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Moduli space of sheaves on fans Hakimi, Koopa 2011

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Moduli Space of Sheaves on Fans by Koopa Hakimi  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April, 2011 c Koopa Hakimi 2011  Abstract A conjecture of H. Hopf states that if M 2n is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic χ(M 2n ), should satify (−1)n χ(M 2n ) ≥ 0. Ruth Charney and Michael Davis investigated the conjecture in the context of piecewise Euclidean manifolds having ”nonpositive curvature” in the sense of Gromov’s CAT(0) inequality. In that context the conjecture can be reduced to a local version which predicts the sign of a ”local Euler characteristic” at each vertex. They stated precisely various conjectures in their paper which we are interested in one of them stated as Conjecture D (see [1]) which is equivalent to the Hopf Conjecture for piecewise Euclidean manifolds cellulated by cubes. The goal of this thesis is to study the Charney - Davis Conjecture stated as Conjecture (D) by using sheaves on fans.  ii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 The Moduli Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Sheaves on Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Construction of a Moduli Space M∆ . . . . . . . . . . . . . . . .  4 4 5  3 Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Tangent Space of the Functor . . . . . . . . . . . . . . . . . 3.2 The Tangent Space of M∆ . . . . . . . . . . . . . . . . . . . . .  10 10 11  4 Irreducibility of the Moduli Space  . . . . . . . . . . . . . . . . .  15  5 Stratification of M∆ . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 k = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 k ≥ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21 21 22  6 Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25  iii  Acknowledgements I would like to express my deepest appreciation to my supervisor, Professor Kalle Karu, for introducing me to the field and allowing me to work on this project. I am grateful for all the help he provided, and his genuine desire to see me succeed. I would like to extend special thanks to Professor Zinovy Reichstein at the University of British Columbia for taking the time to read and evaluate my thesis. Last but not least, I express my sincere appreciation to all the helpful staff of the Mathematics Department.  Koopa Hakimi The University of British Columbia May 2011  iv  Chapter 1 Introduction Definition 1.1 (flag complex). Let K be a simplicial complex. A set V of vertices in K spans a complete graph if any two distinct elements of V span an edge in K. A simplicial cell complex K is a flag complex if any set of vertices which spans a complete graph actually spans a simplex. Thus, a flag complex is a simplicial complex with no ”empty simplices”. Let K be a simplicial complex and fi the number of i-simplices in K. Put λ(K) = 1 +  1 − 2  i+1  fi .  Now let K 2n−1 be a simplicial complex which is homeomorphic to S 2n−1 and λ(K) the quantity defined as above Conjecture 1 (Conjecture D). If K 2n−1 is a flag complex, then (−1)n λ(K 2n−1 ) ≥ 0. Definition 1.2 (simplicial complete fan). A simplicial complete fan ∆ consists of cones over a simplicial complex K ∪{∅}. Actually we think of an element σ ∈ ∆ as a cone over the simplex, with ∅ the zero cone. Then dim(σ) is the dimension of the simplex plus 1. We call the fan ∆ complete if K S 2n−1 . Remark 1. Actually the conjecture can be made more generally for simplicial cell complexes. One can correspond a fan to any piecewise spherical cell complex (see [1, p.120]) by taking a convex polyhedral cone associated to any spherical cell as the cones of the fan. Thus note that in order the fan to be a complete fan the necessary condition is simply our piecewise spherical cell complex to be homeomorphic to S n−1 . Note that what we defined above is an abstract fan, with no embedding in Rn .  