Polyvector fields for Fano 3-folds

We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of Fano 3-folds. This is the first step in understanding the non-trivial Gerstenhaber algebra structure of this invariant, and yields some initial insights in the classification of Poisson structures on Fano 3-folds of higher Picard rank.


Introduction
In this paper we describe the Hochschild cohomology of Fano 3-folds, with the eventual goal of understanding the interesting algebraic structures present on this invariant, and completing the classification of Poisson structures on Fano 3-folds.
Fano 3-folds and the vector bundle method.Fano 3-folds were classified by Iskovskikh [23,24] (for Picard rank 1, where there are 17 families) and Mori-Mukai [39,40] (for Picard rank ≥ 2, where there are 88 families).This classification was obtained by understanding the birational geometry of Fano 3-folds, and the output is a list of 105 deformation families and their numerical invariants c 1 (X) 3 , ρ(X), and h 1,2 (X).Only 12 out of 88 families of Picard rank ≥ 2 are not the blowup of a Fano 3-fold of lower Picard rank.
For the Picard rank 1 case Mukai alternatively described the classification using the vector bundle method in [42], by writing Fano 3-folds of Picard rank 1 as zero loci of vector bundles on homogeneous varieties and weighted projective spaces.In higher Picard ranks this was extended in 2 different ways, by giving a description as On the methods.In Sections 3 and 4 we collect the details for the proof of Theorem A. We will set up the proof so that we can take advantage of computer algebra methods, with some explicit calculations in cases where automated methods fail.We have optimised the automated methods so that only 5/105 deformation families of Fano 3-folds need to be dealt with by hand (2 of which are nearly immediate).
What is interesting to observe is that the homogeneous methods from [8] are very good at determining Hodge numbers (and in particular they are expected to help in classifying Fano 4folds), with only a dozen deformation families of Fano 3-folds not being fully determined.But for twisted Hodge numbers (and in particular the cohomology of T X and 2 T X ) the homogeneous approach gives many underdetermined cases.
The following imprimitive Fano 3-folds admit no Poisson structures: For all other Fano 3-folds there are non-zero global bivectors, and it is necessary to check the self-bracket of a global bivector field.Already for Fano 3-folds of Picard rank 1 this is a highly non-trivial condition [34].
For the imprimitive Fano 3-folds we expect that the birational description of Mori-Mukai together with [44, §8] should allow for a (partial) classification of Poisson structures.In particular, we expect that the second part of Corollary B has a proof using these techniques, but this is outside the scope of the current paper.
Relation to other works.In the representation theory of finite-dimensional algebras the Gerstenhaber algebra structure on Hochschild cohomology is an important invariant, studied in many cases, see [1,14,47,49] to name a few.In algebraic geometry there are (at the time of writing) fewer attempts at giving explicit descriptions of Hochschild cohomology and the Hochschild-Kostant-Rosenberg decomposition.An important case is that of partial flag varieties [7,20].For smooth projective toric varieties (and only the H 0 , not any possible H ≥1 ) one is referred to [21].There are also various cases where the interaction of the Hochschild cohomology of different varieties (and categories) is studied (see e.g.[6,22,31]), with the Kuznetsov components of Fano 3-folds of Picard rank 1 and index 2 being the subject of [32, §8.3].
Some of the results in this paper are standard, whilst for Fano 3-folds of Picard rank 1 results can be found in [26,34].
It would be interesting to understand how mirror symmetry can be used to compute the invariants investigated in this paper, using the symplectic geometry of the mirror Landau-Ginzburg model.For Hodge numbers (and hence Hochschild homology, see Section 2.1) of Fano varieties a recipe for this was conjectured by Katzarkov-Kontsevich-Pantev in [27,Conjecture 3.7], based on the conjectural equivalence from homological mirror symmetry.Here f : Y → A 1 is a (suitably compactified) Landau-Ginzburg model and ω Y an appropriately chosen symplectic form, so that X and (Y, f ) are mirror.Subsequently this was checked by Lunts-Przyjalkowski for del Pezzo surfaces in [37] and by Cheltsov-Przyjalkowski for Fano 3-folds in [13].Hochschild cohomology is also a categorical invariant, and therefore can be computed from either side of (2) (assuming an enhancement of the equivalence).An interesting difference is that Hodge numbers (and hence the dimensions of the Hochschild homology spaces) are constant in families, but this is not the case for Hochschild cohomology.
