K-theoretic computations in enumerative geometry

Transcription

1 K-theoretc computatons n enumeratve geometry Shua Wang Oct 206 Introducton Recently, I m readng Prof. Okounkov s Lectures on K-theoretc computatons n enumeratve geometry, so ths notes s just a readng notes... 2 The localzaton theorem Let X be a smooth projectve scheme wth an smooth acton of the smooth dagonal group D. We want to consder the Grothendeck group K D (X) of D-lnearzed coherent sheaf on X. 2. Basc notatons and constructons Frst recall the followng notatons and constructons K D (X) the Grothendeck group of the category of D-equvarant locally free sheaves on X. K D (X) the Grothendeck group of the category of D-equvarant coherent sheaves on X. K D (X D ) the Grothendeck group of D-equvarant locally free sheaves on the fxed locus X D. K D (X D ) the Grothendeck group of category of D-equvarant locally free sheaves on the fxed locus X D., X(D): character group of D R(D) = Z[ ]. Z-group algebra assocated to the character group = X(T ), or smply the representaton rng of D. It s an ntegral doman. For χ, we usually use e χ to denote the correspondng element n R(D). And let S R(D) be the multplcatve subset generated by ( e χ ) for all non-trval χ, then we know 0 / S. cl(f ), [F ] The class represented by F n K D (X).

2 K D (Spec(k)) = R(D). From ths pont of vew K D (Spec(k)) contans more nformaton than the ordnary K(Spec(k)) = Z. pull-back f!. For f a D-equvarant morphsm, f! = K D (f), whch s nduced by the ordnary pull-back operaton of sheaves. push-forward f!. For f a D-equvarant morphsm, f! = K D (f), whch s nduced by the ordnary push-forward operaton of sheaves, or we can say, t s nduced by the drect mage functor f. ch, tr Let E be a k-lnear representaton of D, then tr(e) := (dm k E χ )e χ R(D). It s just the decomposton of a representaton nto the drect sum of -dmensonal rreducbles. tr X. If D actc trvally on X, then F = F χ. We forget the D-equvarant structure of F χ, and just vew t as an element n K(X), then we can defne a map tr x or tr? tr X : K D (X) K(X) Z R(D) E F χ e χ If x s a (closed) fxed pont of the D=-acton on X, whch means tr z : K D (x) K(x) Z R(D) s the somorphsm we just defned, however, t s just the map tr, snce a class n K(X) s represented by ts dmenson. tr : K D (x) R(D); E m χ E χ. χ λ-operaton. λ : K D (X) K D (X); [F ] [ F ]. λ t : K D (X) + tk D [[t]]; [F ] + + = λ [F ]t. Note that λ s a group homomorphsm. tr X s an somorphsm(we already know the case when X = Spec(k).) A = Sym k E. The symmetrc algebra assocated to the representaon E over k, t s also an ntegral doman. It has a natrual gradng, whch we can secretly take as k[x 0,..., x n ]. D A-graded modules. A D A-graded module M s a A-graded module wth a D acton on each graded pece subjected to d(am) = d(a)d(m); d D, a A, m M. The morphsms n the category of D A-graded modules are those graded of degree 0, A-lnear, D-lnear. 2

3 twsted D A-graded modules. Gven a graded D A-graded module, the twsted D A-graded module M χ s somorphc to M as A-modules, but wth a twsted acton by D: Note that we have d m := χ(d)dm. Hom D A graded (A χ ( n), N) = (N n ) χ. (N n ) χ means the submodule of N n wth the D-acton of weght χ. H (X, F ) s a representaton of D. Consder Čech cohomology and a theorem by Hdeyasu Sumhro whch says that we can fnd a D- onvarant affne open cover of X. There s another way to defne t, namely we have σ F d F F where the second map s gven by the lnearzaton of F, thus we have H (X, F ) H (X, σ F ) H (X, F ). The Lefschtz trace χ D (X, F ). Snce the functor F ( ) trh (X, F ) s addtve, so t nduces the Lefschtz trace Ch(F ) T odd(f ) T odd(x) := T odd(t X ). The Todd class of the tangent sheaf of X. ct D s the composton χ D (X, F ) : K D (X) R(D). tr K D (X) X Ch d R(D) K(X) Z R(D) A(X) Z Q Z R(D) T odd D s the composton K(X) Z R(D) T odd d R(D) A(X) Z Q Z R(D) 2.2 K D (X) and K D (X);! and! Proposton 2. (coherent sheaves v.s vector bundles, K D (X) = K D (X)). The natural of K D (X) nto the Grothendck group of the category of D-lnearzed coherent sheaves on X s an somorphsm. 3