1  Definition 1.3 (flag fan). A simplicial complete fan is flag if the simplicial complex K in the previous definition is a flag complex. Definition 1.4 (Poset of a fan). Let ∆ be a simplicial fan we define a binary relation ”≤” over ∆ as follows: for each τ, σ ∈ ∆ σ ≤ τ if and only if σ ⊂ τ . Clearly this is a partial order over the set ∆. By the definition of a simplicial complete fan we can introduce the new quantity λ (∆) as follows : dim ∆ i 1 λ (∆) = − fi 2 i=0 where fi is the number of i-dimensional cones in ∆ which is equal to the number of (i − 1)-dimensional simplices in K, f0 = 1. Assume ∆ has dimension 2n. Note that since multiplication by a positive constant does not change the sign, by multiplying with 22n , we can introduce a new quantity λ (∆): λ (∆) =  (−2)2n−i fi .  It is therefore convenient to state the following equivalent formulation of Conjecture 1. Conjecture 2. Suppose that ∆ is a simplicial complete flag fan and that λ is defined by the formula above. then: (−1)n · λ (∆) ≥ 0. We approach this conjecture using sheaves on the fan ∆. A sheaf F on a fan consists of vector spaces Fτ for all τ ∈ ∆ and restriction maps Fτ → Fσ for τ ≥ σ, satisfying compatibility conditions spelled out in Definition 2.1. Now consider the following cellular complex of the sheaf F: C • (F, ∆) : 0 → C 0 → C 1 → C 2 → · · · → C dim(∆) → 0 where C i =  Fτ and the boundary maps are the restriction maps with codim(τ )=i  ±signs. Suppose dim(Fτ ) = 2codim τ and consider the Euler characteristic of the given 2  cellular complex C • (F, ∆) χ(C • (F, ∆)) = dim C 0 − dim C 1 + dim C 2 − · · · (−1)i  = i  dim Fτ = λ (∆). codim(τ )=i  Now assume there exists a sheaf F where C • (F, ∆) has only the non-zero cohomology in the middle degree n, thus we conclude that λ (∆) = χ(C • (F, ∆)) = χ (H • (C • (F, ∆))) = (−1)n · dim H n (C n (F)) and this simply shows (−1)n λ (∆) ≥ 0. Thus in view of what mentioned about the sheaf cohomology above, Conjecture 1 is implied by the following Conjecture 3. There exist a sheaf F on ∆ with dim(Fτ ) = 2codim(τ ) which has non-zero cohomology in middle degree only. One guess would be the general point (sheaf) of the given dimension vector is expected to satisfy the above condition. Thus in order to make sense of the general sheaf, we need to introduce an irreducible moduli space of all such sheaves. Thus in proceeding chapters we construct the corresponding moduli space and prove its irreducibility only for 2-dimensional flag fans. In order to prove irreducibility, we will give two proofs, one by studying the tangent space and the embedding dimension and one by stratifying the moduli space and fibre dimensions.  3  Chapter 2 The Moduli Problem 2.1  Sheaves on Fans  From now on we work on the field C. Definition 2.1. Let ∆ be a partially ordered set. By a sheaf on ∆ we mean a data, F, consisting of: (1) a finite dimensional complex vector space Fσ , for each σ ∈ ∆ (2) linear maps (restrictions), resτ,σ : Fτ → Fσ for σ ≤ τ In order to define a (pre)sheaf the maps above should satisfy the restriction morphism conditions. By restriction morphism we mean: (1) for every σ ∈ ∆ the map resσ,σ : Fσ → Fσ is the identity (2) if ζ ≤ τ ≤ σ then (resτ,ζ ) ◦ (resσ,τ ) = resσ,ζ We can define the dimension vector of a sheaf F by dim(F) = (dτ ) where dτ = dim(Fτ ). Definition 2.2. If F and G are sheaves on ∆, a morphism ϕ : F → G consists of a morphism of vector spaces ϕτ : Fτ → Gτ for each τ ∈ ∆, such that whenever τ ≤ σ, the diagram Fσ resσ,τ    Fτ  ϕσ  ϕτ  / Gσ /    resσ,τ  Gτ  commutes, where resσ,τ are the restriction maps in F and G. Example 1. The constant sheaf Cn ∆ is the sheaf comprised of the vector space Cn at each element of ∆, with identity morphisms as the restriction morphisms.  