All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Information Report

Category:
## Technology

Published:

Views: 38 | Pages: 24

Extension: PDF | Download: 0

Share

Description

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

Transcript

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

Recommended

Related Search

Fourier Sine Transformschange for the better mind body and soulEuropean Group for the Study of Deviance and Law for Provision of Free Maternity and ChildTips for men on hair fall and balding causeswant to apply for ba hons in English and Litefor more coloberatif in teaching and knowledgbillboard advertising for d2h adsneon signs advertising for d2h adsoutdoor media partner for d2h ads

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x