ˇ T π ∼ = U (1) n the w eigh t s of N A are of the form e π ( i ) − e j where π ( i ) > j } , so by Prop osition 5.1. (for the rev erse ordering) this orbit has the Incidence Prop ert y . Equiv alent stat ements w ere pro ved in [LS92 ], [Kum96], [G ol01] and [BR04 ] as a c haracteriza- tion of the Bruhat o r der . In fa ct a fo r m ula in [BR04] gav e us the idea to study the relation of incidence with multide gree. 5.2. Quiver representations. In this section w e sho w that for represen t ations corresp onding to quiv ers of Dynkin type all orbits hav e the Incidence Prop erty . The equiv arian t P o incar ´ e duals for these cases were first calculated in [F R02a]. It also con tains a more detailed in tro duction t o the g eometry of these represen tations. Recen tly another a lgorithm was given in [KS06]. Consider an orien ted graph Q and denote by Q 0 the set of its v ertices and by Q 1 the set of its edges. If e is an edge of Q , then let e ′ and e ′′ denote the head and the t a il of e , respectiv ely . If a function d : Q 0 → N (dimension v ector) is give n we ha ve the group G = M i ∈ Q 0 GL( d ( i )) acting on V = M e ∈ Q 1 Hom( C d ( e ′ ) , C d ( e ′′ ) ), b y the form ula M i ∈ Q 0 A i ! · M e ∈ Q 1 ϕ e ! = M e ∈ Q 1  A e ′′ ϕ e A − 1 e ′  The orbits of this repres en tation corresp ond to the represen tatio ns (mo dules) of the path algebra C Q , with dimension v ector d . F or g raphs o f Dynkin t yp e, the se t R ( Q ) of indecomp osable mo dules o f C Q is finite. An y C Q -mo dule can b e decomp osed in to a form of P r ∈ R ( Q ) µ r r for some integer n um b ers µ r b y the Krull-Sc hmidt theorem a nd the n um b ers µ r are w ell defined. The maximal compact stabilizer THE INCIDENCE CLASS AND THE HI ERARCHY OF ORBITS 7 subgroup of M is G M = M r ∈ R ( Q ) U ( µ r ). A normal space is N M = M r,s ∈ R ( Q ) Hom( C µ r , C µ s ) m r s , where m r s = dim Ext C Q ( r , s ), and G M acts o n N M with the rule   M r ∈ R ( Q ) A r   ·   M r,s ∈ R ( Q ) m r s M i =1 ϕ i r s   =   M r,s ∈ R ( Q ) m r s M i =1 A s ϕ i r s A − 1 r   where ϕ i r s ∈ Hom( C µ r , C µ s ) for i = 1 , . . . , m r s . The maximal torus T M of the stabilizer group of M is isomorphic to U (1) P µ r . Let us denote the standard basis of the w eigh t lattice ˇ T M b y { e r,j : r ∈ R ( Q ) , j ≤ µ r } . Then the we igh ts of the represen tation of G M on Hom( C µ r , C µ s ) are e s,i − e r,j for any 1 ≤ i ≤ µ s and 1 ≤ j ≤ µ r . T o apply Prop osition 5.1 it is enough to sho w that there is an ordering ≻ on R ( Q ), suc h that if m r s 6 = 0, then r ≻ s . W e recall the notio n of the Auslander-R eiten tr anslate τ whic h is partial self mapping ma p of R ( Q ) with t he following prop erties. (1) F or ev ery r ∈ R ( Q ), there is a unique n r ∈ N , such that τ n r r is pro jectiv e. (In other w ords: in our case all indecomposables ar e pre-pro jectiv e.) (2) Ext C Q ( r , s ) = Ext C Q ( τ r , τ s ) for ev ery r , s ∈ R ( Q ), where b oth τ r and τ s are defined. (1) implies that there is an ordering ≻ on R ( Q ), for whic h if n r > n s , then r ≻ s . Suppo se that r 6≻ s . Then n r ≤ n s , and consequen tly m r s = dim Ext C Q ( l r , l s ) = dim Ext C Q ( τ n r l r , τ n r l s ) = 0, since τ n r l r is pro jectiv e. This sho ws that indeed if m r s 6 = 0, then r ≻ s , and w e pro v ed Theorem 5.2. F or a r epr esentation c o rr esp onding to a Dynkin typ e quiver al l orbits satisfy the Incidenc