Birational Geometry via Moduli Spaces

Abstract

In this paper we connect degenerations of Fano threefolds by projections. Using mirror symmetry we transfer these connections to the side of Landau–Ginzburg models. Based on that we suggest a generalization of Kawamata’s categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau–Ginzburg models. We suggest a conjectural application to the Hassett–Kuznetsov–Tschinkel program, based on new nonrationality “invariants”—gaps and phantom categories. We formulate several conjectures about these invariants in the case of surfaces of general type and quadric bundles.

Mathematics Subject Classification codes (2000): 14M25, 14H10, 14Q15, 14D07

1 Introduction

1.1 Moduli Approach to Birational Geometry

In recent years, we have witnessed many significant developments in the Minimal Model Program (MMP) (see, e.g., [10]), based on major advances in the study of singularities of pairs. An alternative approach to MMP, relying on the correspondence between Mori fibrations (MF) and semiorthogonal decompositions (SOD), was proposed by Kawamata [67]; it does not involve the study of discrepancies or the effective cone.

In the meantime, a new epoch, the epoch of wall-crossing, has emerged. At present, after papers of Seiberg–Witten [91], Gaiotto–Moore–Neitzke [40], and Cecotti–Vafa [20] and seminal works by Donaldson–Thomas [33], Joyce–Song [62], Maulik–Nekrasov–Okounkov–Pandharipande [79, 80], Douglas [35], Bridgeland [13], and Kontsevich–Soibelman [72, 73], the situation with the wall-crossing phenomenon is quite similar to the situation with the theory of Higgs Bundles after Higgs and Hitchin: it was clear that a general “Hodge type” theory existed and that it needed to be developed. This led to major mathematical applications, for example, in uniformization and in the Langlands program, to mention a few. Similarly, in the framework of wall-crossing, it is also apparent that a “Hodge type” theory needs to be developed in order to reap mathematical benefits, e.g., solve some long-standing problems in algebraic geometry.

Already, several attempts have been made to connect homological mirror symmetry (HMS) to birational geometry. For instance, the paper [1] was motivated by the idea that a proof of HMS is likely to involve a detailed study of birational transformations (including, noncommutative) on the A and B sides of HMS. Later, new ideas were introduced in [4, 27, 28, 45, 68, 70]. These can be summarized as follows:

• 1. The moduli spaces of stability conditions of Fukaya–Seidel categories can be included in a one-parametric family with the moduli space of Landau–Ginzburg (LG) models as a central fiber.

• 2. The moduli space of LG models determines the birational geometry; this was proved in the toric case in [27].

In this paper, we simultaneously consider all Mori fiber spaces, all Sarkisov links and all relations between Sarkisov links, and explore these in terms of the geometry of moduli space of LG models and the moving schemes involved.

We introduce the following main ideas:

• 1. All Fano varieties are connected, via their degenerations, by simple basic links—projections of a special kind. We show this for Fano threefolds of Picard rank 1 (see Table 2). This agrees with their toric Landau–Ginzburg models (see Theorem 3.12).

• 2. All Fano manifolds can be considered together, i.e., there exists a big moduli space of LG models, which includes mirrors of all Fano’s. We demonstrate this partially for two- and three-dimensional Fano’s.

• 3. We introduce an analog of the canonical divisor measure for a minimal model. For us, this is the local geometry of singularities and a fiber at infinity of the LG model. The last one affects the geometry of moduli spaces of LG models, the stability conditions. We propose local models for these moduli spaces (stacks). The correspondence between the usual and the categorical approach to birational geometry is displayed in Table 1.

  Table 1 Extended Kawamata program

  Table 2 Weak Landau–Ginzburg models for Fano threefolds

• 4. Following [28] and the pioneering work [12], we develop the notion of phantom category, emphasizing its connection with the notion of a moving scheme, defined in this paper. Moving schemes determine the geometry of the moduli space of LG models and thus the geometry of the initial manifold. We conjecture:

  • A. For classical surfaces of general type (Campedelli, Godeaux, Burniat, Dolgachev surfaces, quotients of products of higher-genus curves and fake ${\mathbb{P}}^{2}$s) there should exist quasi-phantoms in their SOD, i.e., nontrivial categories with trivial Hochschild homology. These surfaces are clearly not rational as they have nontrivial fundamental groups, but also since, conjecturally, they have quasi-phantom subcategories in their SOD. On the Landau–Ginzburg side, these quasi-phantoms are described by the moving scheme. The deformation of the Landau–Ginzburg models is determined by the moving scheme so the quasi-phantoms factor in the geometry of the Landau–Ginzburg models. On the mirror side, this translates to the fact that the Ext groups between the quasi-phantom and the rest of the SOD determine the moduli space.

  • B. For surfaces of general type with trivial fundamental groups (e.g., the Barlow surface, see [28]), we conjecture the existence of a nontrivial phantom category, a category with a trivial K ⁰ group. The deformation of the Landau–Ginzburg models is determined by the moving scheme, so the phantom factors in the geometry of the Landau–Ginzburg models. Similarly, on the mirror side this translates to the fact that the Ext groups between the phantom and the rest of the SOD determine the moduli space.

  • C. We relate the existence of such phantom categories with nonrationality questions. For surfaces, it is clear that phantoms lead to nonrationality. For threefolds, we exhibit examples (of Sarkisov type) of nonrational threefolds where phantoms conjecturally imply nonrationality. We introduce “higher” categorical invariants detecting nonrationality: gaps of spectra. Conjecturally, these are not present in the Sarkisov examples. These ideas are a natural continuation of [50].

• 5. We introduce conjectural invariants associated to our moduli spaces—gaps and local differentials. We suggest that these numbers (changed drastically via “wall-crossing”) produce strong birational invariants. We relate these invariants to the “Hassett–Kuznetsov–Tschinkel” program (see [46, 77])—a program for studying rationality of the four-dimensional cubic and its “relatives.”

This paper is organized as follows. In Sects. 2 and 3 we relate degenerations of Fano manifolds via projections. Using mirror symmetry in Sect. 3 we transfer these connections to the side of Landau–Ginzburg model. Based on that, in Sect. 4 we suggest a generalization of Kawamata’s categorical approach to birational geometry, enhancing it via the geometry of moduli spaces of Landau–Ginzburg models. We give several applications, most notably a conjectural application to the Hassett–Kuznetsov–Tschinkel program. Our approach is based on two categorical nonrationality invariants—phantoms and gaps. Full details will appear in a future paper.

Notations. Smooth del Pezzo threefolds (smooth Fano threefolds of index 2) are denoted by V _n , where n is its degree with respect to a Picard group generator, except for the quadric denoted by Q. Fano threefolds of Picard rank 1, index 1, and degree n are denoted by X _n . The remaining Fano threefolds are denoted by X _k. m , where k is a Picard rank of a variety and m is its number according to [57]. The Laurent polynomials from the k-th line of Table 2 are denoted by f _k . A toric variety whose fan polytope is a Newton polytope of f _k is denoted by F _k or just a variety number k. The Landau–Ginzburg model for a variety X is denoted by LG(X).

2 “Classical” Birational Geometry

In this section we recall some facts from classical birational geometry of three-dimensional Fano varieties. Our presentation of this geometry is adopted to HMS.

2.1 The Importance of Being Gorenstein

Among singular Fano varieties, those with canonical Gorenstein singularities are of special importance. They arise in many different geometrical problems: degeneration of smooth Fano varieties with a special regard to mirror symmetry (see [6, 7]), classification of reflexive polytopes (see [74, 75]), midpoints of Sarkisov links and bad Sarkisov links (see [22, 24]), compactification of certain moduli spaces (see [81]), etc. Historically, Fano varieties with canonical Gorenstein singularities are the original Fano varieties. Indeed, the name Fano varieties originated in the works of Iskovskikh (see [54, 55]) that filled the gaps in old results by Fano who studied in [36, 37] anticanonically embedded Fano threefolds with canonical Gorenstein singularities without naming them so (cf. [15]).