4  Definition 2.3 (Families of sheaves over S). A family of sheaves F over a scheme S is the families of vector bundles over S consists of vector bundles Fτ over S for each τ ∈ ∆ and homomorphisms of bundles resτ,σ : Fσ → Fτ satisfying the same conditions as in Definition 2.2. A family F has a well defined dimension vector d = (dτ ) where dτ = rank(Fτ ). A morphism ϕ : F → G between families of sheaves over S consists of vector bundle homomorphisms ϕτ : Fτ → Gτ for all τ ∈ ∆ making the diagram in Definition 2.2 commutative. If F, G are families of sheaves over S, then F is a subfamily of G if Fτ ⊂ Gτ is a subbundle for all τ and the restriction maps of F are the restrictions of the restrictions maps of G. When G is a sheaf on ∆, we let G × S be the constant family of sheaves over S, such that (G × S)τ is the trivial vector bundle with fibre Gτ and the maps resτ,σ : (G × S)τ → (G × S)σ are given by resτ,σ : Gτ → Gσ . Definition 2.4 (Families of subsheaves). Fix one sheaf G on ∆. A family of subsheaves of G over S is a subfamilies F ⊂ G × S. Example 2. Let G to be the constant sheaf Cn then by a family of subsheaves of Cn we mean a collection of vector bundles Fσ ⊂ Cn such that ∀σ ≥ τ then Fσ ⊂ Fτ ⊂ Cn , and the following diagram Fσ   Fτ  ϕσ  ϕτ  /  /  Cn   id  Cn  commutes.  2.2  Construction of a Moduli Space M∆  Fix a sheaf G on ∆ and also fix a dimension vector d = (dτ )τ ∈∆ . To define our moduli problem let Sub∆ G,d be a functor from a category of Schemes to Sets, ∆ SubG,d : Schemes → Sets by Sub∆ G,d (S) = families of subsheaves of G such that dim(F) = d .  5  The fundamental question to answer in studying a given moduli problem is whether the functor Sub∆ G,d is representable in the category of schemes i.e. if there is a scheme M∆ and an isomorphism ψ (of functors from Schemes to Sets) between Sub∆ G,d and the functor of points of M∆ . This last is the functor HomSch (S, M∆ ). In this section we will show that this functor is indeed representable. Theorem 2.1. The functor Sub∆ G,d is representable. Example 3. (Grassmannian Functor). For the single point set ∆ = {τ } we define {τ } the grassmannian functor G to be SubG,d . Let W be a finite-dimensional vector space. By G(k, W ) we denote the Grassmannian of k-dimensional linear subspaces in W. Thus, to every such subspace V ⊂ W there corresponds a point [V ] ∈ G(k, W ). It is well known that this functor is representable and in fact G(d, Gτ ) is its fine moduli space. By Uτ we mean a universal bundle over G(dτ , W ) with fibre over [V ] the vector space V ⊂ W . Then Uτ is a sub-bundle of the trivial bundle G(dτ,W ) × W . Example 4. (Flag Functor). Let ∆ = {σ, τ } where σ ≤ τ and G a sheaf on ∆, {τ,σ} and define our Flag functor to be SubG,d . At first let G be the constant sheaf W ∆ where W a finite dimensional vector space. Thus for the families of subsheaf of G over S we have the flag Fτ ⊂ Fσ ⊂ W Fτ         Fσ  /  /  W   id  W .  {τ,σ}  Functor SubG,d is know to be representable by the flag variety F (dσ , dτ , W ). It can be constructed as a closed subvariety of G(dτ , W ) × G(dσ , W ) as follows. Let Uτ and Uσ be the Universal bundles on G(dτ , W ) and G(dσ , W ) respectively and let Qτ as the quotient Qτ = G(dτ , W ) × W/Uτ . Similarly for Uσ denote Qσ the corresponding quotient. Consider the morphisms of vector bundles: ψ  / π∗Q π1∗ Uτ QQ m 2 σ QQQ m m m QQQ mmm QQQ mmm QQQ m m vmm ( G(dτ , W ) × G(dσ , W )  6  where π1 , π2 are the projections to first and second factors. Thus we can write F (dτ , dσ , W ) as a zero locus of ψ. Now in general consider G not necessarily to be a constant sheaf. A subsheaf F ⊂ G is a commutative diagram: Fτ         Fσ  /  /  Gτ   resτ,σ  Gσ  We claim that subsheaves F ⊂ G are again parametrized by a closed subscheme of G := G(dτ , Gτ ) × G(dσ , Gσ ). Let Uτ , Uσ , Qτ , Qσ be as before and let π1 , π2 be the projections: G II II uu IIπ2 uu u II u u II u uz u $ π1  G(dτ , Gτ )  G(dτ , Gτ )  Note that the linear map resτ,σ : Gτ → Gσ induces a homomorphism of trivial vector bundles over G: /  G × Gτ  G × Gσ .  Now construct a homomorphism of vector bundles ψ : π1∗ Uτ → π2∗ Qσ over G as the composition ψ  π1∗ Uτ  /  G × Gτ  /  G × Gσ  /  (  π2∗ Qσ .  