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(1)On the Geometric Genus of Hypersurface Isolated Singularities. by 1. Yoshiaki TASHIRO", Takako SUZUKI*" and Teiichi HIGUCHI"*. .. In this work we show some results of the geometric genus of hypersurface isolated singularities.. Our presentation goes as follows: In Sect. 1, we recall a few preliminaries related to the concept of the geometric genus of normal isolated singularities. In Sect. 2, we prove the pg-formula for non-degenerate hypersurface isolated singularities, which is a generalization of Corollary 1.14 [8, p.72]. In the ]ast section we. show some examples, which teach us how to use the practical means for computlng Pg ln many cases.. g1. Geometric genus of normal isolated singularities. We need to recall a few preliminaries related to the concept of the geometric genus of normal isolated singularities; for more details the reader is referred to Laufer [3] and Yau [10, 11].. Let (X, x) be a normal isolated singularity in the n-dimensional analytic space. X Suppose that V is a (sufficiently small) Stein neighborhood of x in X and K is the canonical line bundle of 1/-{x}. Let T: X.X be a resolution of the singularity (X, x). By the Direct Image Theorem of [1, p.207] Riz*ear are coherent analytic sheaves, which are finite-dimensional vector spaces concentrated at x. '. The dimensions of these vector spaces are determined independently to the choice. i. of a resolution.. ;. .. I i. Definition 1.1. The geometric genus of the singularity (X, x) is defined as. Pg(X, x)=dim R"-irc*e7. 'Definition 1.2. toGZ"(I7-{x}, (e)(to) is called square integrable if. 1. L 1. Svv-{.} tuA(-D<oo. 1. 1 i. i. * Tokyo University of Agriculture and Technology. ** Dept. of Mathematics, Yokohama National University..

(2) 8 Y. TAsHiRo and T. SuzuKi and T. HiGucHi for va any sufficiently small relatively compact neighborhood of x in X.. , Let L2(V-{x}) be the subspace of r(V-{x}, e(K)) consisting of holomorphic. n-form on V-{x}, which are square-integrable near the origin. Then (V-{x}, 0(K))IL2(V-{x}) is a finite dimensional vector space. This integer is determined. independentlytothechoiceoftheSteinneighbothoods. , Theorem 1,3 (Laufer [2], Yau [10], [11]). The geometric genus of a normal isolated singularity (X, x) is. P,(X, x)= dim r(V-{x}, e(K))/L2(V--{x}). Supposen>2. Then dim Riz*e7=dim M(X, e)=dim H{.','(X, e) for lisi.sn-2. where fl?"]i(X, e) are local cohomologies, with support {x}.. Example 1.4. Let (X, x) be a normal surface singularity defined by the poly-. nomial x8+y8+28+(xyg)2. Then P,(X x)=32. Let z: :Sl-X be a resolution of (X, x) and U=,n-'(V) and E=z-i(x). By. Theorem 3.1 [2], we have .. ttt ' L2(l/-{x})=L2((7-E)=r((Z 0(K)).. Then P,(X x)=dim T(U-E, e(K))/](U, O(K)). This formula provides a practical means for computing Pg in many cases.. g2. Geometricgenusofhypersurfaceisolatedsingularities. , Next we consider an isolated singularity defined by a nQn-degenerate holo-. morphicfunctionfSuchsingularitiesarealwaysnormai ' //.. ., In the following we give an effective method for calculating Pij. via the com-. binatorial data of the Newton polyhedron T+(f) of the functipn f, under the as-. sumption that f is nondegenerate with respect to r(f)i The proof is based on L. the computation of the resolution of hypersurface singularities by,using the tech-. niqueoftoricvarieties. ･ ... . For 6., i.e., plurigenera of normal isolated singularities, Watanabe [9] has obtained the generalized Sm-formula of hypersurface isolated singularities defined by. anon-degenerate holomorphic function. 'It is known that there is a canonical resolution by the torus embedding associated with a simplicial subdivision of the dual Newton diagram of f. So we need to recall a few preliminaries related to the concept of the torus embedding asso-. ciated with a simplicial subdivision of the dual Newton diagram of f; for more.