Notation.We will number deformation families of Fano 3-folds as MM ρ.n as in Mori-Mukai [39] (see also [25, §12.2-12.6]),with the caveat that MM 4.13 refers to the blowup of P 1 × P 1 × P 1 in a curve of degree (1, 1, 3), the case which was originally omitted and discovered in [41].
Throughout we work over an algebraically closed field k of characteristic 0.
There exist various approaches to defining the Hochschild cohomology of a variety, which are known to agree in the setting we are interested in.One of the more economical definitions is the following.
Definition 2.1.Let X be a smooth and projective variety.Its Hochschild cohomology is , where ∆ : X → X × X denotes the diagonal embedding.
Hence as a first approximation (disregarding any algebraic structures present on Hochschild cohomology) determining the Hochschild cohomology of a variety reduces to a question in sheaf cohomology.

Remark 2.3.
There is also the Hochschild homology of X, defined as Moreover there is the Hochschild-Kostant-Rosenberg decomposition for Hochschild homology, which now reads for i = − dim X, . . ., dim X. Hence the dimension of the Hochschild homology of X is determined by the Hodge numbers h p,q = h q (X, Ω p X ).These numbers admit symmetries under Serre duality and Hodge symmetry, and therefore are often written down in the form of the Hodge diamond.In particular for Fano 3-folds the Hodge diamond is of the form (9) and it is determined by the invariants from the classification.The dimensions of the Hochschild homology spaces now correspond to different columns in this diamond (as opposed to the rows which describe the dimensions of singular cohomology spaces).
To mimic this economical description of the Hochschild-Kostant-Rosenberg decomposition of Hochschild homology using the Hodge diamond, the first author introduced the polyvector parallelogram.If we denote p p,q := dim k H p (X, q T X ), then for a 3-fold it is given by (10) p 3,3   with an obvious generalisation to other dimensions.There are no symmetries present in the numbers p p,q , and the presentation reflects this absence.
Remark 2.4.Another important difference between the Hodge diamond and the polyvector parallelogram is that the former is constant in families, whilst the latter is not necessarily so.We will explain this for Fano 3-folds in Section 3.1.
Additional structure.There is a rich algebraic structure on Hochschild cohomology HH • (X), and on the polyvector fields p+q=• H p (X, q T X ).Namely there exist: • a graded-commutative product (of degree 0); • a graded Lie bracket (of degree −1) which are related via the Poisson identity, yielding the structure of a Gerstenhaber algebra.
On Hochschild cohomology this structure can be either induced using a localised version of the Hochschild cochain complex of an algebra [29,53], or the general machinery of Hochschild cohomology for dg categories [28].The product corresponds to the Yoneda product on self-extensions in (4), whilst the Gerstenhaber bracket [−, −] does not have a direct sheaf-theoretic interpretation in the definition (4).
For polyvector fields the product structure is given by the cup product in sheaf cohomology together with the wedge product of polyvector fields, whilst the Lie bracket is given by the Schouten bracket [−, −] S .In this case the Gerstenhaber algebra structure is even compatible with the bigrading.
The isomorphism used in Theorem 2.2 is not compatible with the Gerstenhaber algebra structures on both sides.This was remedied by Kontsevich (see [30,Claim 8.4] and [12,Theorem 5.1]) for the algebra structure and Calaque-Van den Bergh [11,Corollary 1.5] for the full Gerstenhaber algebra structure, by modifying it using the square root of the Todd class.We will denote the isomorphism HH • (X) ∼ = p+q=• H p (X, q T X ) of graded vector spaces obtained from Theorem 2.2 by I HKR .
Theorem 2.5 (Kontsevich, Calaque-Van den Bergh).We have an isomorphism of Gerstenhaber algebras (11) By describing the algebraic structure on polyvector fields we can therefore deduce properties of the algebraic structure on Hochschild cohomology of X.