and define the zero locus of the map ψ to be F(dτ , dσ , G) ⊂ G(dσ , Gσ ) × G(dτ , Gτ ) and call it the general flag scheme. Lemma 1. The general flag scheme F(dτ , dσ , G) is the fine moduli space for the {τ,σ} Flag Functor SubG,d . Proof. Let ϕ ∈ Hom(S, F(τ, σ, G)) be given, we need to construct a family of sheaves F on S. let ι : F → G(dσ , Gσ ) × G(dτ , Gτ ) and consider the Universal 7  bundles Uτ and Uσ over the corresponding grassmannians then to give a families of sheaves over S define the vector bundles Fτ := ϕ∗ ι∗ π1∗ (Uτ ) and Fσ := ϕ∗ ι∗ π2∗ (Uσ ). Now by our definition of the general flag F as a zero locus of ψ then we get a morphism ι∗ π1∗ (Uτ ) → ι∗ π2∗ (Uσ ) and consequently a morphism Fτ → Fσ . Now given a family of subsheaves F of a sheaf G over S. Since the grassmannian functor is representable it would imply we would get morphisms for each family Fτ and Fσ over S ϕτ ∈ Hom(S, G(dτ , Gτ )), ϕσ ∈ Hom(S, G(dσ , Gσ )). Now let our morphism be ϕ := (ϕσ , ϕτ ) : S → G(dσ , Gσ ) × G(dτ , Gτ ). Since the general flag was defined as the zero locus of the map ψ then it would imply that ϕ factors through F(τ, σ, G). Definition 2.5 (Moduli Space). Let ∆ = {τ1 , ..., τn } be an arbitrary finite poset and consider the projections G(dτk , Gτk ) → G(dτi , Gτi ) × G(dτj , Gτj )  πi,j : τk  for each two comparable elements τi ≤ τj . Now let M∆ to be the scheme theoretic −1 −1 F(τ, σ, G). intersection of πi,j F(τ, σ, G) i.e. M∆ := i,j πi,j Now we will prove the Theorem 2.1 by showing the M∆ is our moduli space. Proof of Theorem 2.1. To show M∆ is the fine moduli space for our functor Sub∆ G,d . Let ϕ ∈ HomSch (S, M∆ ), To define Fτ , consider the projections π  τ G(dτ , Gτ ) −−−− −→ G(dτ , Gτ ).  τ  Take the universal bundle Uτ over G(dτ , Gτ ). Then define the vector bundles Fτ := ϕ∗ πτ∗ (Uτ ). Now since S maps into πij−1 F then by the restriction map resσ,τ : Gτ → Gσ we get a map Fτ  /  Fσ .  To show the converse correspondence let F be a family of subsheaves of a fixed 8  sheaf G over S, to construct a morphism ϕ : S → M∆ since Fτi ⊂ Gτi from representability of the grassmannian functor by G(dτi , Gτi ) we get morphisms ϕτi : S → G(dτi , Gτi ) for each τi , now let a map ϕ : S → τ G(dτi , Gτi ) to be ϕ := (ϕτi )τi ∈∆ . Now in order to complete the proof we need to show the image of S lies in M∆ . Consider the following composition of morphisms where πi,j is our projection in Definition 2.5. πij  ϕ  S −−→  G(dτi , Gτi ) −→ G(dτi , Gτi ) × G(dτj , Gτj ) τi  thus all we have to show is the image would lie in the F(dτi , dτj , G) ⊂ G(dτi , Gτi ) × G(dτj , Gτj ), and this is because F τi    F τj       / Gτ  /  i  × G(dτi , Gτi )   resτj ,τi  Gτj × G(dτj , Gτj )  commutes and gives a family on F(dτi , dτj , G).  9  Chapter 3 Tangent Space 3.1  The Tangent Space of the Functor  In this section let k be any algebraically closed field. One approach to study irreducibility of the moduli space would be by studying its tangent space. So in these section I would define the tangent space of M∆ and in the proceeding section I will give some applications to irreducibility of the moduli space of 2-dimensional fans. Definition 3.1. Given a functor F : {rings over k} → {sets} and an element x ∈ F (k), we define the tangent space of F at x to be Tx (F ) =  inverse image of x ∈ F (k) under Fk[ ]→k : F (k[ ]) → F (k)  ⊂ F (k[ ])  This has the structure of a vector space over k. Example 5. Let R be any k-algebra. An algebraic variety (or, more generally, a scheme) X determines a functor X : {rings} → {sets} where X(R) is the set of R-valued points of the variety X. Then after taking an affine open cover of X, at each k-valued point x ∈ X(k) the tangent space Tx (X) ⊂ X(k[ ]) in the sense of Definition 3.1.1 coincides with the Zariski tangent space. 