(3) On the Geometric Genus of Hypersurface lsolated Singularities 9. details the reader is referred to [6]. -. Let f(zo, zi, ･･･, z.) be a germ of an analytic function at the origin such that. f(O)=O and f has an isolated critical point at the origin. We assume that f has. a non-degenerate Newton boundary. Let X be a germ of hypersurface f-'(O). Let X"(f) be the dual Newton diagram and let X* be a simplicial subdivision of. T*(f). It is known that there is a canonical resolution A. n: X-X which is associated with 2*.. Let f(zo, 2i, ･･･, zn)-¥ a2g2 be the Taylor expansion of f where z2=2o20･･･ zn2n. Recall that the Newton boundary T(f) is the union of the compact faces of T+(f) where I"+(f) is the convex hull of the union of the subsets {2+(R')""i} for. 2 such that aitO. For any closed face a of T(f), we associate the polynomial fd(g)=,¥, a2z2･ We say that f is non-degenerate if fA has no critical point in (c*)n+i for any ael(f).. Let N' be the space of positive vectors in the dual spaces of R"+'. For any vector A==(ao, ai, ･･･, a.) of IV+, we associate the linear function A(2)=Z aiZz on. P+(f) and let d(A) be the minimal value of A(2) on T+(f) and let a(A)=={2GT+(f);. A('2)=d(A)}. We introduce an equivalence relation -- on IV' by A--B if andonly if A(A)=A(B). For any face A of l".(f), let. A* ={AclV+; A(A)=a}. The collection of A* gives a polyhedral decomposition of Al' which we call the. dual Newton polyhedron of f. We denote it by T"(f). A(A) is a compact face of T(f) if and only if A is strictly positive. We say that a subdivision X* of T"(f) is a simplicial subdivision if the following conditions are satisfied:. E* is a subdivision by the cones over a simplicial polyhedron whose simplexes are spanned by primitive integral vectors with determinant ±1.. Let p: N'-l}, be the natural projection onto an n-dimensional simplex Ih=N'IR>o. Then p(X") gives a Simplicial subdivision of l},.. Since every cone in X* is non-singular, the associated torus-embedding Z is .. non-singular.. Let E(A) be a divisor of Z associated with one-dimensional cone generated by .. AeX*, i.e., using the notations of Oda [5] E(A)=orb (RoA). Since X* is an r.p.p. decomposition of N', Zis a modification of Cn"'. Let z be its birational morphism. fromZto Cn+i. .. More precisely, let X* be a simplicial subdivision of r*(f). For each n-simplex. o=<Ao, ･･･, An>, Ai=(aio, ･･･, aid, ･･･, ain), we associate an (n+1)-dimensional Euclidean space C"+' with coordinates (u.,o, u.,i, ･･･, u.,n) and a birational mapping. n.: C"+i.C"+i which is defined by].