4 Remark (tensor product s not the multplcaton n K D (X)). Snce tensor product s only rght exact, ths s not the multplcaton n K D (X),.e f we have a short exact sequence of coherent sheaves 0 E E 2 E 3 0. Then n K D (X), we have [E 2 ] = [E ] + [E 3 ], however, f F, a coherent sheaf, but not flat, then [E 2 F ] [(E E ) F ]. On the other hand, by the theorem above, we know K D (X) = K D (X) for a smooth projectve scheme X, tensor product s the multplcaton n K D (X), so we know K D (X) does have a rng structure. That s by Hlbert syzygy theorem, we can always fnd a free resoluton then we also have By the defnton of K D (X), we know 0 V n+ V 0 V = E 0 0 V n+ F V 0 F E F 0. [E ][F ] = ( ) [V ][F ]. Then by some general fact from homologcal algebra, we also get [E ][F ] = ( ) T or O X (E, F ). Example 2.2 (transverse ntersecton). Let Y, Y 2 be two closed subschmes of X n general poston, then we have [O Y ][O Y2 ] = [O Y Y 2 ]. Let : Y X be a D-equvarant closed mmerson of smooth projectve schemes wth D-acton. Proposton 2.3 (the projecton formula, push-pull). For any x K D (X), y K D (y), we have Proposton 2.4 (the self-ntersecton formula). Proposton 2.5 (the Cartsan formula). Let! (! x y) = x! y!! y = yλ N Y/X. T Z j j Y X be a Cartsan square of D-equvarant mmersons between smooth projectve schemes wth D-acton. Then there exsts γ T K D (T ), such that for y K D (y) 2.3 basc propertes!(γ T j! y) = j! (! y). Proposton 2.6 (Hdeyasu Sumhro). A normal varety over an algebracally closed feld wth an acton of a torus s covered by nvarant affne open subsets. Remark (normalty s necessary). Consder P /{0, }, or the projectve nodal curve. 4

5 Remark (Torus acton s necessary). Ba lynck-brula and Sweccka s paper On complete orbt spaces of SL(2) actons II. Proposton 2.7 (X D s smooth). Proposton 2.8 (Yur, Mann). Let Z be an smooth projectve varety(?), z Z a closed pont, j z : z Z the ncluson, then we have K(Z) = Z ker(j z ) and ker(j ) s nlpotent. If Z s an rreducble component of X D, consder the somorphsm tr Z, we actually have K D (Z) = (Z ker(j )) Z R(D) = R(D) ker(j! ) and ker(j! ) s nlpotent. the last dentty s due to the fact that K D (Z) s a natural R(D)-algebra, ker(j ) Z R(D) = ker(j! ). Proof. See Mann, Lectures on K-functors n algebrac geometry. 2.4 the localzaton theorem Theorem 2.9 (the localzaton theorem). The ncluson : X D X nduces an R(D)-lnear map! : K D (X) K D (X D ) whch s an somorphsm after localzaton w.r.t S, ts nverse s gven by Before provng ths theorem, we mght ask S K D (X D ) S K D (X) y S! (y(λ N X D /X ) ). why λ N s nvertble n S K D (X D ), n other words, why λ N S? 2.5 compute χ D (X, F ) va χ(x D,! F ) Proposton 2.0 (just lke H (Y, F ) = H (X, F )). Let : Y X be a D-equvarant closed mmerson, then χ D (Y, y) = χ D (X,! (y)). Remark (when H (Y, F ) = H (X, F )?). Proposton 2. (compute χ D (X D, )). If D acts trvally on X, then the followng dagram commutes: K D (X) tr X K(X) Z R(D) χ D (X, ) R(D) χ(x, ) d R(D) The frst proposton tells us f a coherent sheaf s the push-forward of some coherent sheaf on Y, then ther characters are the same. The second proposton tells us f you want to compute the character on the fxed locus, you can just use the ordnary, unlnearzed χ. And the localzaton theorem tells us proposton s always true, f we consder the localzaton w.r.t S, that s Proposton 2.2 (local-global). Let : X D X, x X X D (X, x) = S χ D (X D,! (X) (λ N ) ) n S R(D). 5

