Some notes on Futak nvarant Ch L Analytc Defnton of Futak Invarant Let be an n dmensonal normal varety. Assume t s Fano,.e. ts antcanoncal lne bundle K s ample. If s smooth, then for any Kähler form ω n [c (x)], by -lemma, we have a smooth functon h ω, such that Rc(ω) ω = 2π h ω We call h ω : log ωn h η h the Rcc potental of ω. Let v be a holomorphc vector feld on,.e. v s of type (,0) and v = 0. Then the Futak nvarant s defned to be F c()(v) = v(h ω )ω n () It s a holomorphc nvarant, as a character on the Le algebra of holomorphc vector feld, and ndependent of the choce of the Kähler form n c (). See [Fu]. The necessary condton of exstence of Kähler-Ensten metrc on s that the Futak nvarant vanshes. In [DT], the Futak nvarant s generalzed to the sngular case. When s possbly sngular normal, frst use kk to embed nto projectve spaces, φ k = φ kk : CPN k. h s the Fubn-Study metrc determned by an nner product on H 0 (, kk ). h = (φ k h ) /k s an Hermtan metrc on K. Note that on the smooth part of, Hermtan metrcs on K one-toone corresponds to volume forms. If {z } s a local holomorphc coordnate, denote dz dz n by dz, and d z d z n by d z, the correspondence s gven by h n dz dz n d z d z n dz dz n 2 h = n dz d z dz 2 =: η h h dz 2 h = z zn 2 h s the nduced Hermtan metrc on K by the metrc dual. On the smooth part of, ω h := 2π log h 2π log η h n =: dz d π log η h s a Kähler form, ts Rcc curvature s: Rc(ω h ) 2π log detωh n. Rc(ω h ) ω h 2π log ωn h η h So the Rcc potental s h ωh log ωn h η h. v(log ωn h )ωh n v( ωn h )η h (L v ωh n L vη h ω sm η h sm η h sm η h) n h = dv ηh (v)ωh n = (dv ηh (v) + ω h ) n+ sm n + sm In [DT], t s proved ths s stll a well defned holomorphc nvarant. Note that n local holomorphc coordnate, L v d z = 0, so L v (η h ) = L v(dz) + v(log dz 2 η h dz h )
Note that the frst term on the rght s holomorphc, so dv ηh (v) v log dz 2 h 2π v ω h (2) 2 Calculaton by Log Resoluton Assume s an equvarant log resoluton of sngularty of such that K = π K a E E are exceptonal dvsors wth normal crossngs. v lfts to be a smooth holomorphc vector feld ṽ on, whch s tangental to each exceptonal dvsor E. Let S be the defnng secton of [E ], so E = {S = 0}. Let h be an Hermtan metrc on [E ] and R h = 2π log h be the correspondng curvature form. By lemma (or Hodge theory), there s an Hermtan metrc h on K such that ts curvature form R h = 2π log h 2π log η h satsfes R h = π ω h a R h So π (Rc(ω h ) ω h ) 2π log π ωh n + η h 2π a log S 2 h π h ωh log π ω n h η h + v(h ωh )ωh n = π (v(h ωh ))π ωh n = sm \ E a log S 2 h + C ṽ( π ωh n )η h + \ E η h a ṽ(log S 2 h )π ω n h ṽ( π ωh n ) s a smooth functon on. η h Lemma. θ = ṽ(log S 2 h ) extends to a smooth functon on such that 2π θ ṽr h Proof. It s clearly true away from exceptonal dvsors. Let p E, n a neghborhood U of p, choose a local frame e of [E ], S = f e, and E = {f = 0}. We assume E s smooth at p, so we can take f to be a coordnate functon, say z. Snce ṽ s tangent to E, ṽ s of the form ṽ(z) = z b (z) z + > c (z) z b (z), c (z) are holomorphc functons near p. Now the second term s smooth near p, and s holomorphc near p. Also θ = ṽ(log z 2 ) + ṽ(log e 2 h ) ṽ(log z 2 ) = ṽ(z ) z = b (z) θ = (v(log e 2 h )) v log e 2 h 2π v R h (3) 2