In dimension two, canonical singularities are always Gorenstein, so being Gorenstein is a vacuous condition. Surprisingly, the classification of del Pezzo surfaces with canonical singularities is simpler than the classification of smooth del Pezzo surfaces (see [26, 84]). Fano threefolds with canonical Gorenstein singularities are not yet classified, but first steps in this directions are already have been made by Mukai, Jahnke, Radloff, Cheltsov, Shramov, Przyjalkowski, Prokhorov, and Karzhemanov (see [17, 60, 65, 66, 81, 85]).

2.2 Birational Maps Between Fano Varieties and their Classification

V. Iskovskikh used birational maps between Fano threefolds to classify them. Indeed, he discovered smooth Fano varieties of degree 22 and Picard group $\mathbb{Z}$ by constructing the following commutative diagram:

(1)

where V ₅ is a smooth section of the Grassmannian $\mathrm{Gr}(2,5) \subset {\mathbb{P}}^{9}$ by a linear subspace of codimension 3 (they are all isomorphic), X ₂₂ is a smooth Fano threefold of index 1 and degree 22 mentioned above, i.e., $\mathrm{Pic}(X_{22}) = \mathbb{Z}[-K_{X_{22}}]$ and $-K_{X_{22}}^{3} = 22$, and α is a blowup of a quintic rational normal curve C, ρ is a flop of the proper transforms of the secant lines to C, β contracts a surface to a curve $L \subset X_{22}$ with $-K_{X_{22}} \cdot L = 1$, and ψ is a double projection from the curve L (see [54, 55]). While very powerful, this approach does not always work (see Example 2.1). Iskovskikh gave many other examples of birational maps between smooth Fano threefolds (see [57]); similar examples were found by Takeuchi [94]. Recently, P. Jahnke, I. Radloff, and I. Karzhemanov produced many new examples of Fano threefolds with canonical Gorenstein singularities by using elementary birational transformation between them.

2.3 Birational Maps Between Fano Varieties and Sarkisov Program

Results of V. Iskovskikh, Yu. Manin, V. Shokurov, and K. Takeuchi were used by V. Sarkisov and A. Corti to create what is now known as the three-dimensional Sarkisov program (see [22]). In particular, this program decomposes birational maps between terminal $\mathbb{Q}$-factorial Fano threefolds with Picard group $\mathbb{Z}$ into a sequence of so-called elementary links (often called Sarkisov links). Recently, the three-dimensional Sarkisov program has been generalized to higher dimensions by Hacon and McKernan (see [44]).

Unfortunately, the Sarkisov program is not applicable to Fano varieties with non-$\mathbb{Q}$-factorial singularities, it is not applicable to Fano varieties with nonterminal singularities, and it is not applicable to Fano varieties whose Picard group is not $\mathbb{Z}$. Moreover, in a dimension bigger than two, the Sarkisov program is not explicit, except in the toric case. In dimension three, the description of Sarkisov links is closely related to the classification of terminal non-$\mathbb{Q}$-factorial Fano threefolds whose class group is ${\mathbb{Z}}^{2}$. In general this problem is very far from being solved. But in the Gorenstein case we know a lot (see [11, 25, 58, 59, 63]).

2.4 Basic Links Between del Pezzo Surfaces with Canonical Singularities

The anticanonical linear system $\vert - K_{{\mathbb{P}}^{2}}\vert$ gives an embedding ${\mathbb{P}}^{2} \rightarrow {\mathbb{P}}^{9}$. Its image is a surface of degree 9, which we denote by S ₉. Let $\pi : S_{9} --\rightarrow S_{8}$ be a birational map induced by the linear projection ${\mathbb{P}}^{9} --\rightarrow {\mathbb{P}}^{8}$ from a point in S ₉ (the center of the projection), where S ₈ is surface of degree 8 in ${\mathbb{P}}^{8}$ obtained as the image of S ₉ under this projection. For simplicity, we say that π ₉ is a projection of the surface S ₉ from a point. We get a commutative diagram

where α ₉ is a blowup of a smooth point of the surface S ₉ and β ₉ is a birational morphism that is induced by $\vert - K_{\tilde{S}_{9}}\vert$. Note that S ₈ is a del Pezzo surface and ${(-K_{S_{8}})}^{2} = 8$.

Iterating this process and taking smooth points of the obtained surfaces S _i as centers of projections, we get the following sequence of projections:

(2)

where every S _i is a del Pezzo surface with canonical singularities, e.g., S ₃ is a cubic surface in ${\mathbb{P}}^{3}$ with isolated singularities that is not a cone. Note that we have to stop our iteration at i = 3, since the projection of S ₃ from its smooth point gives a rational map of degree 2.

For every constructed projection $\pi _{i}: S_{i} --\rightarrow S_{i-1}$, we get a commutative diagram

(3)

where α _i is a blowup of a smooth point of the surface S _i and β _i  is a birational morphism that is induced by $\vert - K_{\tilde{S}_{i}}\vert$. We say that diagram (3) is a basic link between del Pezzo surfaces.

Instead of ${\mathbb{P}}^{2}$, we can use an irreducible quadric as a root of our sequence of projections. In this way, we obtain all del Pezzo surfaces with canonical singularities except for ${\mathbb{P}}^{2}$, quadric cone and quartic hypersurfaces in $\mathbb{P}(1,1,1,2)$ and sextic hypersurfaces in $\mathbb{P}(1,1,2,3)$. Note that S ₃ is not an intersection of quadrics (trigonal case), anticanonical linear system of every quartic hypersurface in $\mathbb{P}(1,1,1,2)$, with canonical singularities is a morphism that is not an embedding (hyperelliptic case), and anticanonical linear system of every sextic hypersurface in $\mathbb{P}(1,1,2,3)$ has a unique base point.

Let us fix an action of a torus ${({\mathbb{C}}^{{\ast}})}^{2}$ on ${\mathbb{P}}^{2}$. So, if instead of taking smooth points as projection centers, we take toric smooth points (fixing the torus action), then the constructed sequence of projections (2) and the commutative diagram (3) are going to be toric as well. In this case we say that diagram (3) is a toric basic link between toric del Pezzo surfaces. Recall that there are exactly 16 toric del Pezzo surfaces with canonical singularities. In fact, we can explicitly describe all possible toric projections of toric del Pezzo surfaces from their smooth toric points (this is purely combinatorial problem), which also gives the complete description of all toric basic link between toric del Pezzo surfaces. The easiest way of doing this is to use reflexive lattice polytopes that correspond del Pezzo surfaces with canonical singularities.¹ The answer is given by Fig. 1.

Fig. 1

del Pezzo tree

2.5 Basic Links Between Gorenstein Fano Threefolds with Canonical Singularities

Similar to the two-dimensional case, it is tempting to fix few very explicit basic links between Fano threefolds with canonical Gorenstein singularities (explicit here means that these basic links should have a geometric description like projections from points or curves of small degrees) and describe all such threefolds using these links. However, this is impossible in general due to the following:

Example 2.1 (Iskovskikh–Manin). 

Any Fano threefold with canonical singularities that is birational to a smooth quartic threefold is itself a smooth quartic threefold (see [16, 56]).

However, if we are only interested in classification up to a deformation, then we can try to fix few very explicit basic links between Fano threefolds with canonical Gorenstein singularities and describe all deformation types of such threefolds using these links. Moreover, it seems reasonable to expect that this approach allows us to obtain all smooth Fano threefolds in a unified way.

We can define three-dimensional basic links similar to the two-dimensional case. Namely, let X be a Fano threefold with canonical Gorenstein singularities. Put $g = K_{X}^{3}/2 + 1$. Then, g is a positive integer and ${h}^{0}(\mathcal{O}_{X}(-K_{X})) = g + 1$. Let $\varphi _{\vert -K_{X}\vert }: X \rightarrow {\mathbb{P}}^{g+1}$ be a map given by | − K _X  | . Then,

• 1. Either $\mathrm{Bs}\vert - K_{X}\vert \neq \varnothing$, and all such X are found in [60].

