2 Passing to cohomology Grothendieck ring Cohomological Fourier-Mukai transform Kodaira dimension under derived equivalence 15

Transcription

1 Contents 1 Fourier Mukai Transforms Grothendieck-Verdier Duality Corollaries Fourier Mukai Transforms Composition of FM transforms Orlov s theorem and applications Passing to cohomology Grothendieck ring Cohomological Fourier-Mukai transform Kodaira dimension under derived equivalence 15 4 Nefness Under derived equivalence 19 5 More About Derived Equivalence Derived Equivalence and Birationality

2 2 CONTENTS

3 Chapter 1 Fourier Mukai Transforms Yajnaseni Dutta 1.1 Grothendieck-Verdier Duality Let f : X Y be a morphism of smooth schemes over a field k (any char). Denote the relative canonical bundle by ω X/Y = ω X f ω 1 Y Then, for any F D b (X) and E D b (Y ) there exists a natural isomorphism Rf RHom(F, Lf (E ) L ω X/Y [dim X dim Y ] RHom(Rf F.E ) Since for smooth maps ω X/Y Define, is locally free the tensor product is underived. f! : D b (Y ) D b (X) E Lf (E ) ω X/Y [dim X dim Y ] Then f! Rf Corollaries 1. Taking cohomologies we get, RΓRf RHom(F, Lf (E ) ω X/Y [dim X dim Y ] RΓRHom(Rf F.E ) 3

4 CHAPTER 1. FOURIER MUKAI TRANSFORMS But, RΓ Rf = RΓ and RΓ RHom = RHom. Therefore in degree zero we get, Hom D b (X) (F, Lf (E ) ω X/Y [dim X dim Y ]) Hom D b (Y ) (Rf F.E ) 2. (Serre Duality) Grothendieck Duality applied to f : X k yields classical Serre duality: Hom D b (X) (F [i], ω X [dim X]) Hom k (Rf F [i], k) In particular, for a sheaf we have, F = F. This together with the facts, R i f F = H i (X, F) and Hom D b (X) (F, G[dim X i]) Extn i (F, G), yield Ext n i (F, ω X ) H i (X, F) 3. For F, G D b (X), the derived version of Serre duality gives Hom D b (X) (F, G ω X [dim X]) Hom D b (X) (RHom(G, F ), ω X [dim X]) Hom k (RΓRHom(G, F ), k) Hom D b (X) (G, F ) 1.2 Fourier Mukai Transforms For this section, unless otherwise mentioned the standard notation will always mean derived. For instance, we will write F for F, for L etc. Definition (Fourier-Mukai Transforms). Let P D b (X Y ). Let q : X Y X and p : X Y Y be standard projections. This Fourier- Mukai transform is a functor Φ P : D b (X) D b (Y ) defined by E Rp (q E P) The object P is called the Fourier Mukai kernel of the Fourier-Mukai transform Φ P. Remark. 1. Since q is smooth (hence flat) q is underived. 2. If P is a complex of locally free sheaves, in the formula is also underived. This will be the case in most of the applications. 3. It is a composition of exact functors and hence exact. 4

5 1.2. FOURIER MUKAI TRANSFORMS Examples: 1. (The identity functor) id : D b (X) D b (Y ). Then, id Φ O, where X X is the diagonal. 2. We will show more generally that for a morphism f : X Y, Rf Φ OΓ, where g : X Γ X Y is the graph of the morphism f. For F D b (X), Rf (F) = R(q g) (g p F) = Rq Rg (g p F) = Rq (p F Rg O X ) (projection formula) = Rq (p F O Γ ) (g is an isomorphism.) 3. Let L be a line bundle on X then the functor F F L is isommorphic to Φ i L where, i : X X X is the diagonal embedding. 4. The shift functor T : D b (X) D b (X) is given by Φ O [1]. 5. Let P be a flat coherent sheaf on X Y, then for a closed point x X, Φ P (k(x)) = Rp (q k(x) P) = P x Definition (Theorem). For any object P D b (X Y ) we define the following objects in D b (X Y ) P L = P p ω Y [dim Y ] P R = P q ω X [dim X] where, P = RHom(P, O X Y ). Then, G = Φ PL is the left adjoint and H = Φ PR is the right adjoint of the the fourier mukai transform with kernel P. Proof. G Φ P : For F D b (Y ) and E D b (X), Hom D b (X)(G(F ), E ) = Hom D b (X)(Rq (p F P L ), E ) = Hom D b (X Y )(P L p F, q E ω X Y/X [dim Y ]) (GV Duality; q is underived since q is flat) = Hom D b (X Y )(P L p F, q E p ω Y [dim Y ]) = Hom D b (X Y )(P p F, q E ) = Hom D b (X Y )(p F, P q E ) = Hom D b (X)(F, Rp (P q E )) (Lp Rp ) = Hom D b (X)(F, Φ P ) (p p ) 5