So the Futak nvarant can be wrtten as F c()(v) = ( L ṽη h + a θ )(R h + a R h ) n η h = (dv η (ṽ) + a θ + R h + n + a R h ) n+ Now by (2) and (3), (dv η (ṽ) + a θ + R h + a R h ) s an equvarantly closed form, so we can apply localzaton formula to ths ntegral. See [BGV], [T2] for localzaton formula. Remark. Note that at any zero pont p of ṽ, the dvergence dv η (ṽ) s well defned ndependent of volume forms. Also by the proof of prevous lemma, f p E, θ (p) = b (p) s the weght on the normal bundle of E at p, otherwse θ (p) = 0. In any case, f q = π(p), then dv(ṽ)(p) + a θ (p) s the weght on K q. 3 An example of calculaton We calculate an example from [DT] usng log resoluton. s the hypersurface gven by F = Z 0 Z 2 +Z Z 2 3 +Z3 2. v s gven by λ(t) = dag(, e6t, e 4t, e 3t ). The zero ponts of v are [, 0, 0, 0], [0,, 0, 0], [0, 0, 0, ]. [, 0, 0, 0] s an A-D-E sngular pont of type E 6. Locally, t s C 2 /Γ, Γ s the lftng to SU(2) of the symmetrc group of Tetrahedron n SO(3). Γ = 24. After a (nonlnear) change of coordnate, we change t to the standard form + 2 + z 4 3. The vector feld s gven by v = 6z z + 4 z2 + 3 z3. By vewng the surface as a two-fold coverng of C 2, branched along a sngular curve, we can equvarantly resolve the sngularty by blowup and normalzaton (at the orgn of each step). See [BPV].. + z3 2 + z4 3 = 0. z e 6t z, e 4t, e 3t. 2. s 2 + ( + t 3 ) = 0. t = z2 e t t, s = z e 3t s. 3. s 2 2 + t 2 (t 2 + t 2 ) = 0. t 2 = z3 t e 2t t 2, s 2 = s t e 2t s 2. 4. s 2 3 + t 3(t 3 + t ) = 0. t 3 = t2 t e t t 3, s 3 = s2 t e t s 3. 5. s 2 4 + t 4(t 4 + ) = 0. t 4 = t3 t t 4, s 4 = s3 t s 4. t 2 t 3 t 4 t 2 2 + z 4 3 = 0 + t 3 = 0 t 2 t 3 t t 2 + t 2 = 0 t t + t 3 = 0 t t 4 = t P 6 P 2 P 4 E 3 P 8 E P P 7 E 2 P 3 P 5 E 2 3 E ( 2) E 4 P 9 E 2 2 E 2 ( 2) The ntersecton dagram of Exceptonal dvsors s of type E 6. Assume P 8 P 4 P 6 P 7 P 5 E3 E 4 E3 2 ( 2) ( 2) ( 2) E 2 2 ( 2) K = π K + a E 3
Note that π K E = 0, then K E = j a j E j E By adjont formula, K E = K E E E 2 = 0 Because the ntersecton matrx {E E j } s negatve defnte, we have a = 0. So The zero ponts set of ṽ are: 5 = {P } E 4. K = π K. equaton near P s: u 2 + ( + t 4 ) = 0. u = z e 2t u, t = z3 e t t. 2. equaton near P 2, P 3 s: u 2 2 + t 3 2 z2 3 + = 0. t 2 = t e 2t t 2, e 3t. 3. equaton near P 4, P 5 s: u 2 3 + t 2 3 t 2 + = 0. t 3 = t t 2 e t t 3, t 2 e 2t t 2. 