• 2. Or $\varphi _{\vert -K_{X}\vert }$ is not a morphism, the threefold X is called hyperelliptic, and all such X are found in [17].

• 3. Or $\varphi _{\vert -K_{X}\vert }$ is a morphism and $\varphi _{\vert -K_{X}\vert }(X)$ is not an intersection of quadrics, the threefold X is called trigonal, and all such X are found in [17].

• 4. Or $\varphi _{\vert -K_{X}\vert }(X)$ is an intersection of quadrics.

Thus, we always can assume that $\varphi _{\vert -K_{X}\vert }$ is an embedding and $\varphi _{\vert -K_{X}\vert }(X)$ is an intersection of quadrics. Let us identify X with its anticanonical image $\varphi _{\vert -K_{X}\vert }(X)$.

Let Z be either a smooth point of the threefold X, terminal cDV point (see [88]) of the threefold X, line in $X \subset {\mathbb{P}}^{g+1}$ that does not pass through a non-cDV point, or a smooth irreducible conic in $X \subset {\mathbb{P}}^{g+1}$ that does not pass through a non-cDV point. Let $\alpha : \tilde{X} \rightarrow X$ be a blowup of the ideal sheaf of the subvariety $Z \subset X$.

Lemma 2.2.

Suppose that Z is either a cDV point or a line. Then $\vert - K_{\tilde{X}}\vert$ is free from base points.

Proof.

This follows from an assumption that $\varphi _{\vert -K_{X}\vert }(X)$ is an embedding. □ 

If Z is a smooth point, let β: X → X ^′ be a morphism given by $\vert - K_{\tilde{X}}\vert$.

Lemma 2.3.

Suppose that Z is either a cDV point or a line. Then the morphism β is birational and X ^′ is a Fano variety with canonical Gorenstein singularities such that $-K_{{X}^{{\prime}}}$ is very ample.

Proof.

The required assertion follows from the fact that X is an intersection of quadrics. □ 

If Z is a conic, then we need to impose a few additional assumptions on X and Z (cf. [94, Theorem 1.8]) to be sure that the morphism β is birational, and X ^′ is a Fano variety with canonical Gorenstein singularities such that $-K_{{X}^{{\prime}}}$ is very ample. In the toric case, these conditions can be easily verified.

Let $\pi : X --\rightarrow {X}^{{\prime}}$ be a projection from Z. If Z is not a smooth point, then the diagram

(4)

commutes. Unfortunately, if Z is a smooth point, then diagram (4) does not commute. In this case, we should define the basic link between Fano threefolds in a slightly different way. Namely, if Z is a smooth point, we still can consider the commutative diagram (4), but we have to assume that π is a projection from the projective tangent space to X at the point Z (instead of projection from Z like in other cases). Moreover, if Z is a smooth point, similar to the case when Z is a conic, we must impose a few additional assumptions on X and Z to be sure that the morphism β is birational, and X ^′ is a Fano variety with canonical Gorenstein singularities such that $-K_{{X}^{{\prime}}}$ is very ample. These conditions can be easily verified in many cases—in the toric case or in the case of index bigger than 1, see Remark 2.4.

We call diagram (4) a basic link between Fano threefolds of type:

• II_p if Z is a smooth point

• II_dp (or II_o or II_cDV, respectively) if Z is a double point (ordinary double point or non-ordinary double point, respectively)

• II_l if Z is a line

• II_c if Z is a conic

Moreover, in all possible cases, we are going to call Z a center of the basic link (4) or projection center (of π).

Remark 2.4.

Suppose that Z is a smooth point, and − K _X  ∼ 2H for some ample Cartier divisor H. Put d = H ³. Then the linear system | H | induces a rational map $\varphi _{\vert H\vert }: X --\rightarrow {\mathbb{P}}^{d+1}$ (this follows from the Riemann–Roch Theorem and basic vanishing theorems). If $\varphi _{\vert H\vert }$ is not an embedding, i.e., H is not very ample, then X can be easily described exactly as in the smooth case (see [57]). Namely, one can show that X is either a hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$ or a hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$. Similarly, if H is very ample and $\varphi _{\vert H\vert }(X)$ is not an intersection of quadrics in ${\mathbb{P}}^{d+1}$, then X is just a cubic hypersurface in ${\mathbb{P}}^{4}$. Assuming that $\varphi _{\vert H\vert }(X)$ is an intersection of quadrics in ${\mathbb{P}}^{d+1}$ (this is equivalent to ${(-K_{X})}^{3} > 24$) and identifying X with its image $\varphi _{\vert H\vert }(X)$ in ${\mathbb{P}}^{d+1}$, we see that there exists a commutative diagram

(5)

where π: X −−→ X ^′ is a projection of the threefold $X \subset {\mathbb{P}}^{d+1}$ from the point Z. Then X ^′ is a Fano threefold with canonical Gorenstein singularities whose Fano index is divisible by 2 as well.

Similar to the two-dimensional case, we can take ${\mathbb{P}}^{3}$ or an irreducible quadric in ${\mathbb{P}}^{3}$ and start applying basic links iteratively. Hypothetically, this should give us all (or almost all) deformation types of Fano threefolds with canonical Gorenstein singularities whose anticanonical degree is at most 64 (the anticanonical degree decreases after the basic link).

2.6 Toric Basic Links Between Toric Fano Threefolds with Canonical Gorenstein Singularities

Let X be a toric Fano threefold with canonical Gorenstein singularities. Fix the action of the torus ${({\mathbb{C}}^{{\ast}})}^{3}$ on X. Suppose that − K _X is very ample and X is not trigonal. Then we can identify X with its anticanonical image in ${\mathbb{P}}^{g+1}$, where $g = {(-K_{X})}^{3}/2 + 1$ (usually called the genus of the Fano threefold X). If Z is not a smooth point of the threefold X, then the commutative diagram (4) is torus invariant as well, and we call the basic link 4 a toric basic link. This gives us three types of toric basic links: II_dp if Z is a double point (II_o if Z is an ordinary double point and II_cDV if Z is non-ordinary double point), II_l if Z is a line, and II_c if Z is a conic. When Z is a smooth torus invariant point we proceed as in Remark 2.4 and obtain the toric basic link of type II_p, assuming that the Fano index of the threefold X is divisible by 2 or 3 and ${(-K_{X})}^{3} > 24$.

We can take $X = {\mathbb{P}}^{3}$ and start applying toric basic links until we get a toric Fano threefold with canonical Gorenstein singularities to which we can not apply any toric basic link (e.g., when we get a toric quartic hypersurface in ${\mathbb{P}}^{4}$). Hypothetically, this would give us birational maps between almost all toric Fano threefolds with canonical Gorenstein singularities whose anticanonical degree is at most 64. Similarly, we can take into account irreducible quadrics in ${\mathbb{P}}^{4}$ to make our picture look more complicated and, perhaps, refined. Moreover, we can start with $X = \mathbb{P}(1,1,1,3)$ or $X = \mathbb{P}(1,1,4,6)$, which are the highest anticanonical degree Fano threefolds with canonical Gorenstein singularities (see [85]) to get possibly all toric Fano threefolds with canonical Gorenstein singularities. Keeping in mind that there are 4319 such toric Fano threefolds, we see that this problem requires some computational effort and use of databases of toric Fano threefolds (see [14]).

Let us restrict our attention to toric Fano threefolds with canonical Gorenstein singularities that are known to be smoothable to smooth Fano threefolds with Picard group $\mathbb{Z}$. Starting with ${\mathbb{P}}^{3}$ and with a singular quadric in ${\mathbb{P}}^{4}$ with one ordinary double point and taking into account some toric basic links, we obtain Fig. 2, where we use bold fonts to denote Fano threefolds with Picard group $\mathbb{Z}$. (See the text preceding Table 2 for more explanation of the notation.) Recent progress in mirror symmetry for smooth Fano threefolds (see [21, 23, 42, 43, 86, 87]) shed new light on and attracted a lot of attention to toric degenerations of smooth Fano threefolds (see [6–8, 19, 38, 52]). It would be interesting to understand the relation between toric basic links between smoothable toric Fano threefolds with canonical Gorenstein singularities, basic links between smoothable Fano threefolds with canonical Gorenstein singularities, their toric degenerations, and the geometry of their Landau–Ginzburg models (cf. [86]).