6 CHAPTER 1. FOURIER MUKAI TRANSFORMS Composition of FM transforms Proposition The composition D b (X) Φ P D b (Y ) Φ Q D b (Z) is isomorphic to Φ R : D b (X) D b (Z), where R = Rπ XZ (π XY P π Y Z Q). 1.3 Orlov s theorem and applications Theorem (Orlov). Let X and Y be two smooth projective varieties and let F : D B (X) D b (Y ) be a fully faithful exact functor. If F admits right (and hence left) adjoint functors, then there exists an object P D b (X Y ) unique up to isomorphism such that F Φ P. Proof. Future lecture(maybe) Corollaries: Corollary Let F : D b (X) D b (Y ) be an equivalence between the derived categories of two smooth projective varieties. Then F is isomorphic to a FM transform Φ P associated to a certain object P D b (X Y ) unique up to isomorphism. Moreover, dim X = dim Y and P q ω Y P p ω X. Proof. Note that, equivalence of categories ensures existence of adjoints, namely the quasi-inverse F. Therefore we can apply Orlov s theorem to find a P D b (X Y ) unique up to isomorphism such that the FM transform with kernel P is isomorphic to F. Again by applying the uniqueness of Orlov s theorem to the quasi-inverse F we see that, P L P R. In other words, P P p ω X q ω Y [dim X dim Y ]. Since P is a bounded complex that is not quasi-isomorphic to zero, dim X = dim Y. Corollary Suppose Φ : D b (X) D b (Y ) is an equivalence such that, for any close point x X, there exists a closed point f(x) Y such that Φ(k(x)) k(f(x)). Then, f : X Y defines an isomorphism and Φ simeq(m ) f for some line bundle M Pic(Y ). 6

7 1.3. ORLOV S THEOREM AND APPLICATIONS Definition (Spanning Class). A collection Ω os objects in a triangulated category D is a spanning class of D f for all B D the following condition hold: If Hom(B, A[i]) = 0 for all A Ω and i Z then, B 0 Proof. By Orlov s theorem, Φ Φ P for some object P D b (X Y ). Note that, Φ(k(x)) = q (k(x)) L P k(f(x)). Therefore, for any closed point x X the embedding i : x Y X Y, Li P is also a sheaf. Then the lemma below implies that, P is a coherent sheaf flat over X. Therefore, RΦ(k(x)) = P {x} Y k(f(x)). Hence, SuppP is precisely the graph of f and thus Γ f has a reduced induced scheme structure. Γ f is then a variety isomorphic to X by first projection and f is a composition of this isomorphism with projection to Y. Therefore, f : X Y defines a morphism. Now, we want to show that k(x) spans the category D b (X). To do so, we need to show for F D b (X), Hom(F, k(x)[ i]) 0. We use the spectral sequence, E p,q 2 = Hom(H q F, k(x)[p]) = Hom(F, k(x)[p + q]). Now, let m be the maximal integer such that H m 0. Then all diffrentials from Er 0, m are zero. Moreover, Hom(H q, k(x)[p]) = Ext p (H q, k(x)) = 0 if p < 0. Therefore, all differentials in page r mapping to Er 0, m are zero. Hence, = E 0, m 2 = Hom(H m, k(x) 0 for x Supp(H m ). E 0, m Moreover, since Φ is a equivalence, it is easy to check that, k(f(x)) spans D b (Y ). Therefore, for y Y there is an integer m and x X such that, Hom(k(f(x)), k(y)[m]) 0. This implies that, y = f(x). For injectivity, pick x 1 x 2. then Φ(k(x 1 )) Φ(k(x 2 )). Therefore, f(x 1 ) f(x 2 ). We can use similar argument on the quasi inverse to Φ to show that f has an honest inverse. Now, P SuppP has fibre of dimension 1, therefore, P SuppP is line bundle. Since, Supp(P) Y via the projection p, it gives rise to a line bundle M = p P on Y. From the formula for composition, it is possible to calculate and see that Φ P ( M) f. Lemma Consider a morphism S X. Suppose P D b (S) and assume that for all closed points x X the derived pull back Li xp D b (S x ), for i x : S x S, is a complex concentrated in degree 0, i.e. a sheaf. Then P is isomorphic to a coherent sheaf which is flat over X. 7

