Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences


    We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

    1. Introduction

    Fano manifolds are the basic building blocks of algebraic geometry, both in the sense of the Minimal Model Program [14] and as the ultimate source of most explicit examples and constructions. There are finitely many deformation families of Fano manifolds in each dimension [5]. There is precisely 1 one-dimensional Fano manifold: the line; there are 10 deformation families of two-dimensional Fano manifolds: the del Pezzo surfaces and there are 105 deformation families of three-dimensional Fano manifolds [613]. Very little is known about the classification of Fano manifolds in higher dimensions.

    In this paper, we begin to explore the geography of Fano manifolds in dimension 4. Four-dimensional Fano manifolds of higher Fano index have been classified [6,1422]—there are 35 in total—but the most interesting case, where the Fano variety has index 1, is wide open. We use computer algebra to find many four-dimensional Fano manifolds that arise as complete intersections in toric Fano manifolds in codimension at most 4. We find at least 738 examples, 717 of which have Fano index 1 and 527 of which are new.

    Suppose that Y is a toric Fano manifold and that L1,…,Lc are nef line bundles on Y such that −KYΛ is ample, where Λ=c1(L1)+⋯+c1(Lc). Let XY be a smooth complete intersection defined by a regular section of ⊕iLi. The Adjunction Formula gives that

    so X is Fano. We find all four-dimensional Fano manifolds X of this form such that the codimension c is at most 4.

    Our interest in this problem is motivated by a program to classify Fano manifolds in higher dimensions using mirror symmetry [23]. For each four-dimensional Fano manifold X as above, therefore, we compute the essential ingredients for this program: the quantum period and regularized quantum differential equation associated to X, and a Laurent polynomial f that corresponds to X under mirror symmetry; we also calculate basic geometric data about X, the ambient space Y and f. The results of our computations in machine-readable form, together with full details of our implementation and all source code used, can be found in the electronic supplementary material.

    2. Finding four-dimensional fano toric complete intersections

    Our method is as follows. Toric Fano manifolds Y are classified up to dimension 8 by Batyrev et al.1 [2429]. For each toric Fano manifold Y of dimension d=4+c, we:

    (i) compute the nef cone of Y;

    (ii) find all ΛH2(Y;Z) such that Λ is nef and −KYΛ is ample and

    (iii) decompose Λ as the sum of c nef line bundles L1,…,Lc in all possible ways.

    Each such decomposition determines a four-dimensional Fano manifold XY , defined as the zero locus of a regular section of the vector bundle ⊕iLi. To compute the nef cone in step (i), we consider dual exact sequences

    Inline Graphic

    where the map ρ is defined by the N rays of a fan Σ for Y . There are canonical identifications LH2(Y;Z)Pic(Y), and the nef cone of Y is the intersection of cones

    where Di is the image under D of the ith standard basis vector in (ZN) [30, proposition 15.1.3]. The classes Λ in step (ii) are the lattice points in the polyhedron P=NC(Y)∩(−KY−NC(Y)) such that −KYΛ lies in the interior of NC(Y). Since NC(Y) is a strictly convex cone, P is compact and the number of lattice points in P is finite. We implement step (iii) by first expressing Λ as a sum of Hilbert basis elements in NC(Y) in all possible ways
    Λ=b1++brbi an element of the Hilbert basis for NC(Y),2.1
    where some of the bi may be repeated; this is a knapsack-style problem. We then, for each decomposition (2.1), partition the bi into c subsets S1,…,Sc in all possible ways and define the line bundle Li to be the sum of the classes in Si.