Proposition 2.5 ( [53, Theorem 2.8]). 

Consider a Laurent polynomial $p_{1} = xg_{1}g_{2} + g_{3} + g_{4}/x$ , where g _i are Laurent polynomials that do not depend on x. Let $p_{2} = xg_{1} + g_{3} + g_{2}g_{4}/x$ . Let T _i be a toric variety whose fan polytope is a Newton polytope of p _i . Then T ₂ deforms to T ₁.

Remark 2.6.

In [61, Example 2.3], Jahnke and Radloff considered an anticanonical cone over the del Pezzo surface S ₆ (the rightmost on the fourth line of Fig. 1) of degree 6 and showed that it has two smoothings, to X _2. 32 and X _3. 27. Notice that S ₆ has 4 canonical toric degenerations; all of them are projections from ${\mathbb{P}}^{2}$ and two of them, S ₆ and S ₆ ′ (the third on the fourth line of Fig. 1), are projections of a smooth quadric surface. Cones over these varieties have fan polytope numbers 155 and 121 (according to [21]) correspondingly. Two of these polytopes are exactly the ones having two Minkowski decompositions, each of which gives constant terms series for X _2. 32 and X _3. 27. So we have two smoothings corresponding to two pairs of Minkowski decompositions. The question is why the existence of two deformations to two different varieties corresponds to the fact that the toric varieties are projections from a quadric surface.

Example 2.7.

Consider a Laurent polynomial

$$\displaystyle{p_{1} = xy + xz + xyz + x/y + x/z + x + 1/x.}$$

One can prove that it is a toric Landau–Ginzburg model for X _2. 35. Indeed, one can directly check the period and Calabi–Yau conditions. To prove the toric condition one can observe that

$$\displaystyle{p_{1} = x(z + z/y + 1)(y + 1/z) + 1/x.}$$

So, by Proposition 2.5, the toric variety $T_{p_{1}}$ associated with p ₁ can be deformed to the toric variety associated with

$$\displaystyle{p_{2} = x(z + z/y + 1) + (y + 1/z)/x,}$$

which after toric change of variables coincides with f ₂. We get F ₂ which can be smoothed to X _2. 35 by Theorem 3.12.

Fig. 2

Fano snake

A variety $T_{p_{1}}$ is nothing but a cone over a toric del Pezzo surface S ₇ (the rightmost in the third column of Fig. 1). Consider a basic link—projection from a smooth point on $T_{p_{1}}$. One gets a toric variety—a cone over the del Pezzo surface S ₆. It has two smoothings (see Remark 2.6). Moreover, there are two toric Landau–Ginzburg models,

$$\displaystyle{p_{3} = xy + xz + xyz + x/y + x/z + x/y/z + 2x + 1/x}$$

and

$$\displaystyle{p_{4} = xy + xz + xyz + x/y + x/z + x/y/z + 3x + 1/x,}$$

one for X _2. 32 and another one for X _3. 27. Indeed, as before the period and Calabi–Yau conditions can be checked directly. Notice that

$$\displaystyle{p_{3} = x\left ((yz + 1)/z/y\right )(y + 1)(z + 1) + 1/x}$$

and

$$\displaystyle{p_{4} = x\left ((yz + z + 1)/y/z\right )(yz + y + 1) + 1/x.}$$

After a cluster change of variables by Proposition 2.5 one gets two polynomials associated with F ₃ and F ₄. These varieties by Theorem 3.12 can be smoothed to X _2. 32 and X _3. 27. Nevertheless, the correct target for the basic link between the last two varieties is X _2. 32, since projecting a general X _2. 32 from a point, we always obtain X _3. 27.

Remark 2.8.

The same can be done with another variety, the cone over S ₆ ′ (the third in the fourth column of Fig. 1). Indeed, by Proposition 2.5 a variety S ₆ ′ is a degeneration of S ₆ so cones over them degenerate as well.

3 Classical Theory of Landau–Ginzburg Models

From now on we concentrate on the theory of Landau–Ginzburg models and their moduli. First we recall the classical definition of the Landau–Ginzburg model of a single Fano variety (see, e.g., [86] and the references therein). Let X be a smooth Fano variety of dimension n and $Q{H}^{{\ast}}(X) = {H}^{{\ast}}(X, \mathbb{Q}) \otimes _{\mathbb{Z}}\Lambda$ its quantum cohomology ring, where Λ is the Novikov ring for X. The multiplication in QH ^∗(X), i.e., the quantum multiplication, is given by (genus zero) Gromov–Witten invariants—numbers counting rational curves on X. Let Q _X be the associated regularized quantum differential operator (the second Dubrovin connection)—the regularization of an operator associated with the connection in the trivial vector bundle given by quantum multiplication by the canonical class K _X . Solutions of an equation given by this operator are given by I-series for X—generating series for its one-pointed Gromov–Witten invariants. In particular, one “distinguished” solution is a constant term (with respect to cohomology) of I-series. Let us denote it by $I = 1 + a_{1}t + a_{2}{t}^{2}+\ldots$.

Definition 3.1.

A toric Landau–Ginzburg model is a Laurent polynomial $f \in \mathbb{C}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ such that:

• Period condition: The constant term of ${f}^{i} \in \mathbb{C}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ is a _i for any i (this means that I is a period of a family $f : {({\mathbb{C}}^{{\ast}})}^{n} \rightarrow \mathbb{C}$; see [86]).

• Calabi–Yau condition: There exists a fiberwise compactification (the Calabi–Yau compactification) whose total space is a smooth (open) Calabi–Yau variety.

• Toric condition: There is an embedded degeneration X ⇝ T to a toric variety T whose fan polytope (the convex hull of generators of its rays) coincides with the Newton polytope (the convex hull of nonzero coefficients) of f.

Remark 3.2.

This notion can be extended to some non-smooth cases; see, for instance, [23].

Theorem 3.3 ( [87, Theorem 18] and  [52, Theorem 3.1]). 

Smooth Fano threefolds of Picard rank 1 have toric Landau–Ginzburg models.

Remark 3.4.

Toric Landau–Ginzburg models for Picard rank 1 Fano threefolds are found in  [87]. However they are not unique. Some of them coincide with ones from Table 2. Anyway Theorem 3.3 holds for all threefolds from Table 2; see Theorem 3.12.

Theorem 3.5 ( [34]). 

Let X be a Fano threefold of index i and ${(-K_{X})}^{3} = {i}^{3}k$ . Then fibers of toric weak Landau–Ginzburg model from  [87] can be compactified to Shioda–Inose surfaces with Picard lattice $H \oplus E_{8}(-1) \oplus E_{8}(-1) +\langle -ik\rangle.$

Remark 3.6.

This theorem holds for toric Landau–Ginzburg models for Fano threefolds of Picard rank 1 from Table 2.

This theorem means that fibers of compactified toric Landau–Ginzburg models are mirrors of anticanonical sections of corresponding Fano varieties, and this property determines compactified toric Landau–Ginzburg models uniquely as the moduli spaces of possible mirror K3’s are just ${\mathbb{P}}^{1}$’s.

The discussion above can be summarized to the following mirror symmetry conjecture.

Conjecture 3.7.

Every smooth Fano variety has a toric Landau–Ginzburg model.

Theorem 3.3 shows that this conjecture holds for Fano threefolds of Picard rank 1. Theorem 3.12 shows that the conjecture holds for Fano varieties from Table 2.