8 CHAPTER 1. FOURIER MUKAI TRANSFORMS Proof. For a proof, see [1, pg.82 Lemma 3.31] Corollary (Gabriel). If Φ : Coh(X) Coh(Y ) is an equivalence of categories then a morphism f such that f : X Y and Φ (M ) f, for a line bundle M on Y. Proof. In order to apply the previous corollary, we need to check that, Φ(k(x)) k(f(x)). 8

9 Chapter 2 Passing to cohomology Sebastián Olano 2.1 Grothendieck ring Let K(X) be the Grothendieck group of X. Recall that the elements are coherent sheaves on X with the following equivalence relation: if 0 F F F 0 is an exact sequence, then [F] = [F ] + [F ]. On a smooth projective variety any coherent sheaf admits a locally free resolution, hence any element in the Grothendieck group may be written asa linear combination of locally free sheaves. We define a map [ ] : D b (X) K(X) by [F ] = ( 1) i [F i ]. We also define a ring structure on K(X) by [E 1 ] [E 2 ] = [E 1 E 2 ]. Remark. [F [k]] = ( 1) k [F ]. [F 1 F 2] = [F 1] + [F 2]. [F ] = ( 1) i [H i (F )] K(X). [F 1 F 2] = [F 1] [F 2]. So [ ] is a ring map. We can define the pullback of a morphism f : X Y at the Grothendieck ring in the usual way and it will be a ring homomorphism. Moreover, we will have: 9

10 CHAPTER 2. PASSING TO COHOMOLOGY D b (Y ) [ ] K(Y ) f f D b (X) [ ] K(X) We also want to define a map of Grothendieck rings that commute with the pushforward. Let f! : K(X) K(Y ) be defined by f! [F] = ( 1) i [R i f F]. Using the fact that for a spectral sequence ( 1) p+q [Er p,q ] = ( 1) p+q [E p,q is satisfied and using it for the spectral sequence E p,q 2 = R p f H q E R p+q f E we have that: r+1] D b (X) [ ] K(X) f D b (Y ) [ ] f! K(Y ) Definition (K-theoretic Fourier-Mukai transform). Let e K(X Y ). The K-theoretic Fourier-Mukai transform is the map Φ K e : K(X) K(Y ) defined by Φ K e (g) = p! (e q (g)). Due to the previous remarks we have that: D b (X) Φ P D b (Y ) [ ] [ ] K(X) Φ K [P] K(Y ) 2.2 Cohomological Fourier-Mukai transform We will assume from now that the ground field is C. Consider the ring H (X, Q). The product of two classes α, β H (X, Q) will be denoted by α.β or simply αβ. Let f : X Y be a morphism. The pullback is defined for cohomology rings in the usual way. We assume X and Y are projective varieties and hence Poincaré duality holds for them. The composition of the dual map of the pullback with the isomorphisms 10