10  3.2  The Tangent Space of M∆  Proposition 3.1. The tangent space to the Grassmannian G(r, n) at a point Λ ∈ G(r, n) corresponding to an r-dimensional subspace Λ ⊂ k n is canonically isomorphic to Homk (Λ, k n /Λ). Proof. Consider the grassmannian functor defined in Example 3. To the point [U ] ∈ G(r, n) there corresponds a morphism U    / kn,  and a tangent vector at [U ] is then morphism of k[ ]-modules U  /  k[ ]n  whose reduction module ( ) coincides with U → k n . Let u1 , ..., ur ∈ k n be a basis of U , and let u1 + v1 , ..., ur + vr ∈ k[ ]n be a free basis of U as a k[ ]-module. Since 2 = 0, it follows that u1 , ..., ur ∈ U , and this is a basis of U . This shows that the given tangent vector determines a well-defined linear map U → V = k n /U,  ui → vi  mod U  and this correspondence defines an isomorphism T[U ] G → Hom(U, k n /U ). Remark 2. (Tangent space to the flag variety) For each k, l we consider the flag manifold F (k, l, n), i.e., the incidence correspondence Ω = {(Λ, Γ) : Λ ⊂ Γ ⊂ k n } ⊂ G(k, n) × G(l, n). It is not hard to see that Ω is smooth. We want to identify the tangent space to Ω as a subspace of the tangent space to the product G(k, n) × G(l, n). To do this, let (Λ, Γ) ∈ Ω be any point of Ω. The tangent vector, viewed as an element of the tangent space TΛ,Γ (G(k, n) × G(l, n)) = TΛ (G) × TΓ (G), is (η, ϕ) where η ∈ TΛ (G) and ϕ ∈ TΓ (G) respectively. Now it is well known [2] the tangent space is the  11  space of pairs TΛ,Γ (Ω) =  η ∈ Hom(Λ, k n+1 /Λ) (η, ϕ) : and ϕ|Λ ≡ η mod Γ ϕ ∈ Hom(Γ, k n+1 /Γ)  Proposition 3.2. The tangent space to the general flag scheme at a point (Uτ , Uσ ) ∈ G × G is given by morphisms ϕτ ∈ Hom(Uτ , k n /Uτ ) and ϕσ ∈ Hom(Uσ , k m /Uσ ) such that ϕτ  Uτ   Uσ  ϕσ  /  Gτ /Uτ   (3.1)  / Gσ /Uσ .  Proof. By the same argument in Proposition 3.2 consider the following morphisms Uτ         /  /  Uσ  Gτ = k n   ψ  Gσ = k m  The tangent vector at Uτ , Uσ is then is given by morphisms of k[ ]-modules Uτ        Uσ  /  / Gτ [   ] = k[ ]n  ψ  (3.2)  Gσ [ ] = k[ ]m  Suppose the diagram (3.1) is given. Let u1 , ..., ul , ul+1 , ..., ur ∈ k n be a basis of Uτ and u1 , ..., ul , ul+1 , ..., us ∈ k m be a basis of Uσ where Span{ul+1 , ..., ur } = ker ϕ and ψ(ui ) = ui , and let u1 + v1 , ..., ur + vr ∈ k[ ]n u1 + v1 , ..., us + vs ∈ k[ ]m , Be the basis for Uτ and Uσ respectively then by Proposition 3.2 the tangent space TUτ G is isomorphic to Hom(Uτ , k n /Uτ ) and also similarly TUσ G is isomorphic to Hom(Uσ , k n /Uσ ) via the linear maps  12  Uτ → k n /Uτ , ui → vi mod Uτ  (3.3)  Uσ → k m /Uσ , ui → vi mod Uσ . Since for each i we can write (uj + vj )(aj + bj ) =  ψ(ui + vi ) =  uj aj +  (aj vj + bj uj )  we conclude that ψ(ui ) =  aj uj ,  ψ(vi ) =  (aj vj + bj uj ).  Now for the case where i ≥ l + 1 u_i     / vi _ /  0    0 = ϕ(vi )  from ui → 0 we conclude that aj = 0 for all j thus bj uj ≡ 0 mod Uσ .  ϕ(vi ) =  (3.4)  j  for the case where i ≤ l u_i    ui    / vi _  /v= i  ϕ(vi )  since ui → ui implies ai = 1 and aj = 0 for j = i thus we have ϕ(vi ) = vi +  bj uj = vi  mod Uσ .  (3.5)  From (3.4) and (3.5) we conclude that mod Uσ ϕ(vi ) = [0] for i ≥ l + 1 and ϕ(vi ) = [vi ] for i ≤ l.  13  Thus we get the following commutative diagram of morphisms /  Uτ  Gτ /Uτ     /  Gσ /Uσ  Uσ  Actually the other correspondence direction is straightforward. Suppose the diagram (3.2) is given then by morphisms (3.3) we would get the diagram (3.1) Theorem 3.1. The tangent space of Sub∆ G,d at a point F is Hom(F, G/F). Proof. By the construction of our fine moduli space M∆ as the scheme theoretic intersection of inverse image of each general flag F for each τi ≤ τj by projections G(dτ , Gτ ) → G(dτi , Gτi ) × G(dτj , Gτj )  πij : τ ∈∆  we conclude that TF (M∆ ) ⊂  TF (G(dτ , Gτ ) =  Hom(Fτ , Gτ /Fτ )  is the intersection of (ϕτ )τ ∈∆ , satisfying Fτ   Fσ  ϕτ  ϕσ  /  /  Gτ /Fτ   Gσ /Fσ  14  Chapter 4 Irreducibility of the Moduli Space We will give 2 different proofs of the irreducibility in the 2-dimensional case for certain G and dimension vector d = (dτ ). Definition 4.1 (poset of a 2-dimensional complete fan). The poset of a 2-dimensional complete fan consist of a set ∆ = {0, τ1 , ..., τk , σ1 , ..., σk } and the relations σi , σi+1 ≥ τi for 1 ≤ i ≤ k − 1 and for i = k, τk ≤ σk , σ1 . Also the dimension vector is given by fixing dim τi = 2 and dim σi = 1. Note that as mentioned in the introduction we are interested in such sheaves on ∆ where in this case G = C4∆ , dim(Fσi ) = 1, dim(Fτi ) = 2 and dim(F0 ) = 4 We will fix this G and this dimension vector d = (dτ ) for the rest of the section. Proposition 4.1. The dimension of every component of M∆ is ≥ 3k. Proof. Since M∆ ⊂ (G(1, 4))k × (g(2, 4))k we can consider the projections πτ,σ : M∆ → P3 × G(1, 3) for each τ ≤ σ. now take the flag variety F (1, 2; C4 ) ⊂ G(1, 3) × G(2, 4), since it is nonsingular and has codimension 2 it is defined locally by 2 equations so we get locally 4k equations in total. And since the dimension of the ambient space is 3k + 4k we conclude that every component of M∆ has dimension ≥ 3k + 4k − 4k = 3k  Theorem 4.1. The moduli space M∆ is singular at a point corresponding to F ⊂ G if and only if: (i) Fτi = Fτj for all τi , τj ∈ ∆. (ii) Span{Fσi } = C4 . 15  Proof. First consider in contrary the first condition is not satisfied and we will prove in this case any point is smooth by showing the dimension of the tangent space is exactly 3k. Partition the maximum cones {τ1 , ..., τn } into m sets as follows {τ1 , ..., τn } = ∆1 ∪ ∆2 ∪ ... ∪ ∆m where if τi , τi+1 are neighbours, then they lie in the same ∆j if and only if Fτi = Fτj In what follows for simplicity we shall write Q for the quotient C4 /F. Now order cones τi , σj so that τi , τk lie in different partitions ∆i , τ1 ∩ τk = σk . We choose morphisms ϕτi , ϕτj in the following order τ1 , σ1 , τ2 , σ2 , ..., τk , σk . Choose ϕτ1 arbitrary. By induction, suppose ϕτ1 ϕσ1 , ..., ϕτi are chosen. Let τ = τi , σ = σi and τ = τi+1 . We explain how to choose ϕσ , ϕτ in two cases: Let τ ∈ ∆1 and τ ∈ ∆2 where this two have a common face namely σ. How from the following conditions on morphisms for our corresponding choices of σ, τ and τ: C  Fτ _  C2  FO σ  C  Fτ  ϕτ    ϕσ  ?  ϕ  τ  /  /  /  Qτ   C3  QO σ  C2  Qτ  C3  Case 1. (τ, τ lie in different partition. i.e. Fτ = Fτ ) If ϕτ is given, there are 2 dimension choices for ϕσ , and there is 1 dimension of choice for ϕτ since Fσ = Fτ ⊕ F τ . Case 2. (τ, τ lie in the same partition. i.e. Fτ = Fτ ) Again if ϕτ is given, there are 2 dimension choices for ϕσ , and there is 1 dimension of choice for ϕτ . Thus in both cases we have: dim[Hom(Fσ , C4 /Fσ )] = 2, dim[Hom(Fτ , C4 /Fτ )] = 1 16  Now note that the last step in order to complete the fan is when ϕτ1 , ϕτk are given and we want to choose ϕσk , but as in case 1, ϕτ1 and ϕτk together determine ϕσk uniquely since like in case 1: Fσ = Fτ ⊕ F τ . and as a result we have: dim[Hom(F, C4 /F)] = 3 + (k − 1) × 1 + (k − 1) × 2 = 3k. Now in order to complete the proof let F be any arbitrary sheaf on ∆ and assume Fτi = Fτj for all i, j which would decompose our sheaf F into F = C∆ ⊕ F . Thus we have C4 /F = C4 /C ⊕ F = C3 /F and consequently the tangent space at F would be  TF (M∆ ) = Hom(F, C4 /F) = Hom(C ⊕ F , C3 /F ) = Hom(C, C3 /F ) ⊕ Hom(F , C3 /F ) which would imply dim(TF (M∆ )) = dim[Hom(C, C3 /F )] + dim[Hom(F , C3 /F )]. Note that Fτi = 0 for all τi ∈ ∆. Consider ϕ ∈ Hom(F , C3 /F ), thus for each σ the morphism ϕσ should make the following diagram commutative C  Fσ _  C3  F0    ϕσ  ϕσ  /  Qσ /  C2    0  which would imply that the space of all such morphisms (ϕσ ) would be of dimension 2 and since there are k such σ’s we conclude that dim[Hom(F , Q)] = 2k 17  To compute the dimension of Hom(C, C3 /F ) let ϕ ∈ Hom(C, C3 /F ) consider the commutative diagram C _   ϕτ  /Q  ϕσ  /  CO  ? ϕτ  C  /  τ  C3  QO σ  C2  Qτ  C3    where τ, τ ≥ σ, let ϕτ (1) = qτ ∈ Qτ and ϕτ (1) = qτ ∈ Qτ , subject to conditions: resτ,σ (qτ ) = resτ  ,σ (qτ  ) if τ, τ ≥ σ.  