Principales aditivos de un fluido de perforación

We will now discuss fans whose supports are Gorenstein cones as described above.

Definition 5.2.14. We say a fan Σ is a cone closure of a cone σ if Σ is the set

Take σ to be a maximal dimension, strictly convex, Gorenstein cone σ in the vector space N_R with lattice points in N. Let Σ be the cone closure of σ and Σ∨ be the cone closure of σ∨. We now will take refinements of these fans that will correspond to affine bundles. Recall the following definition.

Definition 5.2.15 (Page 515 of [19]). Take a fan Σ in N_R and a primitive lattice element v ∈ |Σ| ∩N. The generalized star subdivision of the fan Σ at v is the set Σ(v) of the following cones:

1. the conesτ ∈Σ that do not contain the lattice element v and

2. the cones τ(v) = Cone(τ, v) where v /∈τ ∈Σ and {v} ∪τ ⊆ τ0 for some cone

τ0 ∈Σ.

Lemma 5.2.16 (Lemma 11.1.3 of [19]). The fan Σ(v) is a refinement of Σ.

If the setEis a proper splitting ofσ, then every element is primitive inN (due to condition (3) of the definition of proper splitting) and one can iterate the procedure of generalized star subdividing at each element ofE, obtaining new fans Σ(e1), then (Σ(e1)) (e2) and so on. Denote by ΣE the fan obtained after star subdividing by all elements of the set E, which we call the star subdivision of the fan Σ by the proper splitting E.

Given the conesσandσ∨as above and proper splittingsEandF respectively, we obtain the fans ΣE and Σ∨F by taking the cone closures and star subdividing. Now, take the projections πE : NR −→ NR/(E) and πF : MR −→ MR/(F). Construct

two new fans by just taking the collection of cones inN_R/(E) and M_R/(F) that are images of the cones in the fans ΣE and Σ∨F, respectively. Call these new fans ΣE and Σ∨_F. Note that, by construction, the fans ΣE and ΣE (respectively Σ∨F and Σ

∨

F) are compatible under the projection πE (πF) hence they induce toric morphisms

XΣE πE XΣF πF XΣE XΣ∨F

Take the cone closures ΘE and ΘF that are associated to the cones Cone(E) and Cone(F). Note that ΘE ⊂ΣE and ΘF ⊂Σ∨F. Recall the following definition:

Definition 5.2.17 (Definition 3.3.18 of [19]). LetN and N0 be lattices and Σ and Σ0 be fans inN_RandN_R0 respectively. Suppose there is a surjective_Z-linear mapping

φ :N → N0 that is compatible with the fans Σ and Σ0. Take N0 to be the lattice that is the kernel of the map φ and the subfan

Σ0 :={τ ∈Σ :τ ⊆(N0)R}

of Σ. We say that the fan Σ is split by Σ0 and Σ0 if there exists a subfan ˆΣ⊆Σ so that

1. φ_R :N_R →N_R0 maps each cone ˆτ ⊆ Σ bijectively to a coneˆ τ0 ∈Σ0 such that

φ(ˆτ∩N) =τ0∩N0.

2. Given cones ˆτ ∈Σ andˆ τ0 ∈Σ0, the sum ˆτ+τ0 lies in Σ, and every cone of Σ arises this way.

In our context, we can see that ΣE is split by ΣE and ΘE, by construction. The subfan ˆΣE is the set of all cones in the fan ΣE that is disjoint from the set E. By Theorem 3.3.19, this implies that XΣE is a rank k affine bundle over XΣE where k

is the length of the proper splittingE. Analogously, the toric variety XΣ∨

F is a rank ` affine bundle over XΣ∨<sub>F</sub> where ` is the length of the proper splitting F. We now add the last assumption:

Assumption 5.2.18. The affine bundles XΣE and XΣ∨_F are vector bundles over XΣE and XΣ∨F, respectively.

According to [19], Oda notes in [35] that if a toric vector bundle V is a toric variety, then the bundle is a direct sum of line bundles. This is proven by using the classification of toric vector bundles found in [31]. This implies that the vector bundlesXΣE andXΣ∨F both split as a direct sum of line bundles overXΣE andXΣ∨F,

respectively.

Proposition 5.2.19. Each line bundle that is a direct summand in the vector bun- dle XΣE corresponds to a divisor D so that −D is nef.

Proof. The support functions associated to each divisor will be convex, yielding it being associated to an anti-nef divisor.

Due to this proposition, we look at the dual vector bundlesV and Λ toXΣE and XΣ∨_F, respectively. Take generic global sectionsf ∈Γ(XΣE, V) andg ∈Γ(XΣ∨F,Λ) so

the part of the toric variety corresponding to that cone. These zero loci are complete intersections ofkand`polynomials each. Denote the complete intersectionsME :=

Z(f) and WF := Z(g). Note that the variety ME has dimension n−2k and WF has dimension n−2`. We now propose the following question:

Question 5.2.20. AreME andWF mirrors in the sense of Kontsevich’s Homological Mirror Conjecture (Conjecture 2.1.1)?

We now will break down some historical results about some cases of this con- struction. The first result in this theme was that of Batyrev:

Theorem 5.2.21 ([3]). If the cone σ is a Gorenstein cone of index 1, then proper splittings exist and are unique. Moreover, the Calabi-Yau manifolds ME and WF

are mirrors on the level of stringy cohomology, i.e.,

hp,q_st (ME,C) =h

(n−2)−p,q

st (WF,C)

Batyrev and Borisov then generalized this result to Calabi-Yau complete intersec- tions:

Theorem 5.2.22 ([5]). Suppose the coneσ is a dual Gorenstein cone of indexr and one has complete splittings E and F for the cones σ and σ∨ respectively. Then the Calabi-Yau manifolds ME and WF are mirrors on the level of stringy cohomology,

i.e.,

hp,q_st (ME,C) =h

(n−2r)−p,q

Since in our setup we do not necessarily have the Calabi-Yau condition, we state our claims on the level of homological mirror symmetry. We propose a conjecture for what happens when we loosen the hypotheses to those stated above.

Conjecture 5.2.23. Let σ and σ∨ be dual reflexive Gorenstein cones of index r, imposing the assumptions taken in Subsection 5.2.2. Take E to be a complete splitting ofσ andF to be proper splitting ofσ∨ of maximal length. We then obtain varietiesME andWF as above. We claim that they are mirrors in the sense that the dimension of the Hochschild homology of the (derived) Fukaya category Fuk(ME) of

ME is equal to the dimension of the Hochschild homology of the largest Calabi-Yau category CYWF that is a subcategory of the bounded derived categoryD

b_{(coh W} F) of coherent sheaves on WF, i.e.,

dimHH•(Fuk(ME)) = dimHH•(CYWF).

Note that sinceME is a Calabi-Yau variety, we know that dimHHk(Fuk(ME)) =

X

p+q=k

hp,q_st (ME,C).

In the context of our conjecture, we then can focus on cohomology and the B- side of mirror symmetry and sidestep looking at the Fukaya category. While the hypotheses of the conjecture are weaker, the claim is also weaker in that we do not have a bigrading. We are currently in a sense looking at Betti numbers of the Hodge diamond.

We remark that due to the recent work, there is a bigrading of these cohomol- ogy theories that give Hodge-like numbers to these categories due to Katzarkov,

Kontsevich and Pantev [29]. In the next few sections we will be showing how both Batyrev-Borisov and Berglund-H¨ubsch-Krawitz mirror dualities fit into this toric construction.