6 Proof. By the localzaton theorem, after takng the localzaton, we have x =! (! (x) (λ N ) ) then apply the frst proposton above, we get χ D (X, x) = S χ D (X D,! (x) (λ N ) ). Example 2.3 (χ(x, λ Ω X ) = χ(x T, λ Ω T )). Consder the cotangent sequence Snce λ-operaton s a group homomorphsm, we get Note that 0 N X D /X Ω X Ω X D 0. λ [ Ω X ] = λ N λ [Ω X D] λ [ Ω X ] =! [λ Ω X ] by defnton. In general, we don t have ths knd of dentty, but snce X s smooth, Ω X s a vector bundle, thus flat, so s ts restrcton on X D, thus! [Ω X ] contans only the frst term, that s Ω X = f Ω X f O Y O X D. λ Ω X D =! [λ Ω X ] (λ N ) Thus by the local-global proposton above, we get χ D (X, λ Ω X ) = χ D (X D, λ Ω X D) n S R(D). Specally, f D = T s an algebrac torus, then the dentty golds n R(D). So no need to worry about the denomnator, let e χ =, for all χ, then we get χ(x, λ Ω X ) = χ(x D, λ Ω X D). Remark (what s the dfference between D and an algebrac torus?). 2.6 Lefschtz fxed pont theorem Theorem 2.4 (Lefschtz fxed-pont theorem, solated, fnte). If X D s solated and fnte, for a D- equvarant coherent sheaf F on X, we have Proof. Because! [F ] = [F z ]. N z/x = T z/x. ( ) trh (X, F ) = z X D trf z ( ) tr T z X. Remark (! F = F = F z?). Ths s for sure true f F s a vector bundle, here because z s an solated pont, K D (z) = Z, so t s true n general. Example 2.5 (Weyl character formula). 6

7 Theorem 2.6 (the cohomologcal formula). ct D ( F )T odd D (X D ) χ D (X, F ) = X ct D D (λ N X D /X ). Theorem 2.7 (the Woods-Hole formula). Let σ D(k). The evaluaton map ev σ : R(D) k; χ χ(σ) gves us the ordnary trace,.e ev σ (T r(e)) = T r(σ, E), the trace of the σ on E. If σ s a dense(regualr) element(.e χ(σ) for all non-trval character χ), then ev σ can be extended to be a map We have where d z s the dfferental at z. ev σ : S R(D) k. ( ) T r(σ, H (X, F )) = T r(σ, F z ) Det( d z σ) z X D Theorem 2.8 (Specalzaton to the Wtt rng). Assume that char(k) = p 0. For an element σ D(k), the composte of the evaluaton map ev σ and the Techmuller lftng w : k W tt(k) gves a map b σ ; R(D) W tt(k), such that we have b σ (tr(e)) = BT r(σ, E), the Brauer trace for the operaton of σ on E. If we assume further that D s fnte cyclc wth generator d D(k), d s regular(dense), then b σ can be extended to be R(D) S R(D) W tt(k) then we can get a formula of P.Donovan(Thm5.3, The Lefschtz-Remann-Roch formula) 2.7 Some comparsons Example 2.9 (K(X) and P c(x)). Example 2.20 (Lefschtz fxed-pont theorem and Lefschtz hyperplane theorem). Example 2.2 (K(X) and CH(X) = A(X)). 3 Equvarant K-theory of Grassmannans 4 Equvarant K-theory of Flag varetes 5 K-theoretc proof of the Weyl Character formula 6 Comparsons between ntersecton theory and K-theory 7 Exercses n Andre s notes, Chapter 2 Example 7. (Ex2..5). ( s) k χ k V (t) = Π( st µ ) = exp( n sn χ V (t n )). Choose a bass {e,..., e k } for the weght decomposton of V, {e... e r < 2 < r r} s a bass for the weght decomposton of r V, and the second dentty comes from ln( + x) = x x2 2 + x