4. equaton near E 4 (away from P 6, P 7 ) s: s 2 4 + t 4(t 4 + ) = 0. (near P 6, P 7, the equaton s u 2 4 + t 2 4 + = 0) E 4 = {t = 0}. t e t t. So the contrbuton to the localzaton formula of Futak nvarant at pont [, 0, 0, 0] s: 2 + 2 6 + 2 2 + + c ([E]) = 6 the contrbutons from the other two fxed ponts are easly calculated, so the Futak nvarant s: E F c()(v) = 3 ( 6 + ( 5)3 6 + ( 2)3 3 ) 6 Remark 2. The contrbuton of the sngular pont can also be calculated usng the localzaton formula for orbfolds gven n [DT]. Note that the local unformzaton s gven by: π : C 2 C 2 /Γ C 3 (z, ) [, (z 4 + 2 3 2 + z 4 2) 3, 2( 3) 3 4 z (z 4 z 4 2), (z 8 + 4z 4 z 4 2 + z 8 2)] So π v = 2 (z z + z2 ), and 4 Algebrac Defnton (dv(πv)) n+ Γ det( (π v) = 3 T z) 24 /4 = 6 We can transform the expresson of Futak nvarant () nto another form: F c()(v) (S(ω) ω)θ v ω n (4) where S(ω) s the scalar curvature of ω, and θ v s the potental functon of the vector feld v satsfyng v ω = 2π θ v In ths way, the Futak nvarant generalzes to any Kähler class. The vanshng of Futak nvarant s necessary for the exstence of constant scalar Kähler metrc n the fxed Kähler class. Assume there s a C acton on (, L), there are nduced actons on H 0 (, L k ). Let w k be the k th (Hlbert) weght of these actons. For k suffcently large, w k = a 0 k n+ n! + a k n 2n! + O(kn ) (5) 4
d k = dmh 0 (, L k k n ) = b n! + b k n 2 + O(k n 2 ) 2n! At least n the smooth case, one can show that (See [Do]) a 0 = θ v ω n, a = S(ω)θ v ω n (6) b = ω n, b 2 = S(ω)ω n By ths, Donaldson [Do] gves an algebro-geometrc defnton of Futak nvarant: F c(l)(v) a b a 0 b 2 b (7) Remark 3. Assume we can embed nto P(H 0 (, L) ) usng the complete lnear system L such that the C acton s nduced by a one parameter subgroup n SL(d, C). Then we see that, at least n the smooth case, f we normalze θ v, the (normalzed) leadng coeffcent ((n + )a 0 ) n the expanson (5) s the Chow weght of ths C acton. 5 Futak nvarant of Complete Intersectons We wll use the algebrac defnton to calculate. Assume CP N s a complete ntersecton gven by: = r α= {F α = 0} Assume deg F α =, so deg = α Let R = C[Z 0,, Z N ]. has homogeneous coordnate rng R() = C[Z 0,, Z N ]/(I()) = R/I() I() s the homogeneous deal generated by homogeneous polynomal {F α }. It s well known that R() has a mnmal free resoluton by Koszul complex: 0 R( r ) (C F α ) α=0 α r R( d β ) (C (F α F β )) α<β r R( ) (C F α ) R R() 0 Let λ(t) PSL(N +, C) be a one-parameter subgroup generated by A = dag(λ 0,, λ N ), and v be the correspondng holomorphc vector feld. Assume that N =0 λ Z Z F(Z) = µ α F α (C ) 2 acts on S(). Let a k,l = dms() k,l be the dmensons of weght spaces, then ths acton has character: Ch(S()) = (k,l) N Z The k th Hlbert weght s (note t s a fnte sum) α=0 r a k,l t k tl 2 = α= ( tdα tµα 2 ) N =0 ( t t λ 2 ) = f(t, t 2 ) w k = l Z a k,l l 5