We can identify certain interesting substructures: • the self-bracket [α, α] ∈ HH 3 (X) for α ∈ HH 2 (X) measures the obstruction to extending a first-order deformation of the abelian or derived category of coherent sheaves (classified by HH 2 (X), see [35,36]) to higher order, whilst on the polyvector fields and using the finer bigrading we have that: • measures the obstruction to extending a first-order deformation of the variety X to higher order in the Kodaira-Spencer deformation theory of varieties.For a Fano variety the latter obstruction vanishes as H 2 (X, T X ) = 0 by Kodaira-Akizuki-Nakano vanishing, see also Lemma 3.1.By [45] the Lie algebra Lie Aut(X) is non-trivial in many cases, and it would be interesting (but outside the scope of this article) to describe this aspect of the Gerstenhaber algebra structure.
There is also the self-bracket [π, π] S ∈ H 0 (X, 3 T X ) for π ∈ H 0 (X, 2 T X ), which we will now elaborate on.By Kodaira vanishing H 2 (X, O X ) will play no role in this article.
satisfying the axioms of a Poisson bracket; in particular it satisfies the Jacobi identity.It can also be encoded globally as a section π ∈ H 0 (X, 2 T X ), using the equality {f, g} = df ∧ dg, π obtained from the pairing between vector fields and differential forms.The vanishing of the Schouten bracket (12) [π, π] S = 0 ∈ H 0 (X, encodes the Jacobi identity for the corresponding Poisson structure.We will use the following terminology. Definition 2.6.Let X be a smooth projective variety.A Poisson structure on X is a bivector field π ∈ H 0 (X, 2 T X ) such that (12) holds.We denote ( 13) the subvariety of Poisson structures.
In general Pois(X) is cut out by homogeneous equations of degree 2, and one can also consider them up to rescaling, so that one is interested in P(Pois(X)) ⊆ P(H 0 (X, 2 T X )).There can be multiple irreducible components, of varying dimension.For an excellent introduction to Poisson structures, one is referred to [46].Let us just recall that Poisson structures are important to construct deformation quantisations, or noncommutative deformations, as e.g.explained in [9].
The classification of Poisson structures on smooth projective surfaces is done in [2], with the vanishing of the Schouten bracket being automatic for dimension reasons.The classification of Poisson structures Fano 3-folds of Picard rank 1 is summarised in [34, §9, Table 1].We don't need the full classification, let us just mention the following examples.
Example 2.7.By [34, §9, Table 1] we have that • for P 3 there are 6 irreducible components, of varying dimension; • in the family MM 1.10 there exists a unique member for which P(Pois(X)) is non-empty in P(H 0 (X, 2 T X )) ∼ = P 2 , in which case it is a point: the Mukai-Umemura 3-fold X MU for which Aut 0 (X MU ) = PGL 2 ; • in the family MM 1.9 we have for all As mentioned in [46, §3.4], the full classification of Poisson structures on Fano 3-folds of higher Picard rank is still open, and Corollary B gives the first step towards such a classification.

Computing the Hochschild cohomology of Fano 3-folds
In this section we discuss the aspects of the computation of Hochschild cohomology of Fano 3-folds which are common to all cases.After introducing some general results in Section 3.1 we will set up the computation in Section 3.2 and discuss the two approaches in Sections 3.3 and 3.4.For the remaining cases one is referred to Section 4.

General results.
The following lemma is straightforward, but significantly reduces the number of cohomologies one needs to compute for a Fano 3-fold.Lemma 3.1.Let X be a Fano 3-fold.Then ( 14) Proof.This is immediate from the Kodaira-Akizuki-Nakano vanishing In particular, the polyvector parallelogram introduced in Section 2.1 has the form ( 16) Next we describe the Euler characteristic of the vector bundles appearing in Lemma 3.1.Recall that Hirzebruch-Riemann-Roch for a vector bundle E on a 3-fold takes on the following form, where we abbreviate c i = c i (T X ): We obtain the following identities, expressing the Euler characteristic of the bundles we are interested in in terms of the usual invariants ρ, h 1,2 and c 3 1 in the classification of Fano 3-folds.Lemma 3.2.Let X be a Fano 3-fold.