    We found 117 173 distinct triples (X;Y ;L1,…,Lc), with a total of 17 934 distinct ambient toric varieties Y . Note that the representation of a given Fano manifold X as a toric complete intersection is far from unique: for example, if X is a complete intersection in Y given by a section of L1⊕⋯⊕Lc then it is also a complete intersection in Y×P1 given by a section of π1L1π1Lcπ2OP1(1). Thus, we have found far fewer than 117 173 distinct four-dimensional Fano manifolds. We show below, by calculating quantum periods of the Fano manifolds X, that we find at least 738 non-isomorphic Fano manifolds. Since the quantum period is a very strong invariant—indeed no examples of distinct Fano manifolds XX′ with the same quantum period GX=GX are known—we believe that we found precisely 738 non-isomorphic Fano manifolds. Eliminating the quantum periods found in [31], we see that at least 527 of our examples are new.

    Remark 2.1

    There exist Fano manifolds which do not occur as complete intersections in toric Fano manifolds. But in low dimensions, most Fano manifolds arise this way: 8 of the 10 del Pezzo surfaces, and at least 78 of the 105 smooth three-dimensional Fano manifolds, are complete intersections in toric Fano manifolds [32].

    Remark 2.2

    It may be the case that any d-dimensional Fano manifold which occurs as a toric complete intersection in fact occurs as a toric complete intersection in codimension d; we know of no counterexamples. But even if this holds in dimension 4, our search will probably not find all four-dimensional Fano manifolds which occur as toric complete intersections. This is because, if one of the line bundles Li involved is nef but not ample, then the Kähler cone for X can be strictly bigger than the Kähler cone for Y . In other words, it is possible for −KX to be ample on X even if −KYΛ is not ample on Y . For an explicit example of this in dimension 3, see [32, §55].

    3. Quantum periods and mirror laurent polynomials

    The quantum period GX of a Fano manifold X is a generating function

    for certain genus-zero Gromov–Witten invariants cd of X which plays an important role in mirror symmetry. A precise definition can be found in [32, §B], but roughly speaking one can think of cd as the ‘virtual number’ of rational curves C in X that pass through a given point, satisfy certain constraints on their complex structure, and satisfy 〈−KX,C〉=d. The quantum period is discussed in detail in [23,32]; for us what will be important is that the regularized quantum period
    satisfies a differential equation called the regularized quantum differential equation of X:
    where the pm are polynomials and D=t(d/dt).

    It has been proposed that Fano manifolds should correspond under mirror symmetry to Laurent polynomials which are extremal or of low ramification [23], in the sense discussed in §4. An n-dimensional Fano manifold X is said to be mirror-dual to a Laurent polynomial fC[x1±1,,xn±1] if the regularized quantum period of X coincides with the classical period of f:

    If a Fano manifold X is mirror-dual to the Laurent polynomial f then the regularized quantum differential equation (3.3) for X coincides with the Picard–Fuchs differential equation satisfied by πf. The correspondence between Fano manifolds and Laurent polynomials is not one-to-one, but it is expected that any two Laurent polynomials f, g that are mirror-dual to the same Fano manifold are related by a birational transformation φ:(C×)n(C×)n called a mutation or a symplectomorphism of cluster type [3335]: φf=g. We will write such a mutation as fφg. Mutations are known to preserve the classical period [33]: if fφg then πf=πg.

    Remark 3.1

    In the paragraphs above we discuss the regularized quantum differential equation and the Picard–Fuchs differential equation. This involves choices of normalization. Our conventions are that the regularized quantum differential operator is the operator LX as in (3.3) such that:

    (i) the order, N, of LX is minimal;

    (ii) the degree of pN(t) is minimal;

    (iii) the leading coefficient of pN is positive; and

    (iv) the coefficients of the polynomials p0,…,pN are integers with greatest common divisor equal to 1.

    The Picard–Fuchs differential operator is the differential operator Lf such that:

    where the Pm are polynomials and D=t(d/dt), and that the analogues of conditions (i)–(iv) above hold.