(4) 10 ･ Y.TAsHiRoandT.SuzuKiandT.HiGucHi 27'=(u.,o)`iO'j'(za.,i)ai'2'...(u.,n)an,j'. Let Z be the union of CY"i which are glued along the image of z. Let n be the projection and let ff be the closure of T-i(X-{v}), jc the origin. It is known that. nl7: .Xl-X is a resolution of X. Let di :d(Ai) and Ai=A(Pi). We assume that AoDAiD･･･Dan. We define .fZ(za.) and fd,,.(u.) by .1ft(z.(u.))=A(u.)ll(u.,i)di and .11e(na(ua))==fd,,.(u.)ll(u.,i)di. By the definition, l!ir is defined by .f}(za.)=O and. Xn{u.,i= O} is {u.; u.,i =O and fdi,.(u.)=O}･. Note that fdi,.(u.) is a function of en.,i+i, ･･･, u.,n. Thus Xn{u.,i==O} is non-. empty if and only if dim Ai>O. Let E(Ai;a)={u.GX;u.,i=O}. n(E(Ai;o))={O} if and only if Ai is strictly positive. The union of E(Ai;a) for simplexes a which. contain Ai is a divisor of X and we denote it by E(Ai). We say that vertices Ao, ･･･, Ak in X* are adjacent if there is an n-simplex a of E* which contains Ao, ..., A,.. Let (o==Res ((1/f) clzo.clgiA･･･Adgn). co is locally written in the form. =dZiAd22A'''Adznf(opOgo)･ Then tu is a nowhere vanishing holomorphic n-form on XL-{x}. Lemma 2.1. z"(g2(dg!f)) has zeros of-order Ao(Z)+(IAol-1-d(Ao)) at a divisor. E(Ao),whereIAol==aoo+aoi+･･･+aon･' v' Proof. Pick n primitive integral vectors Ai=(ai,o, ai,i, ･･･, ai,.) of X*, by which n-simplex a is spanned. Th' en there exists the associated (n+1)-dimensional Euclidean space C7"' with coordinates (u.,o, ･･･, u.,n) and a birational mapping T.: Ce'i-･C"+' which is defined by zj=(u.,o)"O･j'(u.,i)"'･y'･･･(u.,n)a"･j'. Theh'. z*(ga)=z*(zoRozi2i..･gnRn) = (en.,oae,o...ua,nan'o)2o...(u.,oao,n...u.,nan,n .)2n. =(Ua,o)AO(2)(U.,1)Al(2)'''(U.,n)An(2). z*dz=n*(dzoAdgiA'''AdZn) =(u.,o)IAOI'1'''(u.,n)IAnI-ldUa,oA'''AdUa,n n'Ef=(u.,o)d(AO)'''(ua,n)d(An)f(Ua,o, ''') Ua,n).. The desired result follows immediately from the above equations.. , When (X; x) is defined by a non-degenerate holomorphic fupction, thenumber. p, are expressed in terms of the Newton diagram. We denote by L(f) ,with. cone over r(f) with cone point the origin. ,-". Theorem 2.2. P,(X; x)== 5t(A(f)) where a(f)= {2EAX'Z+i; R+(1, 1, ･･･, I)Gl"-(f)}..

(5) On the Geometric Genus of Hypersurface lsolated Singularities 11 Proof. If 0 is any holomorphic n-form on X-{x}, g=01to is a holomorphie function on X-{x} and hence extends to be holomorphic also at x. The singular point x is normal, so there exists a holomorphic function G(g) in C{zo, xi, ･･･, zn}. such that GIi=g. Expand G(z) in a power series: G=Z C2o,Rl,...,2n Zo20Z121'''Zn2n. Then n*(zRto)El-'(X, Cl)(.K)) if and only if A(Z)+(iAl-1-d(A))llO for any strictly positive integral vector A of X*. So, T"(z2tu)￠l"(X,e(K)) if and only if A(2):S(d(A) -IAI) for some strictly positive integral vector A of X*, i.e., z+(1, 1, ･･･, 1)Gl"-(f).. An arbitrary polynomial F can be uniquely divided into two parts: F=(F)+. +(F)-, where (F)m consists of monomials ga such that 2ea(f), and we denote F-(F)- by (F).. Hence 0--gtu (mod L2(X-{x})), g==(G).ix. Assume moreover gtoGL2(X-{x}). Then, by the result of Watanabe [9], there exists H==H+ in C{zo, gi, ･･･, g.} such. that H+IX=G"IX. Hence GH-H+=P f=(X q,ge2)f=X cA(z2f)=2] c2((zaf).