For (19) we use that ( 23) so that reading off the degree three part of ch( 2 T X )td X gives ( 24) and the identity in (19) follows from the observations made in the previous paragraph.
This observation, together with the classification of infinite automorphism groups of Fano 3-folds (see [33,Theorem 1.1.2]for Picard rank 1, and [45, Theorem 1.2] for Picard rank ≥ 2), makes it straightforward to determine h 0 (X, T X ) and h 1 (X, T X ).Proposition 3.3.Let X be a Fano 3-fold.We have that (25) The computation of Aut 0 (X) can be found in [45, Table 1].It is important to note that the dimension of Aut(X) can vary in families.
For 2 T X we need to determine 3 possibly non-zero cohomologies, and none is known a priori.Some cases are easy (e.g. for toric Fano 3-folds Bott-Steenbrink-Danilov vanishing, see e.g.[43,Theorem 2.4], yields that H ≥1 (X, i T X ) = 0) but others take more effort.

3.2.
Setting up the computation.As discussed in the previous section, it suffices to compute the cohomology of 2 T X to fully determine the Hochschild cohomology of a Fano 3-fold.By Lemma 3.1 we know that its cohomology is concentrated in degrees 0, 1, 2.
To perform this computation we will use suitable descriptions of Fano 3-folds X inside key varieties F provided in [8,15].A key variety will be either a product of Grassmannians or a toric variety.In the former case X is given as the zero locus of a general global section of a homogeneous vector bundle E on F .In the latter case X is given as an intersection of divisors inside a possibly singular F .It turns out that this second description involves non-Cartier divisors only for MM 2.1 and MM 2.3 : this will lead us to deal with these two cases separately in Section 4.
The two methods outlined in this section allow for a near uniform treatment using computer algebra methods.We implemented them using Macaulay2 [19] and Magma [10]; our code is publicly available at [5] and can be used to check our computations.As it turns out, this automated treatment leaves the cohomology of 2 T underdetermined for only 5 Fano 3-folds, which require additional computations by hand (2 of which straightforward).These cases will be treated in Section 4.
Remark 3.4.For many deformation families of Fano 3-folds one can of course envision alternative methods, e.g. using descriptions as a blowup, double cover or product.We will not discuss the details for these alternative methods as they do not allow for an automated approach.One potential benefit (for certain applications) of these methods could be that they give a more intrinsic description of the cohomology.Let us just point out that they are used for 5 explicit instances in Section 4.
Setup and notation.Let us introduce some notation, which is also the notation we use in (the documentation of) the ancillary code.Let X be a Fano 3-fold (not of type MM 2.1 or MM 2.3 ), defined by the vanishing of a global section of a vector bundle E inside a key variety F with codim F X = rank E. By Theorems 3.5 and 3.9 the key variety can be chosen as either a product of Grassmannians or a (possibly singular) toric variety.We wish to compute the cohomology of ( 26) We will do this by using the conormal sequence, using that the ideal sheaf I cutting out X gives (I/I 2 )| X ∼ = E ∨ | X .Since X is smooth and locally complete intersection within F , one has that X ⊂ F sm , hence Ω 1 F | X is locally free.From [50, Tags 06AA and 0B3P] it follows that the conormal sequence (27) 0 X → 0 is an exact sequence of vector bundles on X.We will twist this sequence by the anticanonical bundle ω We are interested in computing the cohomologies of the last term of (28) 0 The first two terms can be resolved by suitable twists of the Koszul complex (29) 0 The whole point of this reduction is that the tensor product of i E ∨ with either of the first two bundles from (28) can now be expressed in terms of vector bundles on F for which good computational methods exist: • for toric varieties we can use the work of Eisenbud-Mustaţă-Stillman [17], as implemented in [48], even when the cotangent sheaf is not locally free; • for homogeneous varieties we can use the Borel-Weil-Bott theorem.

3.3.