    We determined the quantum period GX, for each of the triples (X;Y ;L1,…,Lc) from §2, as follows. For each such triple we found, using the Mirror Theorem for toric complete intersections [36] and a generalization of a technique due to V. Przyjalkowski, a Laurent polynomial f that is mirror-dual to X. This process is described in detail in §5. We then computed, for each triple, the first 20 terms of the power series expansion of G^X=πf using the Taylor expansion

    where αd is the coefficient of the unit monomial in fd. We divided the 117 173 triples into 738 ‘buckets’, according to the value of the first 20 terms of the power series expansion of G^X=πf, and then proved that any two Fano manifolds X, X′ in the same bucket have the same quantum period by exhibiting a chain of mutations fφ0f1φ1φn1fnφng that connects the Laurent polynomials f and g mirror-dual to X and X′.

    For each quantum period GX, we computed the quantum differential operator LX directly from the mirror Laurent polynomial f chosen above, using Lairez’s generalized Griffiths–Dwork algorithm [37]. The output from Lairez’s algorithm is a differential operator L=m=0NPm(t)Dm with P0,,PNQ[t] such that, with very high probability, f≡0. Such an operator L gives a recurrence relation for the Taylor coefficients α0,α1,α2… of πf; using this recurrence relation and the first 20 Taylor coefficients computed above, we solved for the first 2000 Taylor coefficients αk. We then consider an operator

    where the P¯m are polynomials of degree at most R¯, and impose the condition that L¯πf0. The 2000 Taylor coefficients of πf give 2000 linear equations for the coefficients of the polynomials P¯m and, provided that (N¯+1)(R¯+1)2000, this linear system is highly over-determined. Since we are looking for the Picard–Fuchs differential operator (see remark 3.1), we may assume that (N¯,R¯) is lexicographically less than (N,degpN). We searched systematically for such differential operators with (N¯+1)(R¯+1)2000, looking for the operator L¯ with lexicographically minimal (N¯,R¯) and clearing denominators so that the analogues of conditions (iii) and (iv) in remark 3.1 holds. We can say with high confidence that this operator L¯ is in fact the Picard–Fuchs operator Lf, although this is not proven—partly because Lairez’s algorithm relies on a randomized interpolation scheme that is not guaranteed to produce an operator annihilating πf, and partly because if Lf were to involve polynomials Pm of extremely large degree, 2000 terms of the Taylor expansion of πf will not be enough to detect Lf. The operators L¯ that we found satisfy a number of delicate conditions that act as consistency checks: for example they are of Fuchsian type (which is true for Lf, as Lf arises geometrically from a variation of Hodge structure). Thus, we are confident that L¯=Lf in every case.2 Since G^X=πf and LX=Lf by construction, this determines, with high confidence, the quantum period GX and the regularized quantum differential operator LX.

    Remark 3.2

    The use of Laurent polynomials and Lairez’s algorithm is essential here. There is a closed formula [32, corollary D.5] for the quantum period of the Fano manifolds that we consider, and one could in principle use this together with the linear algebra calculation described above to compute (a good candidate for) the regularized quantum differential operator LX. In practice, however, for many of the examples that we treat here, it is impossible to determine enough Taylor coefficients from the formula: the computations involved are well beyond the reach of current hardware, both in terms of memory consumption and runtime. By contrast, our approach using mirror symmetry and Lairez’s algorithm will run easily on a desktop PC.

    Remark 3.3

    The regularized quantum differential equation for X coincides with the (unregularized) quantum differential equation for an anticanonical Calabi–Yau manifold ZX. The study of the regularized quantum period from this point of view was pioneered by Batyrev et al. [39,40] and an extensive study of fourth-order Calabi–Yau differential operators was made in [41]. We found 26 quantum differential operators with N=4; these coincide with or are equivalent to the fourth-order Calabi–Yau differential operators with AESZ IDs 1, 3, 4, 5, 6, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 369, 370 and 424 in the Calabi–Yau Operators Database [42], together with one new fourth-order Calabi–Yau differential operator (which corresponds to our period sequence with ID 469).

    4. Ramification data

    Consider now one of our regularized quantum differential operators

    as in (3.3), and its local system VP1S of solutions. Here SP1 is the set of singular points of the regularized quantum differential equation.