8 Example 7.2 (Ex2..7, Koszul complex). Construct an GL(V )-equvarant exact sequence 2 V S V V S V S V C 0. Example 7.3 (Ex2..9). Consder V = Spec(C[x,..., x n ]) as an algebrac varety on whch GL(V ) acts. Construct a GL(V )-equvarant resoluton of O 0, the struture sheaf of 0 V by vector bundles on V. 0 O (n n) C n O(n ) C n (x,...,x n) O C n O 0 0. But ths s the same as gven by the Koszul complex. We need somethng else. Example 7.4 (Ex 2..2,Ex 2..4 µ = 0 s not a weght of V ). Snce µ = 0 s not a weght of V, we have S V = ( ) rkv detv S V. k 0 k 0 Then S V = ( ) rkv detv S V s nothng but χ Sk V (t) = t µ χ S k V (t) = t µ χ detv (t) = t µ. t t = ( ) t. Wth ths descrpton, we naturally get S (V V 2 ) = S V S V 2, specally S ( V ) = V = ( ) V. Example 7.5 (Ex2..7, 2..8, the map S ). K T K T,localzed S (a b) = b a, a b S q = Π q n b n 0 q n a S a ( q) k+ = Π n 0( q n a) (n+k n ) Here, I thnk a, b represent some vrtual -dmensonal T -representatons wth non-trval character. Then S a = a, S b = b. Together wth the taylor expanson of q and ( q) k+, we get the formula above(note that S turns drect sums nto tensor products). Example 7.6 (nverse of S ). Prove that the nverse to S s gven by χ V (t) = n>0 µ(n) n lnχ S V (t n ) 8

9 where µ s the Mobus fucton µ(n) = { ( ) #prme factors, n square free 0, otherwse. () Let s prove a specal case frst, the -dmensonal representaton χ V (t) = t. We need to prove t = n µ(n) n ln( t n ) plug n t = 0, they are the same. Then we compute the dervatves, = n In the RHS, the coeffcent of t k, k s gven by µ(n) = (n )+rn=k µ(n)t n t n. n (k+) µ(n) = 0. The last equalty follows from the prme factorzaton of k + and the defnton of µ(n). Actually, ths does gve us a proof. By changes of the varable, we have t µ = n µ(n) n ln( t nµ ) thus = n χ V (t) = µ(n) n t µ = n ln(π t ) = nµ n>0 µ(n) n ln( t nµ ) µ(n) n lnχ S V (t n ). Example 7.7 (Balynck-Brula decomposton, page23). We use ths method(nstead of Morse theory), to compute the Poncaré polynomal of several Hlbert schemes of ponts. Example 7.8 (Ex 2.2.3, KG (X) K G(X)). Consder X = Spec(C[x, x 2 ]/(x x 2 )) C 2 wth the natural t 0 acton of the maxmal torus T = GL(2). Let F = O 0 be the structure sheaf of the orgn 0 X. 0 t 2 We want to compute the mnmal T -equvarant resoluton 2 d 2 R R d R 0 d F 0 of F by sheaves(not necessarly vector bundles!) of the form R = O X R where R s a fnte dmensonal T -module. To do ths let R 0 = O X, H 0 (O X ) = span{, x k, x k 2 k } 9