Complete intersections in toric varieties.The majority of the cases will be covered by this method.The starting point is the following theorem, which follows from the case-by-case analysis performed in [15] for Picard ranks 2, . . ., 5, whilst for Picard ranks 1, 6, . . ., 10 it follows from the description using weighted projective spaces and del Pezzo surfaces.Theorem 3.5 (Coates-Corti-Galkin-Kasprzyk).Let X be a Fano 3-fold.Assume its deformation family is not of type Picard rank 1: MM So 90 (resp.92) out of 105 deformation families admit a description in terms of a toric variety F and a vector bundle E (resp.reflexive sheaf) so that we can use the combination of the Koszul sequence and the conormal sequence.We will restrict ourselves to the case where E is a vector bundle, and we will deal with the 2 remaining cases MM 2.1 , MM 2.3 using birational methods in Section 4. We remark that it is certainly possible to find suitable models for them as complete intersections of Cartier divisors in different toric varieties, but we did not manage to fully determine the cohomology of 2 T X in this way.In Table 1 we give an overview of the codimension of X in F , and whether the computational methods can give a fully determined answer for the cohomology of 2 T X .
• The case MM 1.1 can be easily determined from the toric computation together with Kodaira vanishing, see Proposition 4.1.• The case MM 4.13 can be computed using the description as a blowup, see Proposition 4.6.
• The case MM 10.1 readily follows from applying the Künneth formula to P 1 × dP 8 (Proposition 4.7).For MM 9.1 a similar argument using P 1 × dP 7 holds, but we chose to use its description as a homogeneous zero locus, see Table 2.
Remark 3.6.The description in [15] describes F as the GIT quotient of an affine space by a torus.To compute cohomology of coherent sheaves on the toric variety F we need to translate this description to a toric fan, and describe the divisors cutting out X in this language.See [15, §C] for some background.
An example: MM 2.8 .We now describe an example of a toric complete intersection, and the different steps in the computation.We will consider the deformation family MM 2.8 , whose Mori-Mukai description is given by (1) a double cover of Bl p P 3 with anticanonical branch locus B such that B ∩ E is smooth, (2) a double cover of Bl p P 3 with anticanonical branch locus B such that B ∩ E is singular but reduced, where E denotes the exceptional divisor of the blowup Bl p P 3 → P 3 , and the second is a specialisation of the first.Proposition 3.7.Let X be a Fano 3-fold in the deformation family MM 2.8 .Then we have that h i (X, 2 T X ) = 3, 1, 1 for i = 0, 1, 2.
Proof of Proposition 3.7.We want to compute the cohomology of the first two terms in the sequence (27) twisted by ω ∨ F (−2L − 2M ) (which is ω ∨ X before adjunction), so by the Koszul sequence we want to compute the cohomology of the first two terms in the sequences , 0, 0, 0, 0 for i = 0, . . ., 4, which implies the statement after a diagram chase.
Remark 3.8.The homogeneous description from [8] involves a vector bundle on P 2 × P 3 × P 12 which is not completely reducible, making the description as a toric complete intersection much more economical.

3.4.
Zero loci of sections of homogeneous vector bundles.By specialising [8, Theorems 1.1 and 1.2] to the remaining cases we can paraphrase the main result from op. cit.for the relevant subset as follows.
Theorem 3.9.Let X be a Fano 3-fold.Assume its deformation type is not covered by Theorem 3.5, or is MM 9.10 .Then X is the zero locus of a (general) global section of a completely reducible homogeneous vector bundle on a product of Grassmannians.The description is given in Table 2.

Table 2. Description as homogeneous zero loci
As explained in op.cit.this in fact holds for 85 out of 105 deformation families of Fano 3-folds, but we need it only for the 14 cases specified in Theorem 3.9 which are listed in Table 2. See also Remark 3.11.
An example: MM 2.17 .In this subsection we exhibit a detailed example of the computation where the Fano 3-fold does not admit (at least a priori) a model as a complete intersection in a suitable toric variety.We will use the description given in [15, §34] and [8, Table 1] and recalled in Table 2.The deformation family MM 2.17 , originally described by Mori and Mukai as the blow up of the quadric 3-fold in an elliptic quintic, is realised as the zero locus Z (E) ⊂ F := Gr(2, 4) × P 3 where (34) is a rank 4 vector bundle.
