A FORMULA FOR EULER CHARACTERISTICS OF TAUTOLOGICAL LINE BUNDLES ON THE DELIGNE-MUMFORD MODULI SPACES Y.P. Lee U. C. Berkeley Abstract. We compute holomorphc Euler characterstcs of the lne bundles n L d over the modul space M 0,n of stable n-ponted curves of genus 0, where L s the holomorphc lne bundle over M 0,n formed by the cotangent lnes at the -th marked pont. Introducton Let M 0,n be the space of n (ordered) dstnct ponts (x,, x n ) on P modulo PSL 2 (C). There s a natural compatfcaton of ths space by addng n-ponted sngular stable curves to the boundary. The compactfed space M 0,n s called Delgne-Mumford modul space. Each pont x nduces a lne bundle on M 0,n wth fbre Tx P. Ths lne bundle can be extended to M 0,n, whch we denote by L. (For rgorous defntons see the next secton.) Denote χ d,,d n the holomorphc Euler characterstcs of the lne bundles n L d χ d,,d n = k ( ) k dm C H k (M 0,n, n L d ). () Let q s be formal varables. Introduce the generatng functon G(q,, q n ) = = χ( (d,,d n ) n χ d,,d n q d qd n n q L ). (2) Our man result s: Theorem G(q,, q n ) = ( + n q q ) n 3 n q. (3) In fact, several examples show that ths formula mght actually gve the dmensons of the spaces of holomorphc sectons of the lne bundles,.e. all hgher cohomology groups vansh. We wll ndcate some of them n 3.
Here we have several remarks. Frst, ths formula s related to the theory of twodmensonal gravty. Recall that the tree level correlaton functons of two-dmensonal gravty are defned to be [DW]: N d,,d n =< n c (L ) d, M 0,n >. (4) Let us ntroduce another formal varable z, whch s related to q by the formula q = e z. If we defne the generatng functon as F(z,, z n ) = (d,,d n ) 0 then usng the strng equaton [DW] one easly gets N d,,d n z d zd n n = M 0,n ( z c (L )) ( z n c (L n )), F(z,, z n ) = (z + + z n ) n 3. (6) Thus the Euler characterstcs χ d,,d n can be regarded as the correlaton functons n the correspondng K-theory and formula (6) s a lmtng case of our theorem n the cohomology theory. We remark also that our correlaton functons can be wrtten as: χ d,,d n = ch( L d )Td(M 0,n ) (7) M 0,n by the Remann-Roch Formula. Snce the structure of the Todd classes of M 0,n s stll not well understood at ths moment, our formula may provde some nformaton. The dea of usng generatng functon was suggested by A. Gvental and motvated by the fxed pont localzaton technque [G] [Ko] n the equvarant quantum cohomology theory. Notce that the formula (2) can be wrtten as M 0,n Td(M 0,n) ec (L )+z, and the denomnator of the ntergrand s exactly that of the holomorphc Bott-Lefschetz fxed pont formula, wth c + z nterpreted as the equvarant frst Chern class (of the fbrewse U()-acton). In the next secton we wll revew some basc facts on stable curves and ts modul spaces. The proof of the formula (3) wll be gven n 3. Acknowledgements. I wsh to thank Prof. A. Gvental for suggestng ths problem, and for hs gudance durng ths work. I am also grateful to Hsang-Png for her constant support. 2 (5)
2 The stable curves and ther modul spaces Let C be the feld of complex numbers, over whch all schemes are defned. An n-ponted stable curve (C; x,, x n ) of genus zero s a connected and reduced curve of arthmetc genus zero wth at most ordnary double ponts such that C s smooth at x, x x j (for j) and every component has at least 3 specal ponts (marked ponts + sngular ponts). ([Kn]). A famly of n-ponted genus zero stable curves over an algebrac scheme S s a flat, projectve morphsm π : F S wth n sectons x,, x n such that each geometrc fbre (F s ; x (s),, x n (s)) s an n-ponted stable curve of genus zero. Two famles (π : F S; x,, x n ) and (π : F S; x,, x n ) are somorphc f there exsts an somorphsm h : F F over S such that h x = x. Defne the modul functor M 0,n to be a contravarant functor from the category of algebrac schemes to the category of sets, whch assgns to each algebrac scheme S a set M 0,n (S) of somorphsm classes of famles of stable curves over S. In [Kn] Knudsen showed that there exsts a fne modul space M 0,n representng the functor M 0,n. Furthermore M 0,n s a smooth complete varety equpped wth a unversal curve π : U n M 0,n and unversal sectons x,, x n (marked ponts). In addton to representng M 0,n, M 0,n also