    Definition 4.1 ([23])

    Let SP1 be a finite set and VP1S a local system. Fix a basepoint xP1S. For sS, choose a small loop that winds once anticlockwise around s and connect it to x via a path, thereby making a loop γs about s based at x. Let Ts:VxVx denote the monodromy of V along γs. The ramification of V is


    The ramification rf(V) is independent of the choices of basepoint x and of small loops γs. A non-trivial, irreducible local system VP1S has rf(V)2rk(V): see [23, §2].

    Definition 4.2

    Let VP1S be a local system as above. The ramification defect of V is the quantity rf(V)2rk(V). A local system of ramification defect zero is called extremal.

    Definition 4.3

    The ramification (respectively, ramification defect) of a differential operator LX is the ramification (respectively, ramification defect) of the local system of solutions LXf≡0.

    To compute the ramification of LX, we proceed as in [31]. One can compute Jordan normal forms of the local log-monodromies {logTs:sS} using linear algebra over a splitting field k for pN(t). (Every singular point of LX is defined over k.) This is classical, going back to Birkhoff [43], as corrected by Gantmacher [44, vol. 2, §10] and Turrittin [45]; a very convenient presentation can be found in the book of Kedlaya [46, §7.3]. In practice, we use the symbolic implementation of Q¯ provided by the computational algebra system Magma [47,48]. We computed ramification data for 575 of the 738 regularized quantum differential operators, finding ramification defects as shown in table 1; this lends some support to the conjecture, due to Golyshev [23], that a Laurent polynomial f which is mirror-dual to a Fano manifold should have a Picard–Fuchs operator Lf that is extremal or of low ramification. For the remaining 163 regularized quantum differential operators, the symbol pN(t) contains a factor of extremely high degree. This makes the computation of ramification data prohibitively expensive.

    Table 1.Ramification defects for 575 of the 738 regularized quantum differential operators.

    ramification defect 0 1 2 3
    number of occurrences 92 290 167 26

    5. The przyjalkowski method

    We now explain, given complete intersection data (X;Y ;L1,…,Lc) as in §2, how to find a Laurent polynomial f that is mirror-dual to X. This is a slight generalization of a technique that we learned from V. Przyjalkowski3 [50,51], and which is based on the mirror theorems for toric complete intersections due to Givental [36] and Hori et al. [52]. Recall the exact sequence

    Inline Graphic

    from §2 and the elements DiL, 1≤iN, defined by the standard basis elements of (ZN). Recall further that LPic(Y), so that each line bundle Lm defines a class in L. Suppose that there exists a choice of disjoint subsets E,S1,…,Sc of {1,2,…,N} such that:

    — {Dj:jE} is a basis for L;

    — each Lm is a non-negative linear combination of {Dj:jE}; and

    — kSmDk=Lm for each m∈{1,2,…,c};

    and choose distinguished elements smSm, 1≤mc. Set Sm=Sm{sm}. Writing the map D in terms of the standard basis for (ZN) and the basis {Dj:jE} for L defines an (NdN matrix (mji) of integers. Let (x1,…,xN) denote the standard coordinates on (C×)N, let r=Nd, and define q1,…,qr and F1,…,Fc by

    Givental [36] and Hori et al. [52] have shown that
    where W=x1+⋯+xN and Γ is a certain cycle in the submanifold of (C×)N defined by
    Introducing new variables yi for im=1cSm, setting
    xi={11+kSmykif i=smyi1+kSmykif iSm.
    and using the relations q1=⋯=qr=1 to eliminate the variables xj, jE, allows us to write Wc as a Laurent polynomial f in the variables
    {yi:im=1cSm}and{xi:iE and im=1cSm}.
    The mirror theorem (5.1) then implies that G^X=πf, or in other words that f is mirror-dual to X.