Proof.We will follow the strategy used in [8, §3.3] and summarised in Section 3.2.We need to compute the cohomologies of the first two terms in (28), which are resolved by exact complexes of locally free sheaves, namely the twists of the Koszul complex ( 29) by In this case we have ( 35) Each term of the locally free resolutions is a completely reducible vector bundle on F , and we can use the Borel-Weil-Bott theorem to compute its cohomology.It turns out that there are only 2 non-zero cohomology groups for the first two terms of (28) tensored with i E ∨ before restriction, for with i = 0, . . ., 4 = rank E, namely From this we get that the only non-zero cohomologies of the first two terms of ( 28) are (37) h 0 (X, ( and the statement follows.
Remark 3.11.It is possible to apply the description as a zero locus in a homogeneous variety to all Fano 3-folds, but for the purpose of this paper we only do this for the 14 cases listed in Table 2.
The benefit of the toric description is that the codimension is usually (much) lower, making the computation faster and having less places where indeterminacies can occur.E.g. for MM 3.9 the description from [8] has codimension 25, which requires a lengthy Koszul computation.Another complication in the computations in the homogeneous setting is that for some Fano 3-folds the homogeneous bundle used in the description is not completely reducible.

Underdetermined cases
As discussed above there are just a few cases which require additional computations.These are MM 1.1 , MM 2.1 , MM 2.3 , MM 4.13 , and MM 10.1 .We will collect the details for them here.For MM 2.1 , MM 2.3 and MM 4.13 they are somewhat tedious cohomology computations using the birational description.If it were not for the efficiency of the toric and homogeneous computations the majority of the Fano 3-folds would have to be tackled in this way.
The first one is straightforward.
Proof.In this case X is a sextic hypersurface in the weighted projective space P(1 4 , 3).Using the method from Section 3.3 on this description for the toric variety P(1 4 , 3) we immediately obtain that h i (X, 2 T X ) = 0, 0, 35 + a, a for i = 0, 1, 2, 3 for some a ≥ 0. But by Lemma 3.1 we have that h 3 (X, 2 T X ) = 0, so a = 0.Alternatively, one can use that this is a double cover f : X → Y of P 3 = Y with a smooth sextic surface S as branch locus.To do so, recall the short exact sequence (38) 0 ).This allows one to compute H i (X, Ω 1 X ⊗ ω ∨ X ), and the only non-vanishing cohomology lives in degree 2 and is isomorphic to which is 35-dimensional.
For the next three cases we will resort to the birational description by Mori-Mukai.Namely we will consider the situation of a Fano 3-fold X which is the blowup of a complete intersection curve Z inside another Fano 3-fold Y .Let us denote the blowup square as We wish to compute the cohomology of the middle term after twisting by ω e. we will consider the short exact sequence (44) 0 With the setup from (42) we have that Proof.Since Z has codimension 2, we have that which is what we wanted to show.Corollary 4.3.With the setup from (42) we have that Proof.This follows from (44), the vanishing in Lemma 4.2, the isomorphism Rf * (O X (−E)) ∼ = I Z , and adjunction.
We now consider the underdetermined cases MM 2.1 and MM 2.3 .In this case the methods of Section 3.3 don't necessarily apply: both are described as a codimension-2 complete intersection in a singular toric projective variety, and in both cases one of the divisors is not Cartier.Therefore we cannot ensure that the computational (underdetermined) answer is correct2 , so we will combine Corollary 4.3 with Dolgachev's computation of sheaf cohomology on weighted projective spaces.We will repeatedly make use of Dolgachev's formulae for twisted Hodge numbers on weighted projective spaces, cf.[16, §2.3.2-2.3.5].
Proof.Such a variety is described as the blowup of a Fano 3-fold Y of deformation type MM 1.11 in an elliptic curve obtained as complete intersection of two half-anticanonical divisors, and Y is given as a sextic hypersurface in the weighted projective space P := P(1 3 , 2, 3).Alternatively, X is a (1, 1)-section of F := Y × P 1 .We will first use this description to determine h 0 and h 2 − h 1 , and then use the blowup description to determine h 2 (and h 1 − h 0 ), which determines everything.