gves an nterestng compactfcaton of the space of n dstnct ponts on P modulo automorphsms of P. Ths (noncompact) space s contaned n M 0,n as an open subset, whch s sometmes called the fnte part of M 0,n. Knudsen also showed that U n s somorphc to M 0,n+. In partcular, there are n + canoncal morphsms: π () n+ : M 0,n+ M 0,n, =, n +, whch map an (n + )-ponted curve (as a pont n M 0,n+ ) to an n-ponted curve by forgettng the -th pont. They are called forgetful maps. Notce that t mght be necessary to contract unstable components (stablzaton) when forgettng the marked ponts. In the followng dscuson only π (n+) n+ wll be needed. Therefore we wll denote t smply by π n+ and call t the forgetful map. The bundle L s defned as the conormal bundle to the unversal secton x : M 0,n U n. L s a holomorphc lne bundle because the marked ponts are always nonsngular. To compare the dfference of L (n) and πn (L() ) (L () s the correspondng lne bundle on M 0, ) we wll need some mportant dvsors on M 0,n. Defne D to be the dvsor on M 0,n whose generc elements are the curves wth two components, wth {x, x n } on one branch and the remanng marked ponts on the other. It s known ([Kn]) that {D } form a famly of smooth dvsors on M 0,n wth normal crossngs. Wth these defntons we are able to state: Lemma. L (n) = π n (L() ) O(D ). (8) Proof. See [W]. A local holomorphc secton of the locally free sheaf L () s represented by a local holomorphc one form ω on the curve [C] M 0, evaluated at x and πn (ω(x )) vanshes exactly on the dvsor D. 3
Q. E. D. By defnton a stable curve C s locally a complete ntersecton. Then the general theory of dualty [H] wll guarantee the exstence of a canoncal nvertble sheaf K over C. In fact, let f : C C be the normalzaton of C, y,, y l ; z,, z l the ponts on C such that f(y ) = f(z ) = p, =,, l, are the double ponts of C. Then the canoncal sheaf s the sheaf of one forms ω on C, regular except for smple poles at y s and z s and the Res y (ω) = Res z (ω)([dm]). Therefore the Serre dualty reads: for any dvsor D on C. Wth (9) t s easy to see H (C, O(D)) = H 0 (C, K O( D)) (9) Lemma 2. Let D = d D, d 0, be a dvsor on M 0,n. We have H (C, O(D)) = 0. (0) Proof. H (C, O C ) = 0 by the defnton of arthmetc genus. By Serre dualty H (C, O C ) = H 0 (C, K) and H (C, O(D)) = H 0 (C, K O( D)) = 0 because D 0. Q. E. D. 3 The proof of theorem We wll prove the formula (3) by nducton on n. Frst notce that the case n = 3 s trval. Snce M 0,3 s a pont, χ d d 2 d 3 = for every trplet (d, d 2, d 3 ). By the defnton of (2), G(q, q 2, q 3 ) = (d,d 2,d 3 ) 0 qd qd 2 2 qd 3 3. Ths s exactly the RHS of formula (3) as a formal power seres. For the case of general n we need the followng two reducton propostons. Proposton. Let π n : M 0,n M 0, be the forgetful map. Denote L (n) by L and l respectvely, then and L () ch(π n ( )) = ( + q L q )ch( q q l ). () Here π n (α) = R 0 π n (α) R π n (α) are the Grothendeck s hgher drect mages for any element α K(M 0,n ) and the LHS of equaton () s a formal power seres (d,,d ) ch(π n ( L d ))q d qd. Proof. Let α := L d. Our problem s essentally to compute R π n (α). As stated n 2 π n : M 0,n M 0, s n fact the unversal curve, the fbre C x over x M 0, s just an (n )-ponted curve of genus 0. By lemma L = π n (l ) O(D ). Snce 4
πn(l ) s trval on each fbre C x of π n, α Cx = O(D) wth D = d D. By lemma 2 H (C x, O(D)) = 0, we have R π n (α) = 0. It remans to compute R 0 π n (α). Snce H (C x, L ) = 0, H 0 (C x, O( d D )) forms a vector bundle (call t H 0 ) on M 0, by Grauert s theorem. Then π n (α) wll be somorphc to H 0 ( l d ). By defnton, the fbre C x s a tree of P wth n marked ponts x,, x. An element of H 0 (C x, O(D)) s a meromorphc functon wth poles of order no more than d at x. It would be constant f there are no poles. Therefore an element of H 0 (C x, O(D))/(constants) s unquely determned by the polar parts of the functon at the marked ponts. By the ratonalty of each fbre the polar parts are ndependent and the space of polar parts at the marked pont x k s fltered by the degree of poles: F F 2 F dk where F s the subspace of functons wth poles of order no more than. It s easy to see that the graded spaces F + /F are somorphc to Tx k. Therefore the vector bundle H 0 s topologcally somorphc to (C l l 2 l d l l d ). Combnng all above, we have ch(r 0 π n ( L )) =( e d c (l ) )( + e c (l ) + + e d c (l ) + + e c (l ) + + e d c (l ) ). Now multplyng the above by q d qd and summng over (d,, d ) Z n + (Z + := N {0}), we have = (d,,d ) (m,,m ) (d,,d ) e d c (l ) ( + e c (l ) + + e d c (l ) )q d qd e m c (l ) e m c (l ) q d qd, (2) where ether m = d for all or m = d for all except one for whch d m N s arbtrary. Thus (2) s equal to (m,,m ) Z n + (e c (l ) q ) m (e c (l ) q ) m [ + (q k + + qk )] =( + q + + q )( q q q e ) ( c (l ) q e ), c (l ) whch s equal to the RHS of (). 5 k=