    The Laurent polynomial f produced by Przyjalkowski’s method depends on our choices of E,S1,…,Sc, and s1,…,sc, but up to mutation this is not the case:

    Theorem 5.1 ([53])

    Let Y be a toric Fano manifold and let L1,…,Lc be nef line bundles on Y such that −KY−Λ is ample, where Λ=c1(L1)+⋯+c1(Lc). Let X⊂Y be a smooth complete intersection defined by a regular section of ⊕iLi. Let f and g be Laurent polynomial mirrors to X obtained by applying Przyjalkowski’s method to (X;Y ;L1,…,Lc) as above, but with possibly different choices for the subsets E,S1,…,Sc and the elements s1,…,sc. Then there exists a mutation φ such that fφg.

    Example 5.2

    Let Y be the projectivization of the vector bundle O2O(1)2 over P2. Choose a basis for the two-dimensional lattice L such that the matrix (mji) of the map D is

    Consider the line bundle L1Y defined by the element (2,1)L, and the Fano hypersurface XY defined by a regular section of L1. Applying Przyjalkowski’s method to the triple (X;Y ;L1) with E={3,4}, S1={1,2,5} and s1=1 yields the Laurent polynomial
    mirror-dual to X. Applying the method with E={3,4}, S1={1,6} and s1=1 yields
    We have that fφg where the mutation φ:(C×)4(C×)4 is given by

    Remark 5.3

    Observe that, for a complete intersection of dimension n and codimension c, Przyjalkowski’s method requires partitioning n+c variables into c disjoint subsets. If (n+c)/c<2 then at least one of the subsets must have size one and so the corresponding variable, xj say, is eliminated from the Laurent polynomial via the equation xj=1. One could therefore have obtained the resulting Laurent polynomial from a complete intersection with smaller codimension: new Laurent polynomials are found only when (n+c)/c≥2, that is, when the codimension is at most the dimension. In particular, all possible mirrors to four-dimensional Fano toric complete intersections given by the Przyjalkowski method occur for complete intersections in toric manifolds of dimension at most 8.

    6. Examples

    (a) The cubic fourfold

    Let X be the cubic fourfold. This arises in our classification from the complete intersection data (X;Y ;L) with Y=P5 and L=OP5(3). The Przyjalkowski method yields [54, §2.1] a Laurent polynomial

    mirror-dual to X, and elementary calculation gives
    Thus G^X=πf, and the corresponding regularized quantum differential operator is
    The local log-monodromies for the local system of solutions LXg≡0 are
    ( 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0)at t=0 ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0)at t=19 ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0)at the roots of 81t2+9t+1=0 ( 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 )at t=
    and the operator LX is extremal.

    (b) A (3,3) complete intersection in P6

    Let X be a complete intersection in Y=P6 of type (3,3). This arises in our classification from the complete intersection data (X;Y ;L1,L2) with L1=L2=OP6(3). The Przyjalkowski method yields a Laurent polynomial

    mirror-dual to X, and [32, corollary D.5] gives
    The corresponding regularized quantum differential operator LX is
    The local log-monodromies for the local system of solutions LXg≡0 are
    ( 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0)at t=0 ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0)at t=1693 ( 23 1 0 0 0 23 0 0 0 0 13 1 0 0 0 13)at t=136
    and so the operator LX is extremal.

    7. Results and analysis

    We close by indicating how basic numerical invariants—degree and size of cohomology—vary across the 738 families of Fano manifolds that we have found. The degree (−KX)4 varies from 5 to 800, as shown in figures 1 and 2.

    Figure 1.

    Figure 1. The distribution of degrees (frequency plot).

    Figure 2.

    Figure 2. The distribution of degrees (cumulative frequency plot).