11 2.2. COHOMOLOGICAL FOURIER-MUKAI TRANSFORM of Poincaré duality give us a map f : H (X, Q) H (Y, Q) and moreover we know that the map for degree k elements satisfy f : H k (X, Q) H k+2 dim(y ) 2 dim(x) (Y, Q). By definition, this map satisfies the Projection Formula, that is, f (f α.β) = α.f β. Definition (Cohomological Fourier-Mukai). Let α H (X Y, Q). The cohomological Fourier-Mukai transform is the map Φ H α : H (X, Q) H (Y, Q) defined by Φ H α (β) = p (β.q (α)). We will now define a map ch : K(X) H (X, Q) called the Chern character. Let A i (X) be the cycles of codimension i. First we define an element c i (E) A i (X) for a locally free sheaf E. Recall that we also have a map A i (X) H 2i (X, Q). Notice that taking coefficients in C and using the Hodge decomposition, this map will land in H i,i (X). For the current purposes it is enough to give some conditions for c t (E) = c 0 (E) + c 1 (E)t + c 2 (E)t to satisfy, as it will define the elements in a unique way. The conditions are: 1. If E = O X (D) for a divisor D, then c t (E) = 1 + Dt. 2. For f : Y X, c i (f E) = f c i (E) for all i. 3. For an exact sequence 0 E E E 0, c t (E) = c t (E ).c t (E ). It satisfies the following condition: suppose we have a filtration E = E 0 E 1... E r = 0 such that E i /E i+1 = Li an invertible sheaf. Then we have that c t (E) = c t (L i ) and for this we can use condition 1. Suppose now that c t (E) = (1 + a i t). We define ch(e) = exp (a i ). Notice that for L Pic(X), we have ch(l) = c 1 (L) i. We also define the Todd i! class as td(e) = a i. We have that for an exact sequence 0 E 1 e a i E E 0, td(e) = td(e ).td(e ). Definition Let X be a smooth variety. The Todd class of X is defined as td(x) = td(t X ). Theorem (Grothendieck-Riemann-Roch formula). Let f : X Y be a projective morphism of smooth projective varieties. Then for any e K(X) ch(f! (e)).td(y ) = f (ch(e).td(x)) 11

12 CHAPTER 2. PASSING TO COHOMOLOGY For the case Y =Spec(k) we have that the pushforward may only be the nonzero map for H 2n (X, Q) where n = dim(x). We denote by the pushforward in this case. Notice that the map tells us the element in that degree. X Also, notice that f! [F] = ( 1) i [R i f F] = ( 1) i [H i (X, F)] = χ(f). Taking this case we recover the folowing theorem: Theorem (Hirzebruch-Riemann-Roch). For any e K(X) we have χ(e) = X (ch(e).td(x)). Definition The Mukai vector of e K(X) is v(e) = ch(e). td(x). For E D b (X) we define v(e ) = v([e ]) = ch(e ). td(x). Notice that it makes sense to write td(x) as td(x) = then we can construct an element such that its square is td(x). The following is a Corollary of Corollary Let e K(X Y ). Then the following diagram commute: K(X) Φ K e K(Y ) v H (X, Q) Φ H v(e) H (Y, Q) v Proof. We check that the following commute: K(X) q K(X Y ) e K(X Y ) p! K(Y ) v H (X, Q) v td(y ) 1 v td(x) q H (X Y, Q).v(e) H (X Y, Q) p H (Y, Q) v Given P D b (X Y ) we will denote by Φ H P the induced cohomological Fourier-Mukai transform Φ H v(p). As characteristic classes ara in even degree as they come from cycles, we have that Φ H P respects the parity. Indeed, ΦH P is an intersection with an even element followed by a pushforward which respects parity, so we have: Φ H P (Heven (X)) H even (Y ) and Φ H P (Hodd (X)) H odd (Y ).// Using the same notation as for Fourier-Mukai we have the following lemma. 12

13 2.2. COHOMOLOGICAL FOURIER-MUKAI TRANSFORM Lemma Let Φ P : D b (X) D b (Y ) and Φ Q : D b (Y ) D b (Z) be two Fourier-Mukai transforms and let Φ R : D b (X) D b (Z) be the composition. Then Φ H R = ΦH Q ΦH P Proposition Suppose that Φ P : D b (X) D b (Y ) is an equivalence for some P D b (X Y ). Then the induced cohomological Fourier-Mukai transform Φ H P : H (X, Q) H (Y, Q) is an isomorphism of rational vector spaces. Proof. As Φ P is an equivalence then Φ PR Φ P = id = ΦO and Φ P Φ PR = id = ΦO. Due to the previous lemma, we can conclude then that Φ H P R Φ H P = Φ H O and Φ H P ΦH P R = Φ H O. So it is enough to show that Φ H O = id. Let i : X X X. Using Grothendieck-Riemann-Roch we have that ch(o ).td(x X) = ch(i! O X ).td(x X) = i (ch(o X ).td(x)) = i td(x). The last equality is because ch(o X ) = 1. So we get that v(o ) = ch(o ). td(x X) = i (td(x)). td(x X) 1 = = i (td(x).i td(x X) 1 ) = i (td(x).td(x) 1 ) = i (1). We used that i ( td(x X)) = td(x). Finally, we get that Φ H O (β) = p (q (β).v(o )) = p (q (β).i (1)) = = p i (i q (β)) = β. As we are working with X smooth projective variety over C, we have a Hodge structure H k (X, C) = H p,q (X) such that H p,q (X) = H q,p (X) p+q=k and H p,q (X) = H q (X, Ω p X ). As explained before all characteristic classes are of type (p, p), so v() : K(X) H p,p (X) H 2p (X, Q). We have seen that the cohomological Fourier-Mukai does not respect the grading, only the parity. But we can improve that using the Hodge decomposition. Proposition Suppose Φ P : D b (X) D b (Y ) is an equivalence. Then the induced Φ H P yields isomorphisms H p,q (X) = H p,q (Y ) i = dim(x),..., dim(x). p q=i p q=i Proof. Φ H P defines an isomorphism between H (x, Q) and H (Y, Q) as we saw before. So we need to see that its C-linear extension satisfies Φ H P (H p,q (X)) 13