+(g2f)-) for some polynomial P. Therefore G.=XR c(x･2f)-. Thus the proof is complete.. Corollary 2.3. It is the Newton boundary of a non-degenerate holomorphic function that determines P, completely.. This Corollary is due to B. Laufer in the case where dim=2; see [3].. g3. Examples. Example 3.1. Let (Xl x) be a normal surface singularity defined by the poly-. nomial x8+y8+g8+x2y2z2. Then. P,(X, x)=32. In fact the bases are given as follows: w, xto, ytu, za), x2to, y2to, x2tu, xya), yzto, zxte, x3tu, y3tu, z3to, x2ytu, xy2to, y2ztu, yz2to, z2xtu, gx2to, xygto, x4to, y4tu, g4tu, x2y2to, y2g2tu, g2x2to, x3ytu, xy3to, y3xto, yz3tu,. g3xtu, zx3lv. ". Other,numerical data are given as follows:. (1) (tu)=-5E, -5E, -5E,. (2) Ei2==-6, Ei'Ei==2 (iti Pg(Ei)=3 (3) Kl}f2=25E2=-150, E,2= -6, Z.(E)= -18. (4) pto=22, pt+==42, pt-=152, pt=pto+pt++pt-=215･. The formulae 2P,=pto+pt+ and pt+1=K}f2+ZT(E)+12Pg are easily examined. Remark. We can calculate Pg of the above example using the covering methods, i.e,, u==x2, v= y2, w=g2, u4+v4+w4+uvw..

(6) Y. TAsHiRo, T. SuzuKi and T. HiGucHi. 12. REFERENCES ll 1] GRAuERT, H., Remmert, R.: Coherent Analytic Sheves. Grundlehren der mathematischen Wissenschaften 265, A series Comprehensive Studies in Mathematics, Springer-Verlag, Berlin-Hidelberg-New York, 1984. [2] LAuFER, H.B.: On rational singularities. Amer. J. Math., 95 (1972), 597-608. [3] LAuFER, H.B.: On pt for surface singu･larities, Proc. Sym. Pure Math. Vol. 30,Amer. Math. Soc. (1976), 45-49.. [4] MERLE, M., TEissiER, B.: Conditions d'adjonction. Seminaire Demazure-PinkhamTeissier (1976-1977), Centte de Math. Ecole Polytechnique.. [5] ODA, T.: Lectures on torus pmbeddings and applications (Based on joint work with Katsuya MiyAKE), Tata Inst. of Fund. Research 58, Springer-Verlag, Berlin-Hidel-. berg-New York, 1978. [6] OKA, M.: On the resolution of the hypersurface singularities to be submitted to Proc. of the Japan-U.S. Seminar on Complex Analytic Singularities, Tsukuba/ Kyoto, 1984 (T. SuwA and P. Wagreich, eds.), Advanced Studies in Pure Math. 8, Kinokuniya, Tokyo and North-Holand, Amsterdam, New York, Oxford. [7] STEENBRiNK, J.H.M.: Mixed Hodge structures associated with isolated singularities. Proceedings of Symposia in Pure Mathematics Volume 40 (1983), Part 2, 513-535. [8] WATANABE, K.: On plurigenera of normal isolated singularities. L Math. Ann., 250. (1980), 65-94. i. [9] WATANABE, K.: On plurigenera of normal isolated singularities. II. tobesubmitted to Proc. of the Japan-U.S. Seminar on Complex Analytic Singularitjes, Tsukuba/ Kyoto, 1984 (T. SuwA and P. WAGREicH, eds.), Advanced Studies in Pure Math. 8, Kinokuniya, Tokyo and North-Holand, Amsterdam, New York, Oxford. [10] YAu, S.S.T.: Two theorems in higher dimensional singularities. Math. Ann., 231 (1977), 44-59.. [11] YAu, S.S.T.: Sheaf cohomology on 1-convex manifblds. Recent developments in several complex variables. (Annals of mathematics studies; 100). [.12] YuMiBA, Y.: On a classification of isolated hypersurface singularities by their Newton polyhedra (in Japanese). Master's thesis, Saitama University., March, 1983. ,.

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