3.1 The Table

Now we study toric Landau–Ginzburg models for Fano threefolds of Picard rank 1 given by toric basic links from ${\mathbb{P}}^{3}$ and the quadric. We give a table of such toric Landau–Ginzburg models and prove in Theorem 3.12 that Laurent polynomials listed in the table are toric Landau–Ginzburg models of Fano threefolds.

Table 2 is organized as follows. N is the number of a variety in the table. “Var.” is a Fano smoothing numerated following [57]. “Deg.” is a degree of a variety. “Par.” is the number of varieties giving our variety by a projection. “BL” is a type of toric basic link(s). “Desc.” stands for descendants—varieties that can be obtained by projection from given variety. The last column is a toric Landau–Ginzburg model for the variety.

Remark 3.8.

F ₂ is a blowup of $F_{1} = {\mathbb{P}}^{3}$ at one point with an exceptional divisor E. $F_{3} = X_{2.35}$ is a projection from a point lying far from E. If we project from a point lying on E we get another (singular) variety, $F_{3}^{{\prime}}$, with corresponding weak Landau–Ginzburg model

$$\displaystyle{x + y + z + \frac{1} {xyz} + \frac{2} {x} + \frac{yz} {x}.}$$

Remark 3.9.

Variety 24, the toric quartic, has no cDV points or smooth toric lines. So we cannot proceed to make basic links. However, it has 4 singular canonical (triple) points and we can project from any of them. In other words, we can project the quartic

$$\displaystyle{\{x_{1}x_{2}x_{3}x_{4} = x_{0}^{4}\} \subset \mathbb{P}[x_{ 0},x_{1},x_{2},x_{3},x_{4}]}$$

from the point, say, (0 : 0 : 0 : 0 : 1). Obviously, we get variety 1, that is, ${\mathbb{P}}^{3}$ again.

Proposition 3.10.

Families of hypersurfaces in ${({\mathbb{C}}^{{\ast}})}^{3}$ given by Laurent polynomials from Table  2 can be fiberwise compactified to (open) Calabi–Yau varieties.

Proof.

Let f be a Laurent polynomial from the table. Compactify the corresponding family $\{f =\lambda \}\subset \mathrm{ Spec}\,\mathbb{C}[{x}^{\pm 1},{y}^{\pm 1},{z}^{\pm 1}] \times \mathrm{ Spec}\,\mathbb{C}[\lambda ]$ fiberwise using the standard embedding $\mathrm{Spec}\,\mathbb{C}[{x}^{\pm 1},{y}^{\pm 1},{z}^{\pm 1}] \subset \mathbf{Proj}\mathbb{C}[x,y,z,t]$. In other words, multiply it by a denominator (xyz) and add an extra homogenous variable t. For varieties 11, 12, 13 do toric change of variables xy → x, yz → z. We get a family of singular quartics. Thus it has trivial canonical class. The threefold singularities we get are du Val along lines and ordinary double points; the same type of singularities arises after crepant blow-ups of singular lines and small resolutions of ordinary double points. Thus the threefold admits a crepant resolution; this resolution is the Calabi–Yau compactification we need. □ 

Proposition 3.11.

Toric varieties from Table  2 are degenerations of corresponding Fano varieties.

Proof.

Varieties 1–5, 9–15, 25–27 are terminal Gorenstein toric Fano threefolds. So, by [82] they can be smoothed. By [38, Corollary 3.27], the smoothings are Fano’s with the same numerical invariants as the initial toric varieties. The only smooth Fano threefolds with given invariants are listed at the second column. In other words, the statement of the proposition for varieties 1–5, 9–15, 26–28 follows from the proof of [38, Theorem 2.7].

Varieties 6–8 are complete intersections in (weighted) projective spaces. We can write down the dual polytopes to their fan polytopes. The equations of the toric varieties correspond to homogenous relations on integral points of the dual polytopes. One can see that the relations are binomials defining corresponding complete intersections and the equations of a Veronese map v ₂. So the toric varieties can be smoothed to the corresponding complete intersections. For more details see [52, Theorem 2.2].

Variety X _2. 13 can be described as a section of ${\mathbb{P}}^{2} \times {\mathbb{P}}^{4}$ by divisors of type (1, 1), (1, 1), and (0, 2) (see, say, [21]). Equations of ${\mathbb{P}}^{2} \times {\mathbb{P}}^{4}$ in Segre embedding can be described as all (2 ×2)-minors of a matrix

$$\displaystyle{\left (\begin{array}{ccccc} x_{00} & x_{01} & x_{02} & x_{03} & x_{04} \\ x_{10} & x_{11} & x_{12} & x_{13} & x_{14} \\ x_{20} & x_{21} & x_{22} & x_{23} & x_{24}\\ \end{array} \right ).}$$

Consider its section T given by equations $x_{00} = x_{11}$, $x_{11} = x_{22}$, $x_{01}x_{02} = x_{03}x_{04}$. These equations give divisors of types (1, 1), (1, 1), and (0, 2), respectively. They are binomial, which means that T is a toric variety. The equations giving variety 16 are homogenous integral relations on integral points of a polytope dual to a Newton polytope of f ₁₆. It is easy to see that these relations are exactly ones defining T. Thus T = F ₁₆ and F ₁₆ can be smoothed to X _2. 13.

Varieties 17 and 18 correspond to ones from [87]. Thus, by [52, Theorem 3.1] they can be smoothed to corresponding Fano threefolds.

The dual polytope to the fan polytope for variety 19 is drawn on Fig. 3. It obviously has a triangulation on 14 triangles satisfying conditions of [18, Corollary 3.4]. By this corollary variety 19 can be smoothed to the variety we need.

Fig. 3

Polytope defining variety 19

Finally the existence of smoothings of varieties 20–24 to corresponding Fano varieties follows from [19]. □ 

Theorem 3.12.

Laurent polynomials from Table  2 are toric Landau–Ginzburg models for corresponding Fano threefolds.

Proof.

The period condition follows from direct computations; see [21]. The Calabi–Yau condition holds by Proposition 3.10. The toric condition holds by Proposition 3.11. □ 

4 Landau–Ginzburg Considerations

4.1 Categorical Background

Examples in the previous sections suggest a new approach to birational geometry of Fano manifolds. In this section we summarize this approach and develop the relevant technical tools. We proceed by extending Kawamata’s approach described in Table 1 by adding further data to the categorical approach recorded in the geometry of the moduli space of Landau–Ginzburg models. The main points are:

• 1. There exists a moduli space of Landau–Ginzburg models for many (possibly all) three-dimensional Fano manifolds.

• 2. The topology of this compactified moduli space of Landau–Ginzburg models determines Sarkisov links among these Fano manifolds. We conjecture that the geometry of the moduli space of Landau–Ginzburg models gives answers to many questions related to rationality and birational equivalence—we suggest some invariants and give examples.

The geometry of the moduli space of Landau–Ginzburg models was introduced in [27, 28, 68] as an analogy with nonabelian Hodge theory. We describe this analogy.

We build the “twistor” family so that the fiber over zero is the “moduli space” of Landau–Ginzburg models and the generic fiber is the Stability Hodge Structure (SHS) (see below).

Noncommutative Hodge theory endows the cohomology groups of a dg-category with additional linear data—the noncommutative Hodge structure—which records important information about the geometry of the category. However, due to their linear nature, noncommutative Hodge structures are not sophisticated enough to codify the full geometric information hidden in a dg-category. In view of the homological complexity of such categories it is clear that only a subtler nonlinear Hodge theoretic entity can adequately capture the salient features of such categorical or noncommutative geometries. In this section by analogy with “classical nonabelian Hodge theory” we construct and study from such prospective a new type of entity of exactly such type—the SHS associated with a dg-category.

As the name suggests, the SHS of a category is related to the Bridgeland stabilities on this category. The moduli space Stab _C of stability conditions of a triangulated dg-category C is, in general, a complicated curved space, possibly with fractal boundary. In the special case when C is the Fukaya category of a Calabi–Yau threefold, the space Stab _C admits a natural one-parameter specialization to a much simpler space S ₀. Indeed, HMS predicts that the moduli space of complex structures on the mirror Calabi–Yau threefold maps to a Lagrangian subvariety $\mathsf{Stab}_{C}^{\text{geom}} \subset \mathsf{Stab}_{C}$.