14 CHAPTER 2. PASSING TO COHOMOLOGY p q=r s H r,s (Y ). Consider the Künneth decomposition of v(p) = α p,q β r,s with α p,q H p,q (X) and β r,s H r,s (Y ). We know the element is a sum of element of type (t, t) we only consider the part of the sum such that p + r = q + s. Let α H p,q (X). Then Φ H P (α) = p (q (α). α p,q β r,s ) = = (,q X (α.αp ))β r,s H r,s. The last equality is by definition of pushforward and how the pushforward behaves in the Künneth decomposition. Then, it is needed that (p + p, q + q ) = (dim(x), dim(x)). The result follows. Corollary Let E, E be two elliptic curves. Then D b (E) = D b (E ) if and only if E = E. Proof. Let Φ P be the equivalence. As Φ H P respects parity, it induces and isomorphism H 1 (E) = H 1 (E ) and H 0 (E) H 2 (E) = H 0 (E ) H 2 (E ). By the previous proposition we know it also induces an isomorphism on H 0,1 and H 1,0. We also know that E = H 1,0 (E) /H 1 (E, Z) = H 0,1 (E)/H 1 (E, Z). So we need to show that Φ H P (H1 (E, Z)) H 1 (E, Z). As they are elliptic curves we have that td(e E ) = 1 and ch(p) = r + c 1 (P) (c2 1 2c 2 )(P). But the last term does not contribute to H 1. The result follows. 14

15 Chapter 3 Kodaira dimension under derived equivalence Lei Wu Let X be a smooth projective variety over C and let Ω X bundle. The canonical ring of X is defined to be: be its canonical R(X) = i 0 H 0 (X, ω i X), and its kodaira dimension κ(x) is defined to be the tanscendental degree of R(X). The multiplicative structure of R(X) is induced from the tensor product. Proposition (Orlov). Suppose X, Y are smooth projective varieties with D b (X) D ( Y ). Then In particular,κ(x) = κ(y ). R(X) R(Y ). Proof. Assume the equivalence is giving by FM transformation, Φ P : D b (X) D b (Y ), with quasi-inverse Φ Q : D b (Y ) D b (X), 15

16 CHAPTER 3. KODAIRA DIMENSION UNDER DERIVED EQUIVALENCE for some P, Q D b (X Y ). By uniqueness of right and left adjoint functor, we get Φ PL Φ Q Φ PR. Hence P V q ω X [n] P V p ω Y [n]. Here n = dimx = dimy, because of the equivalence. First, we want to prove Φ Q : D b (X) D b (Y ) is also an equivalence. Consider the composition As a composition, we know Φ Q Φ P = Φ R : D b (X) Φ P D b (Y ) Φ P D b (X). R = π 13 (π 12P π 23Q) O, because Φ R is the indentity. Denote τ 12 to be the automorphism of X X which interchanging the two factors, i.e. τ 12 (x 1, x 2 ) = (x 2, x 1 ), and τ 13 the automorphism of X Y X interchanging the two X s. Then Ø = τ 12Ø ) τ 12π 13 (π 12P π 23Q) π 13 τ 13(π 12P π 23Q) π 13 (π 23P π 12Q). But the last one is the kernal of Φ P Φ Q. Hence Swiping P and Q, we get Φ P Φ Q id X. Φ Q Φ P id Y. Hence Φ Q : D b (X) D b (Y ) is an equivalence with quasi-inverse Φ P. Therefore, Φ Q P : D b (X X) D b (Y ) is an equivalence. Denote S := Φ Q P (ι ωx k ), where ι is the diagonal embedding. equivalence Φ S can be computed as the composition Then the 16 D b (Y ) Φ Q D b X Φ ι ω k X D b (X) Φ P D b (Y )