e Pr o p erty. 6. Singularities In t his section w e study incidences of singularities of holomorphic map germs. On singularit y w e mean a con tact o rbit of map germs. The equiv a rian t P oincar ´ e dual in this con text is called Thom p o l yno m ial . W e use [Rim01] and [FR04] as a general reference. The space E ( n, p ) of holomorphic map germs from C n to C p and the contact group K ( n, p ) acting on it are infinite dimensional. T o define G -equiv a rian t cohomology f or infinite dimensional groups some extra care is needed a s it is explained in [Rim01]. Our main in terest lies in studying finite c o dimensional singularities where a finite dimensional reduction is av aila ble. Only finite codimensional singularities hav e Thom p olynomials so it is a natural c hoice. It is p ossible to restrict to infinite co dimensional sin gularities (for example see [FR02b]), but to simplify the situation w e stic k with the finite co dimensional case. Ev ery finite co dimensional singularit y η is finitely determine d , i.e. there is a k ∈ N suc h that for any tw o germs f ∈ η and g ∈ E ( n, p ) w e ha v e that g ∈ η if and only if their k th T aylor p olynomials (or k -jets) j k ( f ) and j k ( g ) are con tact equiv alen t. So w e can reduce to the finite dimensional space of k - jets: J k ( n, p ) ∼ = L k i =1 Hom(Sym i C n , C p ). W e hav e the tr uncating map t k : E ( n, p ) → J k ( n, p ) . Similar ly w e can define K k ( n, p ) , the group of k -jets of the contact group: K k ( n, p ) := { ( ϕ, ψ ) ∈ J k ( n, n ) × × J k ( n + p, n + p ) × : ψ | C n × 0 = Id C n , π C n ψ = ψ π C n } . This group a cts on J k ( n, p ) and for the tr uncating homomorphism π k : K ( n, p ) → K k ( n, p ) the map t k is π k -equiv arian t. The homomorphism π k induces isomorphism on equiv ariant cohomo lo gy 8 L. M. FEH ´ ER A ND ZS. P A T AKF AL VI and for any t w o k -determined singularities η and ξ : [ η ] | ξ = [ t k η ] | t k ξ . W e don’t pro v e this statemen t—as we a v oided defining the left ha nd side—but use as a motiv ation to study the represen tation of the algebraic group K k ( n, p ) on the jet space J k ( n, p ) . W e cannot exp ect that all orbits of this represen tatio n ha v e the Incidence Prop erty . There are man y singularities ξ with small symmetry: the maximal torus of their stabilizer group is trivial. It implies that the restriction map is automatically trivial. T o ha v e a c hance for the Incidence Prop ert y to b e satisfied w e ha v e to require the ex istence of at least a U (1) symmetry . Since the diagonal torus U (1) n × U (1) p is maximal in K k ( n, p ) and all maximal tori are conjugat e, this requiremen t implies that a represen ta tiv e f = ( f 1 , . . . , f p ) of the orbit ξ with diago na l symmetry can b e c hosen. Then f is a weig h te d homo gene ous p olynomial, i.e. t here are weigh ts a 1 , . . . , a n , b 1 , . . . , b p ∈ Z suc h that f j ( t a 1 x 1 , . . . , t a n x n ) = t b j f j ( x 1 , . . . , x n ) for j = 1 , . . . , p . These p olynomials are also called quasi-homogeneous. The Incidence Prop ert y can fail ev en for w eighted homogeneous p olynomials, whic h can b e detected b y restricting them to themselv es. According to the definition. Na mely if ξ has the Incidence Prop ert y then e ( ξ ) = [ ξ ] | ξ 6 = 0. Example 6.1. F or ξ = ( x 2 + 3 y z , y 2 + 3 xz , z 2 + 3 xy ) the normal Euler class e ( ξ ) = 0. Remark 6.2. This