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Relative Fourier-Mukai transforms for Weierstrass fibrations, Abelian schemes and Fano fibrations A. C. López Martín D. Sánchez Gómez C. Tejero Prieto Departamento de Matemáticas and Instituto Universitario

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Relative Fourier-Mukai transforms for Weierstrass fibrations, Abelian schemes and Fano fibrations A. C. López Martín D. Sánchez Gómez C. Tejero Prieto Departamento de Matemáticas and Instituto Universitario de Física Fundamental y Matemáticas Universidad de Salamanca The locally free geometry seminars Salamanca, 17/05/2012 C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Outline 1 Relative integral functors Definition Properties Base change Moduli spaces 2 Fourier-Mukai groups Overview of the absolute case Relative case Goals Key results Weierstrass Fibrations] C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Relative Integral functors Definition Proper morphisms over T (arbitrary scheme: finite type, separated) X p q T Y Kernel K D (X T Y ) (bounded above cplxs, qc cohomology) Projections X T Y π 1 X Y Relative integral functor with kernel K π 2 Φ K X Y : D (X) D (Y ) E Rπ 2 (Lπ 1 (E ) L K ) Relative Fourier Mukai transforms: relative integral functors that are equivalences. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Relative Integral functors Properties A) Via the immersion ι: X T Y X Y, the relative integral functor Φ K X Y coincides with the absolute integral functor with kernel ι K. B) Smooth projective world: Serre s theorem = Perf (X) Dc b (X). No longer true for singular schemes: Perf (X) Dc b (X) (x X singular, O x is not perfect). If X p is locally projective and K Dc b (X T Y ), a result of Spaltenstein shows that the integral functor extends to the whole derived category Φ K X Y : D(X) D(Y ). If we want a better behaved Φ K X Y we have to impose a technical condition to the kernel. Let f : Z T be a morphism of schemes. An object E in Dc b (Z ) is said to be of finite homological dimension over T if E L Lf G is bounded for any G in D b c (T ). C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Relative Integral functors Properties Examples 1 T = Spec(k), any E D b c (Z ) has fhd T. 2 f = Id T, then E has fhd T E is a perfect complex. 3 f : Z T projective, O(1) very ample, then E has fhd T Rf (E (n)) is a perfect complex n Z. Now one can prove that the integral functor Φ K X Y : D(X) D(Y ): a) maps Dc b (X) to Dc b (Y ) K has fhd X. If in addition K has fhd Y then it maps Perf (X) to Perf (Y ). b) has a right adjoint integral functor satisfying a) K has fhd X,Y Moreover, if X p T, Y q T are projective and Φ K X Y is a relative Fourier-Mukai transform, then K has fhd X,Y. Therefore, in the case of projective morphisms a Fourier-Mukai transform sends perfect complexes to perfect complexes. FM T (D b c (X)) is the relative Fourier-Mukai group of X T, i.e the subgroup of of the group of autoequivalences of D b c (X) given by relative Fourier-Mukai transforms. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Relative Integral functors Base change An arbitrary base change B X f p T q Y gives a commutative diagram (X T Y ) B f X T Y X T Y with (X T Y ) B X B B Y B and X B = B T X, Y B = B T Y are the base changes of X and Y, respectively. The kernel K B := Lf X T Y K D (X B B Y B ) defines an integral functor Φ B := Φ K B X B Y B : D (X B ) D (Y B ) called the base change of the integral functor Φ := Φ K X Y. B f T C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Relative Integral functors Base change If either X p T or B f T is flat and K has fhd X then K B has fhd XB If p and q are flat, then for every E D b c (X) one has the compatibility formula Φ B (Lf X E ) Lf Y Φ(E ), where f X and f Y are the base change morphisms: X B f X X Y B f Y Y B f T B f T C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Relative Integral functors Moduli spaces (Mukai, Bridgeland, Bartocci-Bruzzo-Hernádez Ruipérez,... ) Let Φ = Φ K X Y : D(X) D(Y ) be an integral functor such that 1 Φ transforms sheaves into sheaves (up to translation): E Coh(X) = Φ(E) Ê[ i] with Ê Coh(Y ). 2 Φ preserves (semi)stability: E Coh(X) (semi)stable with Hilbert polynomial P = Ê Coh(Y ) (semi)stable with Hilbert polynomial ˆP then Φ induces a morphism of relative moduli spaces over T φ M X/T (P) M Y /T (ˆP) T In general φ is neither injective nor surjective. However if Φ is an equivalence and Φ 1 satisfies 1) and 2) then φ is an isomorphism. This provides good motivation to study relative Fourier-Mukai transforms. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Overview of the absolute case (T = Spec(k)) X projective variety over Spec(k). The group of autoequivalences Aut(D b c (X)) at least contains the subgroup Aut 0 (D b c (X)) of trivial autoequivalences, generated by Aut(X), Pic(X), Z where an automorphism ϕ Aut(X) acts by direct image, a line bundle L Pic(X) acts by twisting L ( ) and an integer n Z acts by shifting complexes [n]. Representability: If X is smooth, Orlov proved that Aut(D b c (X)) = FM(D b c (X)). Ballard has extended this result to any projective scheme and Orlov gave a generalization for quasi projective schemes. Fano (ω X ample) or anti-fano ( ω X ample) then Aut(D b c (X)) = Aut 0 (D b c (X)). In the smooth case this is due to Bondal-Orlov. In the Gorenstein case this has been proved by Ballard and by Carlos and Fernando Sancho. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Overview of the absolute case (T = Spec(k)) Aut((D b c (X)) is interesting for X Calabi-Yau, i.e ω X = O X. 1 For elliptic curves, the structure of Aut(D b c (X)) was completely determined by Hille-van den Bergh 2 For integral curves of arithmetic genus 1 the description of Aut(D b c (X)) was given by Burban and Kreusler. 3 For abelian varieties the first results were due to Polishchuck and the complete dtermination of Aut(D b c (X)) was obtained by Orlov. 4 For K3 surfaces there is some work done by Huybrechts, Macrì and Stellari. 5 For toric surfaces the group Aut(D b c (X)) was determined by Ploog and Broomhead C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Relative case (T Spec(k)) Given X p T we say that an autoequivalence F Aut(D b c (X)) is T -linear if for any E D b c (X) and any N D b c (T ), one has F(E L Lp N ) F(E ) L Lq N. We denote by Aut T (D b c (X)) the group gnerated by the T -linear autoequivalences. By the Projection Formula any relative Fourier-Mukai transform is a T -linear autoequivalence, i.e FM T (D b c (X)) Aut T (D b c (X)). In general it is a hard problem to decide wether FM T (D b c (X)) and Aut T (D b c (X)) are equal or not. Concentrate on the first step: determination of FM T (D b c (X)). C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Goals 1 Description of FM T (D b c (X)) when X p T is a Weierstrass fibration, an abelian scheme or a Fano or anti-fano fibration. 2 To show that even if we know the structure of the Fourier-Mukai groups FM T (D b c (X t )) of the fibers of X p T it is not a trivial task to determine FM T (D b c (X)), but in any case try to give a general machinery for acomplishing this task. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Key Results A) Base change formula: Lets consider a Cartesian diagram B T X f B g g X T f For any E there is a natural morphism Lg Rf E R f L g E. If E has quasi coherent cohomology and either f or g is flat then it is an isomorphism. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Key Results Given a point t T we have a cartesian diagram: X t Y t {t} α t X T Y T flat Thus we can apply the previous result. Let K Dc b (X T Y ) be a kernel such that it has fhd X,Y and let Φ = Φ K X Y : Db c (X) Dc b (Y ) the corresponding integral functor. We define K t = Lαt K and the associated integral functor Φ t = Φ K t X t Y t : D b c (X t ) D b c (Y t ). We have the natural inmmersions i t : X t X, j t : Y t Y and we have the following compatibility relations Lj t Φ(E ) Φ t (Li t E ), E D b c (X) Φ(i t F ) j t Φ t (F ), F D b c (X t ) C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Key Results B) Theorem (Hernández Ruipérez, F. Sancho, A. C. López Martín) X p T locally projective, K has fhd X,Y. Φ is an equivalence t T Φ t is an equivalence. Corollary If Φ is an equivalence Φ B is an equivalence for any base change B f T. Thus from Aut(D b c (X t ) and K a sheaf up to translation we can try to give a description of FM T (D b c (X)). C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Key Results C) Proposition Let K D b c (X T Y ) be a kernel and assume that X is connected. If t T K t K t [n t ] with K t a sheaf on X t Y t flat over X t K K[n] with K a sheaf on X T Y flat over X and n Z. 1 Weierstrass fibrations: K t Poincaré sheaf 2 Abelian schemes: K t semihomogeneous sheaf 3 Fano fibration: K t structural sheaf All of them are flat C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Key Results D) Proposition