The space S ₀ is the fiber at 0 of this completed family and conjecturally $\mathsf{S} \rightarrow \mathbb{C}$ is one chart of a twistor-like family $\mathcal{S}\rightarrow {\mathbb{P}}^{1}$ which is by definition the Stability Hodge Structure associated withC.

SHSs are expected to exist for more general dg-categories, in particular for Fukaya–Seidel categories associated with a superpotential on a Calabi–Yau space or with categories of representations of quivers. Moreover, for special non-compact Calabi–Yau threefolds, the zero fiber S ₀ of a SHS can be identified with the Dolbeault realization of a nonabelian Hodge structure of an algebraic curve. This is an unexpected and direct connection with Simpson’s nonabelian Hodge theory (see [92]) which we exploit further, suggesting some geometric applications.

We briefly recall nonabelian Hodge theory settings. According to Simpson (see [92]), we have a one-parametric twistor family such that the fiber over zero is the moduli space of Higgs bundles and the generic fiber is the moduli space—M _Betti —of representations of the fundamental group of the space over which the Higgs bundle lives. By analogy with the nonabelian Hodge structure we have:

Conjecture 4.1 ([68]). 

The moduli space of stability conditions of Fukaya–Seidel category can be included in one-parametric “twistor” family.

In other words SHS exists for Fukaya–Seidel categories. Parts of this conjecture are checked in [45, 68].

We give a brief example of SHS.

Example 4.2.

We will give a brief explanation of the calculation of the “twistor” family for the SHS for the category A _n recorded in the picture above. We start with the moduli space of stability conditions for the category A _n , which can be identified with differentials

$$\displaystyle{{e}^{p(z)}dz,}$$

where p(z) is a polynomial of degree n + 1.

Classical work of Nevanlinna identifies these integrals with graphs (see Fig. 4)—graphs connecting the singularities of the function given by an integral against the exponential differentials. Now we consider the limit

$$\displaystyle{{e}^{p(z)/u}dz.}$$

Fig. 4

Taking limit

Geometrically, the limit differential can be identified with polynomials, e.g., with Landau–Ginzburg models (see Fig. 4)—for more see [45].

4.2 The Fiber Over Zero

The fiber over zero (described below) plays an analogous role to the moduli space of Higgs Bundles in Simpson’s twistor family in the theory of nonabelian Hodge structures. As alluded to earlier, an important class of examples of categories and their stability conditions arises from HMS—Fukaya–Seidel categories. Indeed, such categories are the origin of the modern definition of such stability conditions. The prescription given by Batyrev–Borisov and Hori–Vafa in [9, 47] to obtain homological mirrors for toric Fano varieties is perfectly explicit and provides a reasonably large set of examples to examine. We recall that if Σ is a fan in ${\mathbb{R}}^{n}$ for a toric Fano variety X _Σ , then the homological mirror to the B model of X _Σ is a Landau–Ginzburg model $w: {({\mathbb{C}}^{{\ast}})}^{n} \rightarrow \mathbb{R}$, where the Newton polytope Q of w is the fan polytope of Σ. In fact, we may consider the domain ${({\mathbb{C}}^{{\ast}})}^{n}$ to occur as the dense orbit of a toric variety X _A where A is $Q \cap {\mathbb{Z}}^{n}$ and X _A indicates the polytope toric construction. In this setting, the function w occurs as a pencil $V _{w} \subset {H}^{0}(X_{A},L_{A})$ with fiber at infinity equal to the toric boundary of X _A . A similar construction works for generic nontoric Fano’s. In this paper we work with the directed Fukaya category associated to the superpotential w—Fukaya–Seidel categories. To build on the discussion above, we discuss here these two categories in the context of stability conditions. The fiber over zero corresponds to the moduli of complex structures. If X _A is toric, the space of complex structures on it is trivial, so the complex moduli appearing here are a result of the choice of fiber $H \subset X_{A}$ and the choice of pencil w, respectively. The appropriate stack parameterizing the choice of fiber contains the quotient $[U/{({\mathbb{C}}^{{\ast}})}^{n}]$ as an open dense subset where U is the open subset of ${H}^{0}(X_{A},L_{A})$ consisting of those sections whose hypersurfaces are nondegenerate (i.e., smooth and transversely intersecting the toric boundary) and ${({\mathbb{C}}^{{\ast}})}^{n}$ acts by its action on X _A . To produce a reasonably well-behaved compactification of this stack, we borrow from the work of Alexeev (see [2]), Gelfand–Kapranov–Zelevinsky (see [41]), and Lafforgue (see [78]) to construct the stack $\mathcal{X}_{Sec(A)}$ with universal hypersurface stack $\mathcal{X}_{Laf(A)}$. We quote the following theorem which describes much of the qualitative behavior of these stacks:

Theorem 4.3 ( [29]). 

• (i) The stack $\mathcal{X}_{Sec(A)}$ is a toric stack with moment polytope equal to the secondary polytope Sec(A) of A.

• (ii) The stack $\mathcal{X}_{Laf(A)}$ is a toric stack with moment polytope equal to the Minkowski sum Sec(A) + Δ _A where Δ _A is the standard simplex in ${\mathbb{R}}^{A}$.

• (iii) Given any toric degeneration $F : Y \rightarrow \mathbb{C}$ of the pair (X _A ,H), there exists a unique map $f : \mathbb{C} \rightarrow \mathcal{X}_{Sec(A)}$ such that F is the pullback of $\mathcal{X}_{Laf(A)}$.

We note that in the theorem above, the stacks $\mathcal{X}_{Laf(A)}$ and $\mathcal{X}_{Sec(A)}$ carry additional equivariant line bundles that have not been examined extensively in existing literature but are of great geometric significance. The stack $\mathcal{X}_{Sec(A)}$ is a moduli stack for toric degenerations of toric hypersurfaces $H \subset X_{A}$. There is a hypersurface $\mathcal{E}_{A} \subset \mathcal{X}_{Sec(A)}$ which parameterizes all degenerate hypersurfaces. For the Fukaya category of hypersurfaces in X _A , the compliment $\mathcal{X}_{Sec(A)} \setminus \mathcal{E}_{A}$ plays the role of the classical stability conditions, while including $\mathcal{E}_{A}$ incorporates the compactified version where MHS come into effect.

To find the stability conditions associated with the directed Fukaya category of (X _A , w), one needs to identify the complex structures associated with this model. In fact, these are described as the coefficients of the superpotential, or in our setup, the pencil $V _{w} \subset {H}^{0}(X_{A},w)$. Noticing that the toric boundary is also a toric degeneration of the hypersurface, we have that the pencil V _w is nothing other than a map from ${\mathbb{P}}^{1}$ to $\mathcal{X}_{Sec(A)}$ with prescribed point at infinity. If we decorate ${\mathbb{P}}^{1}$ with markings at the critical values of w and ∞, then we can observe such a map as an element of $\mathcal{M}_{0,V ol(Q)+1}(\mathcal{X}_{Sec(A)},[w])$ which evaluates to $\mathcal{E}_{A}$ at all points except one and ∂X _A at the remaining point. We define the cycle of all stable maps with such an evaluation to be $\mathcal{W}_{A}$ and regard it as the appropriate compactification of complex structures on Landau–Ginzburg A models. Applying techniques from fiber polytopes we obtain the following description of $\mathcal{W}_{A}$:

Theorem 4.4 ( [29]). 

The stack $\mathcal{W}_{A}$ is a toric stack with moment polytope equal to the monotone path polytope of Sec(A).

The polytope occurring here is not as widely known as the secondary polytope but occurs in a broad framework of “iterated fiber polytopes” introduced by Billera and Sturmfels.