17 Since Φ ι ω X k Sk X [ kn] (S X is the Serre functor), and since the Serre functor commutes with any equivalence, we get Φ S SY k [ kn] Φ. ι ω Y k Hence, thanks to the uniqueness of the FM transformation S = Φ Q P (ι ωx) k ι ωx. k Since Φ Q P is an equivalence, we obtain Hom X (ω k X, ωl X ) Φ Q P = Hom Y (ω k Y, ωl Y ). ι ι Hom X X (ι ω k X, ι ω l X ) = Hom Y Y (ι ω k Y, ι ω l Y ) Two vertical arrows are isomorphism because ι is fully faithful. k = 0, l 0, we have H 0 (X, ωx) l H 0 (Y, ωy l ). Picking Since H 0 (X, ωx l ) Hom X(O X, ωx l ), multiplication of R(X) is just composition of morphisms, which definitely compatible with any functors. So R(X) R(Y ). Remark. Construct the bigraded ring containing R(X), HH(X) := i,l HA i,l (X), where HA i,l (X) := Ext i X X (ι O X, ι (ωx l )). Define Hochschild cohomology as HH (X) := HA i,0 (X), i and Hochschild homology as HH (X) := i HA i,1 (X). Then exactly the same as the above proof, we get HH(X) HH(Y ) 17

18 CHAPTER 3. KODAIRA DIMENSION UNDER DERIVED EQUIVALENCE Corollary If furthermore ω X and ω Y are ample (or anti-ample), then X Y. In fact, only ampleness (or anti-ampleness) of ω X is needed, because ampleness (or anti-ampleness) of ω X will imply ampleness (or anti-ampleness) of ω Y (See Proposition 4.11 in [1]). 18

19 Chapter 4 Nefness Under derived equivalence Lei Wu Let Φ P : D b (X) D b (Y ) be a F-M transformation with kernal P D b (X Y ). Then, suppp = suppp V = suppp R = suppp L. From now on, we always assume Φ P is also an equivalence. Therefore, and in particular, P q (ω X ) P p (ω Y ), H i (P) q (ω X ) H i (P) p (ω Y ). Lemma The morphism supp(p) X is surjective. Proof. Assume x / q(supp(p)). Hence supp(q k(x)) and supp(p) are disjoint. The spectral sequence E r,s 2 = T or r (H s (P), q k(x)) = T or r s (P, q k(x)). Since T or is local, T or i (H s (P), q k(x)) is 0 for all i. Hence P L q k(x) = 0. So Φ P (k(x)) = 0. But this can not happen because Φ P : D b (X) D b (Y ) is an equivalence. 19

20 CHAPTER 4. NEFNESS UNDER DERIVED EQUIVALENCE Corollary There exists an integer i Z and an irreducible component Z of supp(h i (P)) that projects onto X. Before we prove our main theorem in this section, let s look at some technical lemmas. Lemma Let Z be a normal variety over k and let F be a coherent sheaf on Z generically of rank r. If L 1, L 2 P ic(z), such that then L r 1 L r 2. F L 1 F L 2, Proof. By dividing out its torsion part we can assume F is torsion-free. Since Z is normal, F is locally free of rank r on some open U whose complement is of codimension at least 2. Hence By taking determinant, F U L 1 U = F U L 2 U. (L 1 U ) r (L 2 U ) r. Since Z is normal again, this isomorphism will extends to an isomorphism globally, i.e. (L 1 ) r (L 2 ) r. Corollary Let Z supp(p) be a closed irreducible subvariety with normalization µ : Z Z. Then there exists an integer r 0, such that where π X = q µ, and π Y = p µ. π Xω r X π Y ω r Y, Proof. Assume Z supp(h i (P)). Then µ (H i (P) Z ) is coherent of generical rank > 0. By the above lemma, we are done. Immediately we get, Corollary q ω X supp(p) is numerically equivalent to p ω Y supp(p) By lemma 4.3, we can also get 20