Let K D b c (X T Y ) be a kernel and assume that X is connected. If t T K t K t [n t ] with K t a sheaf on X t Y t flat over X t K K[n] with K a sheaf on X T Y flat over X and n Z. 1 Weierstrass fibrations: K t Poincaré sheaf 2 Abelian schemes: K t semihomogeneous sheaf 3 Fano fibration: K t structural sheaf All of them are flat C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Key Results D) Proposition Let Φ = Φ K X Y : Db c (X) D b c (Y ) be an integral functor such that K D b c (X T Y ) has fhd X. If for every x X we have Φ(O x ) = O y for some closed point y Y X f Y and L Pic(X) such that Φ( ) = Rf (( ) L). T C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Weierstrass Fibrations X p T Weierstrass fibration proper flat morphism whose fibres are Gorenstein integral curves of arithmetic genus 1. We assume that there is a section T σ X whose image does not contain singular points. Therefore C = X t is an elliptic curve or a rational curve with a node or a cusp. K (C) is the Grothendieck group: free abelian group generated by objects of D b c (C) modulo expressions generated by triangles. We have the Euler form E C : K (C) K (C) Z given by E C ([E ], [F ]) = χ(e, F ) whenever E or F is perfect (K (C) is generated by perfect objects). K (C): = K (C)/rad E C (rk,deg) Z 2 an the induced bilinear form ẼC is symplectic. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Weierstrass Fibrations Any integral functor Φ: D b c (X) D b c (X) induces a group morphism φ: K (X) K (X) such that the following diagram commutes D b c (X) [ ] K (X) Φ φ D b c (X) [ ] K (X). Moreover if Φ is an equivalence then φ induces an automorphism of K (C) that preserves the symplectic form ẼC. Therefore we get a group morphism Aut Dc b (C) ch SL 2 (Z) such that if F is an object in D b c (C), then ( rk(φ(f ) ( ) rk(f ) ) = ch(φ) deg(φ(f ) deg(f ) To determine ch(φ) it is enough to compute (rk, deg) of Φ(O C ) and Φ(O x ) with x C a smooth point. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Weierstrass Fibrations Aut Dc b (C) ch SL 2 (Z) is surjective: The ideal of the diagonal I C and the line bundle O C (x 0 ), wherex o is a non singular point, ( define ) two 1 1 autoequivalences Φ 1, Φ 2, respectively and ch(φ 1 ) =, 0 1 ( ) 1 0 ch(φ 2 ) = and they are generators of SL (Z). Theorem (Hille-van den Bergh, Burban-Kreussler) Let C be an integral projective curve with arithmetic genus one. 1 The following exact sequence holds 1 Ãut0 (D b c (C)) Aut D b c (C) ch SL 2 (Z) 1, where Ãut0 (D b c (C)) = Aut(C) (2Z Pic 0 (C)). 2 For any Φ K Aut(D b c (C)) K = K[n], K a sheaf on C C flat over both factors. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Weierstrass Fibrations Results B) + C): If Φ K FM T (D b c (X)) K K[n], K a sheaf on X T X flat over both factors. ech Result B): FMT (Dc b (X)) SL 2 (Z) ρ t ch Xt Aut Dc b (X t ) Claim: ch is well defined and surjective. 1 ch(φ) := ch Xt (Φ t ) does not depend on t. To prove that it is enough to see that (rk, deg) of Φ t (O Xt ) and Φ t (O x ) (x smooth) do not depend on t. We use the following Lemma Let X T be a flat morphism and E be a sheaf on X flat over T. The restriction E t to the fibre X t is WIT i -Φ t for every t T if and only if E is WIT i -Φ and Ê is flat over T. Moreover, in this case (Ê) t Êt for every t T. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Weierstrass Fibrations Applying the Lemma to the structure sheaf of the diagonal E = O, taking into account that (O ) x = K[n] = O x = is WIT 0 Φ x x X we see that O is WIT 0 Φ X and Φ X (O ) = Ô is flat over X and K x [n] = (Ô ) x (O ) x = Φ t (O x ) therefore (rk, deg)(φ t (O x )) is independent on t. 2 ch is surjective: consider the relative Fourier-Mukai transforms given by I and δ O X (σ(t )), where I is the ideal sheaf of the relative diagonal immersion of X. We have seen before ( that the ) 1 1 matrices corresponding to these equivalences are and 0 1 ( ) 1 0 respectively. Since these matrices are a pair of 1 1 generators for the group SL 2 (Z) the result follows. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24 Fourier-Mukai groups Weierstrass Fibrations Theorem There exists an exact sequence 1 Aut 0 T Db c (X) FM T (D b c (X)) ech SL 2 (Z) 1, with Aut 0 T Db c (X) = Aut(X/T ) (2Z Pic 0 (X)), where Pic 0 (X) = {L Pic(X); deg(l t ) = 0, for any t T }. C. Tejero Prieto (Universidad de Salamanca) Relative Fourier-Mukai Transforms Salamanca, 17/05/ / 24

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