Now we can write this condition as: (resτ,σ , − res  τ ,σ  )  (qτ , qτ ) ∈ ker(Qτ ⊕ Qτ −−−−−−−−−−−−−→ ⊕Qσ ) Putting these together: ψ  (qτ )τ ∈ ker(⊕Qτi −−−→ ⊕Qσi ) where the map ψ is ± resτ,σ . The signs are chosen as in the cellular complex, depending on the chosen orientation of τi , σi . Thus we conclude that ψ  dim[Hom(C, Q)] = dim(ker(⊕Qτi −−−→ ⊕Qσj )). Now since dim(⊕Qτi ) = 3k and dim(⊕Qσj ) = 2k we conclude that dim[Hom(C, Q)] ≥ k. Note that since TF (M∆ ) is the Zariski tangent space for any point F ∈ M∆ , dim(TF (M∆ ) ) ≥ dim(M∆ ), with equality if and only if F is nonsingular. Now from above we can derive that a point is singular if and only if dimension of Hom(C, Q) is strictly greater that k and consequently this is the case if and only if ψ is not surjective. Now in order to complete the proof we will show non-surjectivity of ψ is equivalent  18  to condition (II). From the following exact sequences /  0  0  /  ⊕Fτi   ⊕Fσj  /  ⊕C3τi   /  ⊕Qτi  α  / ⊕C3  σj  /0  ψ    / ⊕Qσ  j  /0  we get a long exact cohomology sequence: 0 → ker α −→ ker ϕ −→ ⊕Fσj −→ coker α = C3 −→ coker ψ → 0. Thus coker ψ = 0 iff (⊕Fσj → C3 ) is not surjective iff (⊕Fσj → C4 ) is not surjective. This gives, in particular, that dim(coker ψ) = codim(Span{Fσj }) ⊂ C3 = codim(Span{Fσj }) ⊂ C4 . Remark 3. Before proving irreducibility by studying the tangent space, now that we know which points are singular lets say more about the dimension of singular points. Consider the singular point F. We know from condition (I), Fτi = Fτj . Consider 2 types of singular points: 1 dim Span{Fσi } = 3 2 dim Span{Fσi } = 2 which is equivalent to Fσi = Fσj for all i, j. The second type of singular points are the most singular points with tangent space dimension 3k + 2 coming from the fact that dim TF (M∆ ) = 2k + dim(ker ψ) = 3k + dim(cokerψ). Lemma 2. The dimension of Sing(M∆ ) is 5 + k. Proof. Since we know a point F is singular if and only if the two conditions above is satisfied we can simply compute the singular locus as follow: Choose a 3-dimensional space V ⊂ C4 . The space of all such V is of dimension 3 (The dimension of G(3, C4 )). Then choose a 1-dimensional space Fτ ⊂ V = C3 . The corresponding space is of dimension 2 (The dimension of P2 ) and at last for each σ we have to choose a 2-dimensional space Fσ such that Fτ ⊂ Fσ ⊂ V or  19  equivalently Fσ /Fτ = C ⊂ V /Fτ  C2  which has dimension k, i.e. The dimension of (P1 )k . Theorem 4.2 (Affine Dimension Theorem). Let Y, Z be varieties of dimensions r, s in An . Then every irreducible component W of Y ∩Z has dimension ≥ r+s−n. Proof. (see [3], p. 48.) Lemma 3. M∆ is connected. Proof. Let U4 be the set of all 4×4 upper triangular matrices with diagonal entries 1:   1 ∗ ∗ ∗   0 1 ∗ ∗   0 0 1 ∗   0 0 0 1 U4 acts on C4 , hence acts on M∆ . Since U4 is solvable, every projective variety on which U4 acts has a fixed point. Thus M∆ has a unique fixed point F where Fτi = Span{e1 }, Fσj = Span{e1 , e2 }. This fixed point must lie in every component of M∆ . Theorem 4.3. For 2-dimensional complete fans the moduli space is irreducible for k ≥ 4. Proof. Suppose M∆ has two irreducible components then each component is of dimension ≥ 3k. Now let F be the fixed point of U4 as in Lemma 3. then TF (M∆ ) = 3k + 2. Now since locally (analytically) we can embed M∆ to its tangent space thus by the ”Affine Dimension Theorem” we have 5 + k = dim(Sing(M∆ )) ≥ dim Y ∩ Z ≥ 3k + 3k − (3k + 2) which would imply k ≤ 3.  20  Chapter 5 Stratification of M∆ Another way to prove Theorem 4.3 would be by means of stratification of M∆ and calculating the fibre dimensions of each stratas. Consider the map G(dτ , C4 ) = (P3 )k .  π : M∆ → τ  We can stratify M∆ according to the combinatorial type of the image of F ∈ M∆ .  