In addition to applications of these moduli spaces to stability conditions, we also obtain important information on the directed Fukaya categories and their mirrors from this approach. In particular, the theorem above may be applied to computationally find a finite set of special Landau–Ginzburg models $\{w_{1},\ldots,w_{s}\}$ corresponding to the fixed points of $\mathcal{W}_{A}$ (or the vertices of the monotone path polytope of Sec(A)). Each such point is a stable map to $\mathcal{X}_{Sec(A)}$ whose image in moment space lies on the 1-skeleton of the secondary polytope. This gives a natural semiorthogonal decomposition of the directed Fukaya category into pieces corresponding to the components in the stable curve which is the domain of w _i . After ordering these components, we see that the image of any one of them is a multi-cover of the equivariant cycle corresponding to an edge of Sec(A). These edges are known as circuits in combinatorics and we study the categories defined by each such component in [29].

Now we put this moduli space as a “zero fiber” of the “twistor” family of moduli family of stability conditions.

Theorem 4.5 (see [68]). 

The fiber over zero is a formal scheme F over $\mathcal{W}_{A}$ determined by the solutions of the Maurer–Cartan equations for a dg-complex

A Sketch of the Proof. The above complex describes deformations with fixed fiber at infinity. We can associate with this complex a Batalin–Vilkovisky algebra. Following [68] we associate with it a smooth stack. In the case of Fukaya–Seidel category of a Landau–Ginzburg mirror of a Fano manifold X the argument above implies that the dimension of the smooth stack of Landau–Ginzburg models is equal to h ^1, 1(X) + 1.

We also have a ${\mathbb{C}}^{{\ast}}$ action on F with fixed points corresponding to limiting stability conditions.

Conjecture 4.6 (see [69]). 

The local completion of fixed points over X has a mixed Hodge structure.

In the same way as the fixed point set under the ${\mathbb{C}}^{{\ast}}$ action plays an important role in describing the rational homotopy types of smooth projective varieties we study the fixed points of the ${\mathbb{C}}^{{\ast}}$ action on F and derive information about the homotopy type of a category.

Similarly we can modify the above complex by fixing only a part of the fiber at infinity and deforming the rest. Similar Batalin–Vilkovisky algebra technique allows us to prove

Theorem 4.7 ( [68]). 

We obtain a smooth moduli stack of Landau–Ginzburg models if we fix only a part of the fiber at infinity.

This means that we can allow different parts of the fiber at infinity to move—we call this part a moving scheme. The geometrical properties of the moving scheme contain deep birational, categorical, and algebraic cycle information. We record this information in new invariants, mainly emphasizing the birational content.

4.3 Birational Applications

In this subsection we look at the data collected from Sects. 2 and 3 from a new categorical prospective. We apply the theorems above to the case of Landau–Ginzburg models for del Pezzo surfaces—this gives a new read of Sect. 2.4. The basic links among del Pezzo surfaces can be interpreted as follows.

Theorem 4.8 ( [29]). 

There exists an 11-dimensional moduli stack of all Landau–Ginzburg models of all del Pezzo surfaces. This moduli stack has a cell structure with the biggest cells corresponding to the del Pezzo surfaces of big Picard rank. The basic links correspond to moving to the boundary of this stack.

Proof.

The proof of this statement amounts to allowing all points at the fiber at infinity to move (the case of rational elliptic fibration) and then fixing them one by one (for the first step see Table 3). □ 

Theorem 4.8 suggests that we can extend the construction to rational blowdowns. We associate a moduli space of Landau–Ginzburg models to a rational blowdown of a rational surface by fixing corresponding subschemes of the fiber at infinity. This is a new construction in category theory, where the compactified moduli spaces of Landau–Ginzburg models play the role of the moduli space of vector bundles in the Donaldson theory of polynomial invariants. As a result we get a tool for studying the SODs by putting a topological structure on them based on the compactification of moduli spaces of Landau–Ginzburg models. We conjecture the following (see also  [28]).

Conjecture 4.9 (see [69]). 

The derived categories of the Barlow surface and of the rational blowdown described above contain as a semiorthogonal piece a phantom category, i.e., a nontrivial category with trivial K ⁰ group.

This conjecture is rather bold and will make the study of algebraic cycles and rationality questions rather difficult. Some evidence for it has appeared in [50].

Table 3 Moving points

We summarize our findings in Table 4. In the leftmost part of this table we consider different surfaces. In the second part we describe the fiber at infinity with the corresponding moving scheme. In the last part we comment what is the moduli of Landau–Ginzburg models and what are some of its invariants. In most cases this is the fundamental group of the non-compactified moduli space. In case of rational blowdown this fundamental group suggests the appearance of new phenomenon a nontrivial category with trivial K ⁰ group—a phantom category, which we will discuss later. This also appears in the Barlow surface. The connection with the Godeaux surface (see [12]) suggests that the fundamental group of the non-compactified moduli space differs from the fundamental group of the non-compactified moduli space of LG models for del Pezzo surfaces of degree 1.

Remark 4.10.

Figure 2 suggests that different Fano’s are connected in the big moduli of Landau–Ginzburg models either by wall-crossings or by going to the boundary of such a moduli space.

4.4 High-Dimensional Fano’s

We concentrate on the case of high-dimensional Fano manifolds. We give the findings in Sect. 2.6 in the following categorical read.

Table 4 Moduli of Landau–Ginzburg models for surfaces

Conjecture 4.11.

There exists a moduli stack of Landau–Ginzburg models of all three-dimensional Fano’s. It has a cell structure parallel to the basic links from Sect. 2.6.

This conjecture is based on the following implementation of the theory of Landau–Ginzburg models. In the same way as in the case of del Pezzo surfaces we can allow moving different subschemes at the fiber at infinity. The first three-dimensional examples were worked out in [1, 64]. In these cases the moving scheme at infinity corresponds to a Riemann surface so modified Landau–Ginzburg models correspond to Landau–Ginzburg mirrors of blown up toric varieties. In higher dimensions of course the cell structure is more elaborate. By fixing different parts at the divisor at infinity we can change the Picard rank of the generic fiber. Modifications, gluing, and conifold transitions are needed in dimension three and four. These lead to the need for Landau–Ginzburg moduli spaces with many components.

The next case to consider is the case of the three-dimensional cubic. In this case the moving scheme is described in [50]. A similar moving scheme is associated with the threefold X ₁₄.

The next theorem follows from [29].

Theorem 4.12.

The moduli space of the Landau–Ginzburg mirrors for the smooth three dimensional cubic and X ₁₄ are deformations of one another.

As it follows from [29] this would imply their birationality since it means that some Mori fibrations associated with the three-dimensional cubic and X ₁₄ are connected via Sarkisov links.

The A side interpretation of this result is given in [5]. It implies that the SODs of the derived categories of three-dimensional cubic and X ₁₄ have a common semiorthogonal piece and differ only by several exceptional objects—a result obtained by Kuznetsov in [77].

Similar observations can be made for other three-dimensional Fano manifolds, whose Landau–Ginzburg models can be included in one big moduli space. So studying and comparing these Landau–Ginzburg models at the same time brings a new approach to birational geometry. The material described in Sect. 2.6 suggests that there are many other three-dimensional Fano manifolds related as the three-dimensional cubic and X ₁₄, that is, related by one only non-commutative cobordism. Moving from one Landau–Ginzburg model associated to one Fano threefold to another can be considered as a certain “wall-crossing.” As the material of Sect. 2.6 suggests we can include singular Fano threefolds as boundary of the moduli space of Landau–Ginzburg models — i.e.,“limiting stability conditions” on which even more dramatic “wall-crossing” occurs. The experimental material from Sect. 2.6 and Conjecture 3.7 also brings the idea that studying birational geometry of Fano threefolds and proving HMS for them might be closely related problems.

Similar picture exists in higher dimension. We give examples and invariants connected with moduli spaces of Landau–Ginzburg models associated with very special four-dimensional Fano manifolds—four dimensional cubics and their “relatives.” For these fourfolds we look at the Hassett–Kuznetsov–Tschinkel program from the Landau–Ginzburg perspective.