First step: weighted projective space computation.Consider the conormal sequence for the inclusion X → F twisted by the anticanonical line bundle, which by the adjunction formula is ( 48) together with (49), so let us prove this.Consider the twisted Koszul sequence (51) 0 F is the direct sum of the pullbacks of the cotangent bundles of Y and P 1 , one readily gets from the Hodge numbers of P 1 and MM 1.11 that h i (F, Ω 1 F ) = 0, 2, 21, 0, 0 for i = 0, 1, Thus it remains to compute the cohomologies of the middle term of (51).To do that, we apply the Künneth formula to ( 52) ; the second term is acyclic since O P 1 (−1) is, whilst for the first one we get (53) ).To compute the latter, we consider the twisted conormal sequence for Y , seen as a sextic hypersurface in P: | Y can be obtained as usual by means of a twisted Koszul complex (56) yields the cohomologies for the first two terms, whence we deduce that (57) We plug the last equalities into (55) and get for some integer α ≥ 0. By ( 53) and ( 51) we get (59) for some integers α, β ≥ 0. But by (49) and Lemma 3.1 we get 2α = h 3 (X, 2 T X ) = 0, hence the claim.
Second step: blowup computation.We consider the short exact sequence cutting out Z inside Y and tensor it with The cohomology of the middle term is determined by the methods in Section 3.3 and is given by To compute the cohomology of the third term, consider the conormal sequence for As Z is cut out by two half-anticanonical divisors, we obtain that ⊕2 .It now suffices to compute the cohomology of the line bundles O Z (1) and O Z (2), which is concentrated in degree 0 by degree reasons, and is determined by a Hilbert series computation on the weighted projective space.We obtain h i (Z, (Ω 1 Y ⊗ ω ∨ Y )| Z ) = 4, 0 for i = 0, 1. Combining this with (61) we get that Together with (50) and Corollary 4.3 this finishes the computation.
Next we tackle the underdetermined case MM 2.3 in exactly the same way.In the proof we will explain which details of the computation change.
Proof.Such a variety is described as the blowup of a Fano 3-fold Y of deformation type MM 1.12 in an elliptic curve obtained as complete intersection of two half-anticanonical divisors, and Y is given as a quartic hypersurface in the weighted projective space P := P(1 4 , 2).Alternatively, X is a (1, 1)-section of )) is analogous to the two-step procedure from Proposition 4.4, so we only summarise the relevant differences in the numerology appearing.
First step: weighted projective space computation (abbreviated).This goes along the same lines as the first step in the proof of Proposition 4.4, except that the claim (50) now becomes From the Hodge numbers of P 1 and MM 1.12 we now get h i (F, Ω 1 F ) = 0, 2, 10, 0, 0 for i = 0, 1, 2, 3, 4. The twisted conormal sequence for Y , which is a quartic hypersurface in P, gives h 3 (Y, O Y (−3)) = 4.We obtain ) = α for some integer α ≥ 0. As in the proof of Proposition 4.4 we get (66) for some integers α, β ≥ 0. But by Lemma 3.1 we get 2α = h 3 (X, 2 T X ) = 0, hence the claim.
Second step: blowup computation.This goes along the same lines as the second step in the proof of Proposition 4.4, except that (61) now becomes (67) 0, 1 for i = 0, 1, 2 and that the Hilbert series computation is performed for an elliptic curve in the weighted projective space P(1 4 , 2).We obtain Combining this with (67) we get that (68) Together with (64) and Corollary 4.3 this finishes the computation.
The next underdetermined case is the Fano 3-fold which was originally missed in the classification [41].The method is similar to what we did for MM 2.1 and MM 2.3 using the birational description, but requires less work.Proposition 4.6.Let X be a Fano 3-fold in the deformation family MM 4.13 .Then we have that h i (X, 2 T X ) = 4, 0, 0 for i = 0, 1, 2.
Proof.Such a variety is described as the blowup of the Fano 3-fold Y = P 1 × P Denote by p i,j : Y → P 1 × P 1 the three projections.Using the isomorphism ( 69) and the projection formula we want to compute the cohomology of (70) In the case of jumping of h i (X, T X ), we have given the lowest value, and indicated with a * next to the value h 0 (X, T X ) that jumping occurs.This information (and more) is also available in a more interactive way at [3].