Euler class can b e directly calculated using unfolding—see the second part of this section—but there is a deep er reason fo r the v anishing of e ( ξ ). This orbit is a mem b er of a one-para meter family of orbits: f λ = ( x 2 + λy z , y 2 + λxz , z 2 + λxy ). (This is a famous example ([W al77]), the smallest co dimensional o ccurence of a family in the equidimensional case.) The tangen t direction of the family in the normal space N ξ has 0 w eigh t, whic h implies t ha t e ( ξ ) = 0. In f act we don’t kno w a ny example when e ( ξ ) = 0 but ξ is n ot in a family . The following conjecture is a sligh tly mo dified v ersion of R im´ a n yi’s from [Rim01]: Conjecture 6.3. The singularity ξ ⊂ J k ( n, p ) has the In c idenc e Pr op erty iff e ( ξ ) 6 = 0 . Remark 6.4. Defining the Incidence Prop ert y (Definition 4.4) in suc h a w a y that w e require the condition e ( ξ ) = [ ξ ] | ξ 6 = 0 to b e satisfied seems f o rmal, we kno w that ξ is a subset of ξ anyw ay . Ho w ev er the v anishing of e ( ξ ) sometimes can b e related to the v anishing of a prop er incidence [ η ] | ξ where η 6 = ξ but ξ ⊂ η . Consider the follo wing abstract example: Example 6.5. Let GL (1) act on C 2 b y z ( x, y ) = ( x, z y ). The orbits of this represen tation are f λ = { ( λ, 0 ) } and g λ = { ( λ, y ) : y 6 = 0 } . The fixed p oint f λ has a U (1) symmetry and f λ ⊂ g λ but the incidence [ g λ ] | f λ is zero. W e hav e a geometric reason for this: [ g µ ] | f λ = 0 if µ 6 = λ since f λ 6⊂ g µ . But [ g µ ] dep ends con tin uously o n µ so it is constant. W e can also calculate directly: g λ is a line so [ g λ ] | f λ is the Euler class of the complemen t a ry in v ar ia n t line { ( λ, 0) : λ ∈ C } . But this is exactly the set of fixed p o in ts so the Euler class is zero. W e b eliev e that suc h situation is not uncommon f or singularities, i.e. there are con ta ct orbits f λ and g λ dep ending on the parameter λ contin uously suc h that co dim( f λ ) = codim( g λ ) + 1, f λ ⊂ g λ and f λ has a U (1) symmetry . The same reasoning sho ws that the incidence [ g λ ] | f λ is zero in this case. Unfortunately w e w ere unable to v erify the existenc e of suc h a n example. THE INCIDENCE CLASS AND THE HI ERARCHY OF ORBITS 9 Theorem 4.3 implies that Prop osition 6.6. If a sin g ularity ξ ⊂ J k ( n, p ) is p ositive i.e the norm al action of the s tabil i zer gr oup i s p ositive, then ξ has the Incidenc e Pr op erty. Complicated singularities are usually not p o sitiv e: Example 6.7. The singularities ( x a , y b ) for 2 ≤ a ≤ b are p ositiv e exactly for ( x 2 , y 2 ), ( x 2 , y 3 ) and ( x 2 , y 4 ). On the other hand we kno w man y p ositiv e cases: Example 6.8. All Σ 1 and Σ 2 , 0 singularities (type A n , I a,b , I I I a,b , I V a , V a , for the notation see [Mat71]) a re po sitiv e. Remark 6.9. In fact all the incidences o f the singularities in Example 6.8 can b e explicitly calculated since their adjacencies are well understoo d (see [Rim99]). F or example in t he equidi- mensional case for 2 ≤ a ≤ b , ( a, b ) 6 = (2 , 2), 2 ≤ c ≤ d and c + d < a + b w e ha v e [Rim]: [ I c,d ] | I a,b = ( a − 1)!( b − 1)!  a d b c ( a − c )!( b − d )! + a c b d ( a − d )!( b − c )!  · 1 δ cd + 1 · g , where k ! = ∞ f or k negative and δ cd = 1 if c = d and 0 otherwise. The maximal torus of the stabilizer group of I a,b for ( a, b ) 6 = (2 , 2) is U (1) and g denotes the g enerato r of Z ∼ = H D U (1) ( pt ; Z ) where D is the degree of the cohomology class [ I c,d ]. W e ha v e one result whic h go es b ey ond the p o sitivit y condition. T o state it, w e recall the definition o f Σ i classes: Definition 6.10. Σ i := Σ i ( n, p ) = { f ∈ J k ( n, p ) : dim k er d 0 f = i } . W e supressed t he dep endence on k in the notation as the Σ i -class of f dep ends only on t he linear term j 1 f . Theorem 6.11. Supp o s e that the c ontact orbit ξ ⊂ Σ 0 ∪ Σ 1 ∪ Σ 2 has at le ast a U (1) symmetry (i.e. ther e i s a weighte d homo gene ous p olynomial in the orbit). Then the i n cidenc e [Σ 2 ] | ξ vanishes if and only if ξ 6∈ Σ 2 . Notice that the classification of Σ 2 singularities is not known, so case by case c heck ing is not a v ailable here. T o prepare the pro of we mak e some remarks. Remarks 6.12. (i) If the contact orbit ξ is in Σ 0 ∪ Σ 1 ∪ Σ 2 then ξ has a represen tative f suc h that f depends only on the first tw o v ariables so w e will assume that n = 2. If f ∈ J k (2 , p ) then f ∈ Σ 2 if and only if f has no linear terms. (ii) Σ 2 alw a ys con tains a con tact orbit which is op en in Σ 2 . If k = 1 then Σ 2 is a n orbit. If k ≥ 2 a nd p = 2 then I 2 , 2 —the orbit of ( x 2 , y 2 )—is op en in Σ 2 . If p > 2 then I I I 2 , 2 —the orbit of ( x 2 , xy , y 2 , 0 , . . . , 0)—is op en in Σ 2 . So for example for the latter case [ I I I 2 , 2 ] = [Σ 2 ] (see [Mat71]). 10 L. M. FEH ´ ER A ND ZS. P A T AKF AL VI (iii) By the Thom-P or t eous-G iam b elli formula [P or71] w e hav e [Σ 2 (2 , p )] = Y ( β i − α 1 )( β i − α 2 ) in terms of Chern ro o ts. In other w ords α 1 , α 2 , β 1 , . . . , β p denote the generators of H 2 T ( pt ) where T = U (1) 2 × U (1) p is the maximal tor us acting on J 1 (2 , p ). (It can also b e seen directly since J 1 (2 , p ) ∩ Σ 2 (2 , p ) = 0.) (iv) Consequen tly for the w eigh ted ho mogeneous p olynomial f ∈ J k (2 , p ) with one U (1) sym- metry , i.e. with stabilizer subgroup of rank one, and with w eights a 1 , a 2 and b 1 , . . . , b p w e ha v e (1) [Σ 2 ] | f = Y ( b i − a 1 )( b i − a 2 ) g 2 p where g is the generator of the cohomology ring H 2 U (1) ( pt ; Z ) ∼ = Z [ g ] of the symmetry gr o up. In other w ords the incidence is zero if and only if there is a tar g et w eigh t equal to a source w eigh t. If f has symmetry group U ( 1 ) × U (1), then w e can still restrict [Σ 2 ] | f further to the subtorus U of U (1) × U (1) corresp o nding to t he w eigh ting of f . Then we get form ula (1 ) for ( [Σ 2 ] | f ) | U . Hence if no tar g et w eigh t is equal to a source w eight w e still g et that the incidence is not zero. (v) No t ice that (1) holds only if f has no additional symmetry . F or example if f is con tact equiv alent to a monomial germ then the rank of the symmetry group is at least 2 . If f ∈ Σ 2 (2 , p ) is monomial then the incidence [Σ 2 ] | f is not zero since w e can r estrict to the U (1) symmetry corresp onding to the source we igh ts a 1 = a 2 = 1 and see that b i ≥ 2 for all i ≤ p since f ∈ Σ 2 (2 , p ) implies that there are no linear terms. These remarks imply that to prov e Theorem 6.11 it is enough to pro v e: Prop osition 6.13. L et f ∈ J k (2 , p ) b e a weighte d homo gen e ous p o l yno m ial with weights a 1 , a 2 and b 1 , . . . , b p . If a 1 = b 1 then f is c o ntact e quival e nt to a monomial germ. Pr o of. W e distinguish three cases and in a ll cases w e will use the fact that if the ideals ( f 1 , . . . , f p ) and ( f ′ 1 , . . . , f ′ p ) agr ee then f is con ta ct equiv alen t to f ′ . In the first tw o cases w e don’t need the a 1 = b 1 assumption. Case 1: If a 1 = 0 or a 2 = 0. If, say , a 2 = 0 then f i = x b i /a 1 g i ( y ) for some p o lynomials g i ( y ). If the lo w est p ow er of y in g i ( y ) is k i then g i ( y ) = y k i ( h i + y g i ( y ) for some po lynomials g i ( y ) and non-zero num b ers h i . The functions h i + y g i ( y ) are units in the algebra C [ y ] / ( y k +1 ) whic h implies that t he ideal ( f 1 , . . . , f p ) is equal to ( x b 1 /a 1 y k 1 , . . . , x b p /a 1 y k p ) i.e. f is con tact equiv alent to a monomial germ. Case 2: If a 1 > 0 and a 2 < 0. Supp ose that deg x u y v = deg x u ′ y v ′ and u ≥ u ′ . Then v ≥ v ′ .It implies that f i ( x, y ) = x u i y v i ( h i + g i ( x, y )) for some u i , v i non-negativ e integers and h i 6 = 0 constan t. Therefore ( f 1 , . . . , f p ) = ( x u 1 y v 1 , . . . , x u p y v p ). Case 3: If a 1 > 0 and a 2 > 0. Then a 1 = b 1 implies that f 1 = y k for a 1 = k a 2 . Without lo ss of generalit y w e can assume that a 1 = k a nd a 2 = 1. F rom here w e don’t use t he a 1 = b 1 assumption. There is a unique decompo sition b i = u i k + d i for 0 ≤ d i < k . Then ( f i , y k ) = ( x u i y d i , y k ) for 2 ≤ i ≤ p , therefore ( f 1 , . . . , f p ) = ( y k , x u 2 y d 2 , . . . , x u p y d p ).  THE INCIDENCE CLASS AND THE HI ERARCHY OF ORBITS 11 6.1. C alculations of the E xamples 6.7 and 6.8. Let us recall no w that the normal space of the con tact orbit of the singularit y ξ = ( f 1 , . . . , f p ) is isomorphic to C n ⊕ U ξ where U ξ = J k ( n, p ) / ( f i e j , ∂ f /x l : i, j ≤ p, l ≤ n ) is the unfolding s p ac e of ξ . Here e j denotes the constant map (0 , . . . , 0 , 1 , 0 , . . . , 0 ) with 1 in the j th co ordinate. (see e.g. in [AGZV88] or [FR04]). ( x a , y b ) : The unfolding space is spanned b y the monomials { ( x i y j , 0 ) : i < a − 1 , j < b } and { (0 , x i y j ) : i < a, j < b − 1 } (exc ept for i = 0 , j = 0—constan ts are no t in J k ( n, p ) ). The maximal torus of the symmetry is a U (1 ) 2 = { ( α, β ) : α , β ∈ U (1) } acting via  α 0 0 β  ,  α a 0 0 β b  on Hom ( C 2 , C 2 ) therefore the w eigh ts of the normal space are { ( i, j ) : i = 2 , . . . , a ; j = − b + 1 , . . . , 0 } \ { ( a, 0) } and { ( i, j ) : i = − a + 1 , . . . , 0 ; j = 2 , . . . , b } \ { ( 0 , b ) } . By sk etch ing the distribution of the w eigh ts on the co or dina t e plane w e can see tha t (0 , 0) is in the conv ex hull unless a = 2 and b = 2 , 3 or 4. T o demonstrate that Σ 1 and Σ 2 , 0 singularities are p ositiv e w e pic k the series I I I a,b . the other calculations are similar but simpler since the symmetry is U (1). I I I a,b = ( x a , x y , y b ) : The unfo lding space is spanned b y { ( x j , 0 , 0 ) : j = 1 , . . . , a − 2 } ∪ { ( y j , 0 , 0 ) : j = 1 , . . . , b − 1 }∪ { (0 , 0 , x j ) : j = 1 , . . . , a − 1 } ∪ { (0 , 0 , y j ) : j = 1 , . . . , b − 2 } ∪ { ( ax a − 1 , y , 0) , (0 , x, by b − 1 ) } . 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Math. Pr o c. Cambridge Philos. So c. , 81(3):351 – 364, 1977 . Dep ar tment of Anal ysis, Eotv os University Budapest, H ungar y, R ´ enyi Institute Budapest, Hungar y E-mail addr ess : lfeher@ renyi .hu Dep ar tment of Ma thema tics, U n iv ersity of W ashington, S ea ttl e, W A-98195 , USA E-mail addr ess : pzs@mat h.was hington.edu