Q. E. D. Snce π n : M 0,n M 0, s a proper morphsm of smooth varetes, we can apply the Grothendeck-Remann-Roch theorem to any holomorphc lne bundle α : Proposton can be restated as: ch(π n (α))td(m 0, ) = π n (ch(α)td(m 0,n )). (3) G(q,, q n ) qn =0 = ( + q q )G(q,, q ). (4) Proposton 2. On M 0,n the generatng functon (2) satsfes the followng dentty: I {,,n}( ) I I ( q ) G(q) {q =0, I} = 0. (5) Proof. The above dentty reads: n Td(M 0,n )( ( M 0,n q e )) = 0. c (L ) q Snce dm C (M 0,n ) = n 3 < n, the dentty follows from the fact that ( qe c ) ( q ) s dvsble by c and dmenson countng. Q. E. D. Now G(q,, q n ) wth (at least) one q = 0 was calculated n Proposton as (constant)g(q,, q ), whch s known by nducton hypothess. Combnng (4) and (5) we have G(q,, q n ) n = ( + q ) n 3 n ( + q j= q j j<k n + + ( ) q. j k q q ) n 3 Thus we are left to prove the combnatoral dentty (a := q q ) : ( + n a ) n 3 = n ( + a ) n 3 j j = j <j 2 ( + 6 n q j j 2 a ) n 3 + + ( ),
whch s easly verfed by elementary algebra. For example t s the fnte dfference verson of the dentty: n x x n ( + x + x 2 + + x n ) n 3 = 0 at (0, 0,, 0). Remarks : () The formulas of Euler characterstcs (coeffcent of G(q,, q n )) are rather complcated. It s remarkable that they can be packed n a very elegant generatng functon. () As we have mentoned n the ntroducton, there are some evdences whch support our guess on the vanshng of H (M 0,n, L d ) for. Frst, n the case of M 0,4 and M 0,5 t can be explctly computed. Second, f the lne bundle n L d conssts of only n bundles,.e. some d = 0, then the vanshng of hgher cohomology groups can be proved by the same arguments n the proof of proposton 2 and smple spectral sequence argument. Even n the case when some d = t can be proved by the same method (some modfcatons are necessary) plus the resdue theorem. But when mn{d } 2 the computaton becomes very complcated and we don t know how to proceed. () The above proof s based on a suggeston by A. Gvental. Our orgnal proof was done by a term by term argument,. e. computng the correlaton functons nstead of the generatng functon tself. In fact we can derve a reducton formula relatng the correlaton functons () on M 0,n n terms of those on M 0,, smlar to the dervaton [DW] of the correlaton functons (4) n the theory of two-dmensonal gravty. Not surprsngly we have to deal wth very complcated combnatoral denttes and cancellaton process. One of the mprovement of the present proof s that we derve the generatng functons drectly, and therefore drastcally reduce the complexty of combnatoral denttes. Refrences [DM] P. Delgne, D. Mumford, The rreducblty of the space of curves of gven genus, Inst. Hautes. Ètudes Sc. Publ. Math. 45 (969) 75-09. [DW] E. Wtten, On the structure of the topologcal phase of two-dmensonal gravty, Nucl. Phys. B340 (990) 28-332. R. Djkgraaf, E. Wtten, Mean feld theory, topologcal feld theory, and mult-matrx models, Nucl. Phys. B342 (990) 486-522. [G] A.Gvental, Equvarant Gromov-Wtten nvarants, IMRN 3 (996) 63-63. [H] R. Hartshorne, Resdues and dualty, Sprnger lecture notes, 20, 966. [Ke] S. Keel, Intersecton theory of modul space of stable n-ponted curves of genus zero, Trans. Amer. Math. Soc. 330 (992) 545-74. [Kn] F. Knudsen, Projectvty of the modul space of stable curves II, Math. Scand. 52 (983) 225-65. [Ko] M. Kontsevch, Enumeraton of ratonal curves va torus actons, In: The modul space of curves. R. Djkgraff, C. Faber, G. van der Geer (Eds.). Progress n Math., 29, 7
Brkhäuser, 995, 335-68. [W] E. Wtten, Two-dmensonal gravty and ntersecton theory on modul space, Surveys n Dff. Geo. vol. (99) 243-30. 8