21 Proposition (Kawamata). Let X and Y be smooth complex projective varieties with equivalent derived categories D b (X) and D b (Y ). Then the (anti)-canonical bundle of X is nef iff the (anti)-canonical bundle of Y is nef. Proof. We only need the following fact: If f : X Y is surjective morphism of projective scheme, then for a line bundle L P ic(y ) L is nef iff f L is nef. Definition Let L be a line bundle on X. ν(x, L) := max{m [ψ L] m.[w ] 0, for some ψ : W Xwith dimw = m} The following lemma is very basic in algebraic geometry. Lemma Let π : Z X be a morphism of projective schemes, and let L P ic(x). (i) Then ν(x, L) ν(z, π L) (ii) If π is surjective, then ν(x, L) = ν(z, π L) With the help of the above lemma, similarly to Proposition 4.6 we get Proposition (kawamata). Let X and Y be smooth projective varieties with D b (X) D b (Y ). Then ν(x, ω X ) = ν(y, ω Y ). 21

22 CHAPTER 4. NEFNESS UNDER DERIVED EQUIVALENCE 22

23 Chapter 5 More About Derived Equivalence Yajnaseni Dutta Throughout this chapter we will assume that k is algebraically closed and κ(x) will denote the Kodaira dimension or Itaka dimension of ω X. 5.1 Derived Equivalence and Birationality Theorem (Kawamata). Let X and Y be two smooth projective varieties over k with equivalent derived categories. If in addition, κ(x) = dim(x) = n or κ(x, ωx ) = n, we get that X and Y are birational and there exists a normal variety Z such that there is a birational correspondence. Moreover, p X ω X = p Y ω Y. Z p X p Y X Y Proof. We will only treat the case when κ(x) = dim(x). By Orlov s theorem in Chapter 1, we know, the derived equivalence is given by Φ P : D b (X) D b (Y ) for some P D b (X Y ). 23

24 CHAPTER 5. MORE ABOUT DERIVED EQUIVALENCE Let H X be a smooth hypersurface section in X. following exact sequence: Then we have the 0 ω l X( H) ω l X ω l X H 0 This induces a left exact sequece of global sections: 0 H 0 (ω l X( H)) H 0 (ω l X) H 0 (ω l X H ) Since, κ(x) = dim(x), H 0 (ωx l ) grows as ln and H 0 (ωx l H) grows as l n 1 for large enough n, we get that, there is a section of H 0 (ωx l ( H)) defining an effective divisor D such that, ωx l = O(D) O(H). Due to Corollary , there is a component in the supp(p) surjecting onto X. Let Z be the normalisation of that component. Then, by corollary , we have that, p X ωr X = p Y ωr Y for some integer r. Birationality: We claim that p Y : Z \ p 1 X (D) Y has finite fibres. Suppose on the contrary there is a curve C, not in p 1 X (D), contracted by p Y. Then deg p Y ω Y C = 0. Since, p X ωr X = p Y ωr Y, we get, deg p X ω X C = 0. But, deg p X ω X C > 1 l deg O(H) C > 0. The first inequality is a consequence of D being effective and therefore intersecting p X (C) in finitely many points. The second inequality is by the numerical criterion of ampleness. Hence, Z Y is generically finite and hence dim(z) dim(y ). On the other hand, since Z X is a surjection, dim(z) dim(x). But, since X and Y have equivalent derived categories, dim(x) = dim(y ) by corollary Hence X p X Z p Y Y maps generically finitely onto X and Y In the following lemma, we will show that fibres of the surjection Supp(P) X is connected. Assuming the lemma, let W be an irreducible component of Supp(P), other than Z, surjecting onto X. Then a fibre of W over x X must contain the points in Z lying above X, by connectivity. Hence, Z is the only component dominating X and moreover the general fibre consists of one point. Therefore, Z is birational to X. p X ω X = p Y ω Y : 24

25 Bibliography [1] Huybrechts, Daniel, Fourier Mukai Transforms in Algebraic Geometry, 25