and w k t k = f t 2 t2= k N Lemma 2. Let then Proof. So when k, α (µ αt dα β α ( td β )) ( t ) N+ + ( α (µ αt dα β α ( + + td β )) r α= ( t ) N+2 r + λt f(t) = g(t) ( t) n+ = r =0 a t + ( t) n+ = b k t k k=0 b k = kn kn g() + n! 2(n )! ((n + )g() 2g ()) + O(k n 2 ) =0 r ( ) n + j f(t) = ( a t ) t j n =0 r ( ) n + k b k = a = n =0 r = kn a + kn n! 2(n )! =0 j=0 r (n + k ) (k + ) a n! r a (n + 2) + O(k n 2 ) =0 = kn kn g() + n! 2(n )! ((n + )g() 2g ()) + O(k n 2 ) r α= ( tdα λ )t ) ( t ) N+2 ( + + tdα ) ( t ) N+2 r (8) Let g(t) α (µ αt dα β α ( + + td β )) + λt r α= ( + + tdα ), n = N + r, let µ α = µ α λ N+, then µ s nvarant when λ(t) dffers by a dagonal matrx. by the lemma, we can get g() α µ α β αd β +λ α α (N r + 2)g() 2g () α α ( β (N + β (N + β µ β d β λ) α d β ) γ d β ) γ β µ γ d γ β µ γ d γ β µ β λ (N + r) d β N + µ β λ(n β d β ) (9) µ β λ N + (N r)(n + α ) w k α µ β k N+ r d β (N + r)! α β (N + β d β ) γ µ γ d γ β µ β kn r 2(N r)! + O(kN r ) + λ N + k dmh0 (, O(k)) (0) By (7), we can get the Futak nvarant F c(o())(v) α β µ β N + γ d γ N + r µ β β d β 6
Remark 4. In hypersurface case, the above formula becomes (d )(N + ) F c(o())(v) (µ λ N N + d) Apply ths to the example n secton 3, where d = 3, N = 3, λ = 6 + 3 + 4 = 3, µ = 2, O() = K, then we get the same result as before. F c()(v) 2 4 3 (2 3) 6 3 4 Remark 5. We can calculate drectly the leadng coeffcent of w k n (0)usng the Lelong-Poncáre equaton. Also see [Lu]. Lemma 3 (Poncáre-Lelong equaton). Assume L s a holomorphc lne bundle on, s s a nonzero holomorphc secton of L, D s the zero dvsor of s,.e. {s = 0} counted wth multplctes. h s an Hermtan metrc on L, R h = 2π log h s ts curvature form. Then n the sense of dstrbuton, we have the dentty 2π log s 2 h R h D.e., for any smooth (2n 2) form η on, we have (log s 2 2π h) η = D η R h η Let 0 = CP N, a+ = a {F a = 0}, then 0 r =. θ v = v ω = θ 2π v. On a, by the lemma, we have 2π F a 2 log ( Z 2 ) d a ω da a a a So θ v ω N a a = d a θ v ω N a+ + θ v a 2π log a Usng ntegraton by parts, the second ntegral on the rght equals F a 2 θv log 2π a ( Z 2 ) d ωn a = v ω log a F a 2 v(log N a + a ( Z 2 )ωn a+ da ) (µ a d λ Z 2 a N a + a Z 2 )ωn a+ N a + µ a deg( a ) + d a N a + F a 2 ( Z 2 ) d ωn a λ Z 2 Z 2, then F a 2 ( Z 2 ωn a da ) θ v ω N a+ a So (N a + ) θ v ω N a µ a deg( a ) + d a (N a + 2) θ v ω N a+ a a Whle (N + ) θ v ω N = (N + ) λ Z 2 0 CP N Z 2 ωn = λ = λ By nducton, we get (N r + ) θ v ω N r r α Ths s the same as g(), (9). β µ β d β + λ α = α β µ β λ + (N + r) d β N + 7
References [BGV] Berlne, N., Getzler, E., Vergne, M.: Heat kernels and Drac operators, Sprnger-Verlag Berln-Hedelberg-New York, 992 [BPV] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berln-Hedelberg-New York: Sprnger 984 [DT] Dng, W. and Tan, G.: Kähler-Ensten metrcs and the generalzed Futak nvarants. Invent. Math., 0, 35-335 (992) [Do] Donaldson, S.: Scalar curvature and stablty of torc varetes, J. Dff. Geom., 62 (2002), 289-349 [Fu] Futak, A.: An obstructon to the exstence of Ensten-Kähler metrcs. Invent. Math., 73, 437-443 (983) [Lu] Lu, Z.: On the Futak nvarants of complete ntersectons, Duke Math. J., Volume 00, Number 2 (999), 359-372 [T] Tan, G.: Kähler-Ensten metrcs wth postve scalar curvature, Invent. Math., 37 (997), -37 [T2] Tan, G.: Canoncal metrcs n Kähler geometry, Lectures n Mathematcs ETH Zürch, Brkhäuser Verlag, 2000 8