10 Smlarly, we defne R = O X (Ce Cf ), wt(e ) = t, wt(f ) = t 2 d : t n e t n t, t n f t n t 2. It s straghtforward to check that Now we know R k = O X (Ce k Cf k ), wt(e k ) = t 2 t k, wt(f k ) = t t k 2 d k : t n e k (t n t ) f (k ), t n f k (t 2 t n ) e (k ), ker(d k ) = span{t n 2 e k, t n f k n } = m(d (k+) ). R k F = (O X F ) R = R, χ R Tor (F, F ) := H (R F ) = R. = t 2 t k + t t k 2. The last equalty s because of Schur s lemma and the fact that dfferent R s have dfferent weghts. Ths already tells us F s not n the mage of K T (X) K T (X). We also know ( ) χ(f )2 χ Tor (F,F) = χ(o X ) = 2 t + t 2 ( t )( t = t t 2 Remark ( ( ) χ Tor (F,F) = χ(f)2 χ(o X )). In the derved category of coherent sheaves, we have F L G F G If we assume the free resolutons are T -equvarant and of the form O X F, O X G, for some T -modules F and G. We get 2 ). χ(f L G ) = ( ) χ Tor (F,G ) = ( ) χ( m+n= (O X F m ) (O X G n )) = thus n ths stuaton, we have χ(o X ) ( ) ( m+n= = χ(f L G ) = χ(o X F m )χ(o X G n )) χ(f )χ(g ), χ(o X ) what we want n the example s above s just a specal case. χ(f )χ(g ), χ(o X ) Remark (every coherent sheaf on a smooth varety s perfect). Example 7.9 (Ex 2.2.6). Generalze the last dentty above to the case X = Spec(C[x,..., x d ]/I) where I I are monomal deals. F = C[x,..., x n ]/I 0

11 Example 7.0 (Ex 2.2.0, 2.3.4, compute χ(p n, O(k)) by localzaton). Take X = P n = P(C n+ ) and GL(n + ) naturally acts on X. {D + (x )} n+ = s a T -nvarant Čech coverng of Pn. We denote O(k) by F n+=k t t t n+ n+ k 0 χ(p n, O(k)) = 0 n < k < 0 (t t 2... t n ) +... n+= k n t t tn+ k n (2) fxed ponts weght of the stalk. p = [0,..., 0, x =, 0,..., 0] wt(f p ) = t k weght of λ N p /P n, use the fact that N p/p n = T P n p, the localzaton theorem tells us that wt(n p /P n) = j t t j (λ N p /P n) = Π j ( t t j ) χ(p n, O(k)) = n+ = t k Π j ( t t j ). to convnce yourself ths s s the correct answer, check for example χ(p, O(k)) = t k t k 2 + t t 2 t2 = t k 2 t t k t 2 = t k 2 t ( ( t2 t ) k+ ) t t 2 t t 2 = t k 2 (t t 2 )( + t2 t + + ( t2 t ) k ) t t 2 = t t j 2. +j=k We note that ths only checks the k 0 cases, because we need the factorzaton of x k+, but t s not a problem at all, use the factorzaton of x k y k nstead. Ths also gves us a way to check the two expresson of χ(p n, O(k)), we leave t as an exercse. Remark (Euler sequence, tangent sequence are not T -equvarant ). We know N p/p n = O() n, but we can not use ths somorphsm to compute the character of the normal bundle, because t s not a T -equvarant somorphsm. To be more precse, as vector bundles they re the same, but the T -acton on the normal bundle n ths exercse doesn t agree wth the natural T -acton on O() n. Remark (S k C n s rreducble). We can use the computaton of the character to prove that S n C n s rreducble as a representaton of GL(n). Example 7. (Ex 2.3.5, χ(g/b, L λ )). t