5.1  k=3  By Proposition 4.1 the dimension of M∆ is 9. Let (P3 )3 = U  S1,2  S1,3  S2,3  S1,2,3  be the stratification of P3 , where Si,j is the set consisting of 3 choices of points namely pi := P(Fτi ) in P such that the two points pi = pj and the third point is different from the other two and S1,2,3 is a set consisting of three such points (p1 , p2 , p3 ) ∈ (P3 )3 where all three are equal and U would be a set where pi ’s are all distinct. Now consider the restriction of π on each inverse image of these stratas. Since we have M∆ = π −1 (U) π −1 (S1,2 ) π −1 (S1,3 ) π −1 (S2,3 ) π −1 (S1,2,3 ) where each piece is irreducible. Since S1,2,3 is an irreducible closed set in (P3 )3 (it is the diagonal set in the product) the inverse image of it under π is a closed set in M∆ . The dimension of the fiber is 6 since by fixing one point in P3 we are looking for all the possible 3 lines passing through this one point which the space is of dimension (P2 )3 by fibre dimension  21  theorem we conclude that dim(π −1 (S1,2,3 )) = 6 + 3 = 9 Consider the strata S1,2 . We need to choose 2 points p1 , p3 in P3 thus dim S1,2 = 6. Now in order to compute the fibre dimension note that by fixing two points p1 , p3 these two points uniquely determine a line passing through them thus we only need to choose one line so the fibre would be P2 . Thus we conclude that dim(π −1 (S1,2 )) = 2 + 6 = 8. Similarly for strats S1,3 and S2,3 we have: dim(π −1 (S1,3 )) = 2 + 6 = 8,  dim(π −1 (S2,3 )) = 2 + 6 = 8.  Thus M∆ is reducible since it has 2 components. Actually the other component is π −1 (U) since each fiber on U is a single point (there exists only one line passing through 2 points) thus dim(π −1 (U)) = 0 + dim(U) = 0 + 9 = 9.  5.2  k≥4  In order to define the stratas consider the partitions of the cones {τ1 , ..., τk } introduced in The proof of Theorem 4.0.2. in to m pieces. Thus we can associate a cyclic partition of the set {1, ..., k}, namely {i1 , ..., im } to each partition of {τ1 , ..., τk }, as follows : {1, · · · , k} = Σ1  Σ2  ···  Σm  = {i1 , i1 + 1, · · · , i2 − 1}  {i2 , i2 + 1, · · · , i3 − 1}  ···  where m = |{i1 , · · · , im }| is equal the number of partitions of cones {τ1 , ..., τk } = ∆1 ∪ ... ∪ ∆m . Now by the same argument for the case k = 3, consider the following stratification of (P3 )3 (P3 )3 = U  S{1}  S{i1 ,i2 }  ...  S{i1 ,...,im }  ...  S{1,2,...,k} . 22  Then if 1 < m < k dim π −1 (S{i1 ,...,im } ) = dim(fibre) + dim(SΣ ) = dim(P2 ) i |Σi | −m + dim(P3 )m = 2(k − m) + 3m = 2k + m < 3k If m = 1, then dim π −1 (S{1} ) = dim(fibre) + dim(SΣ ) = dim(P2 )k + dim(P3 ) < 3k Now since in each case the dimension of each inverse image of components are strictly less than 3k we conclude that M∆ is irreducible and its only component is π −1 (S{1,2,...,k} ).  23  Chapter 6 Conclusion Following the construction of our moduli space and its irreducibility in 2 dimensional case we prove Conjecture 3 for 2 dimension case by showing the general sheaf F on ∆ = {0, τ, σ} with dimension vectors dim(Fτ ) = 1, dim(Fσ ) = 2 and dim(F0 ) = 4 has non-zero cohomology in middle degree only. Theorem 6.1. Let F be a general sheaf in M∆ for k ≥ 4. Then the cellular complex C • (F, ∆) has nonzero cohomology in degree 1 only. Proof. By Theorem 4.1. we need that (1). Span{Fσ } = C4 . (2).  ψ  τ  Fτ −−−→  σ  Fσ is injective.  If (fτ ) ∈ ker ψ, then resτ,σ (fτ ) = resτ  ,σ (fτ  )  for all τ, τ ≥ σ.  Thus if Fτ = Fτ , then fτ = fτ = 0 Remark 4. Note that C • (F, ∆) has nonzero cohomology in middle dimension only if and only if both conditions of Theorem 4.1 are false Example 6. For the case where k = 3 the component π −1 (U) satisfies condition (ii) in the Theorem 4.1., hence C • (F, ∆) has cohomology in degree 2. The component π −1 (S1,2,3 ) satisfies the condition (I), and C • (F, ∆) has cohomology in degree 0.  24  Bibliography [1] R. Charney and M. Davis, The Euler characteristic of a non-positively curved, piecewise Euclidean manifold, Pacific J. Math. 171 (1995), pp. 117-137, [2] J. Harris, Algebraic Geometry: A First Course. GTM No. 133. SpringerVerlag, first edition, 1992. [3] R. Hartshorne, Algebraic Geometry. GTM, vol. 52, Springer Science + Business Media, LLC, 2006. [4] S. Mukai, An Introduction to Invariants and Moduli, Cambridge Studies in Advanced Mathematics, 81. Cambridge Univ. Press, Cambridge, 2003.  25  

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