    We do not have direct access to the size of the cohomology algebra of our Fano manifolds X, as many of the line bundles occurring in the complete intersection data (X;Y ;L1,…,Lc) are not ample and so the Lefschetz theorem need not apply. But the order N of the regularized quantum differential operator is a good proxy for the size of the cohomology. N is the rank of a certain local system—an irreducible piece of the Fourier–Laplace transform of the restriction of the Dubrovin connection (in the Frobenius manifold given by the quantum cohomology of X) to the line in H•(X) spanned by −KX—and in the case where this local system is irreducible, which is typical, N will coincide with the dimension of H(X). For our examples, N lies in the set {4,6,8,10,12}. Figure 3 shows how N varies with the degree (−KX)4, with darker grays indicating a larger number of examples with that N and degree.

    Figure 3.

    Figure 3. The distribution of degrees with N.

    The isolated example on the right of figure 3, with N=6 and degree 800, is the blow up of P(1,1,1,1,3) at a point. Figure 4 again shows how N varies with the degree (−KX)4, but this time with toric Fano manifolds highlighted in red. Figure 5 shows how the Euler number χ(TX) varies with the degree (−KX)4, with darker grays indicating a larger number of examples with that Euler number and degree. The three examples with the largest Euler number χ are a quintic hypersurface in P5, with χ=825; a complete intersection of type (2,4) in P6, with χ=552; and a complete intersection of type (3,3) in P6, with χ=369. The three examples with the most negative Euler number are P1×V43 where V43 is a quartic hypersurface in P4, with χ=−112; P1×V63 where V63 is a complete intersection of type (2,3) in P5, with χ=−72; and P1×V83 where V83 is a complete intersection of type (2,2,2) in P6, with χ=−48.

    Figure 4.

    Figure 4. The distribution of degrees with N, with toric Fano manifolds highlighted.

    Figure 5.

    Figure 5. The distribution of degrees with Euler number.

    Ethics statement

    This research did not involve human or animal subjects.

    Data accessibility

    The Electronic Supplementary Material contains the results of our computations, in machine readable form, together with full source code written in Magma [47]. See the files called README.txt for details. The source code and data, but not the text of this paper, are released under a Creative Commons CC0 license (Creative Commons CC0 license, and see the files called COPYING.txt for details. If you make use of the source code or data in an academic or commercial context, you should acknowledge this by including a reference or citation to this paper.


    The computations underlying this work were performed using the Imperial College High Performance Computing Service and the compute cluster at the Department of Mathematics, Imperial College London. We thank Simon Burbidge, Matt Harvey and Andy Thomas for valuable technical assistance. We thank John Cannon and the Computational Algebra Group at the University of Sydney for providing licenses for the computer algebra system Magma. We thank Alessio Corti for a number of very useful conversations, Pierre Lairez for explaining his generalized Griffiths–Dwork algorithm and sharing his code with us, and Duco van Straten for his analysis of the regularized quantum differential operators with N=4.

    Author contributions

    T.C., T.P. and A.K. performed the research and wrote the paper.

    Funding statement

    This research was supported by a Royal Society University Research Fellowship; the Leverhulme Trust; ERC Starting Investigator Grant number 240123; EPSRC grant EP/I008128/1; an EPSRC Small Equipment grant; and an EPSRC Prize Studentship.

    Competing interests

    We have no competing interests.


    1 Øbro’s method, which applies in all dimensions, starts with the observation that without loss of generality one cone in the fan for Y is the positive orthant and the sum of the primitive generators v1,…,vN for the rays of the fan lies in this cone; he then uses the facts that Y is smooth and Fano to give bounds on the coordinates of the vi, and defines an order on such fans with the property that if F1<F2 then the toric Fano varieties defined by F1 and F2 are non-isomorphic. By constructing the fans in increasing order, one never needs to check for fan isomorphism, and thus one never needs to refer back to fans already found.

    2 This could be proved in any given case using methods of van Hoeij [38]; cf. [37, §8.2.2].

    3 Przyjalkowski informs us that he learned this, for the case of the cubic threefold, from Katzarkov [49] and Orlov (V. Przyjalkowski 2014, personal communication).

    © 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License, which permits unrestricted use, provided the original author and source are credited.