12 Example 7.2 (χ(x a,b, O(k)), character of sheaves on a weghted projectve space). For a, b > 0, consder the weghted projectve lne z a 0 X a,b := C 2 \ {0}/, z C. 0 z b Then D + (x), D + (y) are two orbfold charts. Lke any C -quotent, t nherts an orbfold lne bundle O(k) whose sectons are functons φ on C 2 \ {0} such that φ(z x) = z k φ(x). These sectons are just the vector space spanned by monomals of the form Then we easly get the character formula χ(x a,b, O(k))s k = k 0 x y j, a + bj = k. ( t sa )( t 2 sb ). We can also get ths result by applyng the localzaton theorem(for orbfolds), qute smlar lke the P case, we have χ(x a,b, O(k)) = t k + t k 2 COMPUTING TORUS-EQUIVARIANT K-THEORY OF SINGULAR VARIETIES A celebraton of algebrac geometry Example 7.3 (Ex Projecton formula). f (F f E ) = f (F ) E. Example 7.4 (fractonal power of the canoncal bundle). Let X be a proper nonsngular varety wth a nontrval acton of T = C, Assume that a fractonal power K p, 0 < p < of the canoncal bundle K X exsts n Pc(X). Replacng T by a fnte cover, we can make t act on K p. Show that χ(x, K p ) = 0. What does ths say about projectve spaces? Concretely, whch are the bundles K p, 0 < p <, for X = P n and what do we know about ther cohomology? Serre dualty, K s t = O P n( s(n+) t ), f we assume (s, t) =, we then need t (n + ). Example 7.5 (χ(x, λ Ω X ) = χ(x T, λ Ω T )). Consder the cotangent sequence Snce λ-operaton s a group homomorphsm, we get Note that 0 N X D /X Ω X Ω X D 0. λ [ Ω X ] = λ N λ [Ω X D] λ [ Ω X ] =! [λ Ω X ] by defnton. In general, we don t have ths knd of dentty, but snce X s smooth, Ω X s a vector bundle, thus flat, so s ts restrcton on X D, thus! [Ω X ] contans only the frst term, that s Ω X = f Ω X f O Y O X D. 2

13 λ Ω X D =! [λ Ω X ] (λ N ) Thus by the local-global proposton above, we get χ D (X, λ Ω X ) = χ D (X D, λ Ω X D) n S R(D). Specally, f D = T s an algebrac torus, then the dentty golds n R(D). So no need to worry about the denomnator, let e χ =, for all χ, then we get χ(x, λ Ω X ) = χ(x D, λ Ω X D). Example 7.6 (Ex 2.4.2, localzaton formula for χ(x, λ Ω X )). Let X be proper and smooth wth an acton of a connected reductve group G. Wrte a localzaton formula for the torus acton of T G on ( z) p χ(x, Ω p ) p and conclude that every term n ths sum s a trval G-module. Wthout losng of generalty, we may assume X T s an rreducble subvarety(otherwse, the localzaton formula s just the summaton over all the components, whch makes no essental dfference). From the example above, we know χ(x, λ Ω X ) = χ(x T, λ Ω X T ), and ths s essentally what we need, we can wrte down the localzaton theorem ( z) p χ(x, Ω p ) = χ(x, λ z Ω X ) = tr(λ zω X X T ) tr(λ N X T /X ) p = tr(λ zn X T /X )tr(λ zω X T ) tr(λ N X T /X ). Note that the torus acton on X T s just the dentty acton, thus the formula tr(λ z N X T /X)( z)dmxt tr(λ N X T /X ) Let tr(n X T /X ) = w + + w k,where w are Laurent polynomals w.r.t t,..., t dmt, then we fnally have ( z) p χ(x, Ω p ) = χ(x, λ z Ω X ) = ( Π k z)dmxt = ( zw ) Π k = ( w. ) p No matter what knd of lmt we take w.r.t t, the formula above s always well-defned, ths tells us that t s actually of the form ( z) p χ(x, Ω p ) = a p z p, a p Z 0. p p Ths tells us exactly every term Ω p s a trval G-module. Remark. compact Kähler + Hodge theory gves trvalty of G-acton on each 8 Chapter 3 H q (X, Ω p ) H p+q (X, C). Example 8. (Ex3.3.3, 3.3.4, 3.3.5, Spn representatons of SO(V )). 3