It is expected that there are many analogs in dimension four to the behavior of three-dimensional cubics and X ₁₄, namely, they have a common semiorthogonal piece and differ only by several exceptional objects. We indicate several of them.

Four-dimensional cubics have been studied by many people: Voisin, Beauville, Donagi, Hassett, and Tschinkel. On the level of derived categories, a lot of fundamental work was done by Kuznetsov and then Addington and Thomas. On the Landau–Ginzburg side, calculations were done by [51, 70]. We extend our approach in the case of three-dimensional Fano threefolds to some fourfolds. Recall the following theorem by Kuznetsov.

Theorem 4.13 ( [77]). 

Let X be a smooth four-dimensional cubic. Then

$$\displaystyle{{D}^{b}(X) =\langle {D}^{b}(K3),E_{ 1},E_{2},E_{3}\rangle.}$$

Here D ^b(K3) is the derived category of a noncommutative K3 surface. This noncommutative K3 surface is very non-generic. Moving the subscheme at infinity corresponding to a generic K3 surface we obtain the moduli spaces of Landau–Ginzburg models associated to four-dimensional X ₁₀. This suggests

Conjecture 4.14.

Let X be a smooth four-dimensional variety X ₁₀. Then

$$\displaystyle{{D}^{b}(X) =\langle {D}^{b}(K3),E_{ 1},E_{2},E_{3},E_{4}\rangle.}$$

Here D ^b(K3) is the derived category of a generic noncommutative K3 surface. We expect that this conjecture will follow from some version of homological projective duality.

Similarly to the three-dimensional case, there is an overlap between the Landau–Ginzburg moduli spaces of the four-dimensional cubic and four-dimensional X ₁₀.

Conjecture 4.15.

There is an infinite series of moduli of Landau–Ginzburg models associated with special (from the Noether–Lefschetz loci) four-dimensional cubics and four-dimensional X ₁₀’s, which can be deformed one to another and therefore they are birational (see Table 5).

This series corresponds to cases when the moving scheme at infinity is associated with commutative K3 surface. According to [29] such a deformation between moduli of Landau–Ginzburg models implies the existence of a Sarkisov links between such cubics and X ₁₀. We will return to rationality questions in the next subsection. We would like to mention here that a generalization of homological projective duality of Kuznetsov’s arrives at

Conjecture 4.16.

Let X be a smooth four-dimensional Kuechle manifold (see  [76]) Then

$$\displaystyle{{D}^{b}(X) =\langle {D}^{b}(K3),E_{ 1},E_{2},\ldots,E_{n}\rangle.}$$

Here D ^b(K3) is the derived category of a generic noncommutative K3 surface. As a consequence we have

Conjecture 4.17.

There is an infinite series of moduli of Landau–Ginzburg models associated with special four-dimensional cubics, four-dimensional X ₁₀s, and Kuechle manifolds, which can be deformed one to another.

This conjecture suggests that the rationality question for Kuechle manifolds can be treated similarly as the questions for four-dimensional cubic and X ₁₀—see the next subsection.

Table 5 Noncommutative Sarkisov program

4.5 Invariants

In this subsection we introduce two types of invariants, which are connected with SHS and moduli spaces of Landau–Ginzburg models. The first type is a global invariant—the Orlov spectra of a category. It was conjectured in [4] that it is an invariant measuring nonrationality. Its relation to Landau–Ginzburg models was emphasized in [50, 51, 70]. In this subsection we relate it to the Hassett–Kuznetsov–Tschinkel program—a program relating the Noether–Lefschetz components to rationality of four-dimensional cubic.

The second type of invariant is of local nature—the local singularity of the Landau–Ginzburg models. We relate these invariants to stability conditions. We suggest that they play the role of discrepancies and thresholds in the Kawamata’s correspondence described in Table 1. In other words, these invariants measure if two Landau–Ginzburg moduli spaces can be deformed one to another and according to [29] if there are Sarkisov links connecting the Fano manifolds from the A side.

Noncommutative Hodge structures were introduced by Kontsevich, Katzarkov, and Pantev in [68], as means of bringing techniques and tools of Hodge theory into the categorical and noncommutative realm. In the classical setting, much of the information about an isolated singularity is recorded by means of the Hodge spectrum, a set of rational eigenvalues of the monodromy operator. The Orlov spectrum (defined below) is a categorical analog of this Hodge spectrum appearing in the works of Orlov [83] and Rouquier [89]. The missing numbers in the spectra are called gaps.

Let $\mathcal{T}$ be a triangulated category. For any $G \in \mathcal{T}$ denote by $\langle G\rangle _{0}$ the smallest full subcategory containing G which is closed under isomorphisms, shifting, and taking finite direct sums and summands. Now inductively define $\langle G\rangle _{n}$ as the full subcategory of objects, B, such that there is a distinguished triangle, X → B → Y → X[1], with $X \in \langle G\rangle _{n-1}$ and $Y \in \langle G\rangle _{0}$.

Definition 4.18.

Let G be an object of a triangulated category $\mathcal{T}$. If there is some number n with $\langle G\rangle _{n} = \mathcal{T}$, we set

$$\displaystyle{t(G) := \text{min}\{n \geq 0\ \vert \ \langle G\rangle _{n} = \mathcal{T}\}.}$$

Otherwise, we set t(G) : = ∞. We call t(G) the generation time of G. If t(G) is finite, we say that G is a strong generator. The Orlov spectrum of $\mathcal{T}$ is the union of all possible generation times for strong generators of $\mathcal{T}$. The Rouquier dimension is the smallest number in the Orlov spectrum. We say that a triangulated category, $\mathcal{T}$, has a gap of length s if a and a + s + 1 are in the Orlov spectrum but r is not in the Orlov spectrum for a < r < a + s + 1.

The first connection to Hodge theory is given by the following theorem.

Theorem 4.19 (see [4]). 

Let X be an algebraic variety with an isolated hypersurface singularity. The Orlov spectrum of the category of singularities of X is bounded by twice the embedding dimension times the Tjurina number of the singularity.

The following conjecture plays an important role in our considerations.

Conjecture 4.20 (see [4]). 

Let X be a rational Fano manifold of dimension n > 2. Then a gap of spectra of D ^b(X) is less or equal to n − 3.

After this brief review of theory of spectra and gaps we connect them with the SHS and moduli of Landau–Ginzburg models.

Conjecture 4.21 (see [68]). 

The monodromy of the Landau–Ginzburg models for Fano manifold X determines the gap of spectra of D ^b(X).

This conjecture was partially verified in [50, 51, 70]. We record our findings in Table 6. It is clear from our construction that the monodromy of Landau–Ginzburg models depends on the choice of moving scheme. This suggests that the classical Hodge theory cannot distinguish rationality. We employ the geometry of the moduli spaces of Landau–Ginzburg models in order to do so. These moduli spaces measure the way the pieces in the SODs are put together—this information computes the spectra of a category.

Table 6 Summary

This was first observed in [50, 51, 70]. Applying the theory of Orlov’s spectra to the case of four-dimensional cubic, four-dimensional X ₁₀, and Kuechle manifolds we arrive at the following conjecture suggested by the Hassett–Kuznetsov–Tschinkel program.

Conjecture 4.22.

The four-dimensional cubic, four-dimensional X ₁₀, and Kuechle manifolds are not rational if they do not contain derived categories of commutative K3 surfaces in their SODs (see  [77] for cubic fourfold).

In other words this conjecture implies that the generic fourfold as above is not rational since the gap of their categories of coherent sheaves is equal to two. In case the SODs contain the derived category of a commutative K3 surface—the issue is more delicate and requires the use of a Noether–Lefschetz spectra—see [50].

Remark 4.23.

We are very grateful to A. Iliev who has informed us that some checks of Conjecture 4.22 were done by him, Debarre, and Manivel.

We proceed with a topic which we have started in [50]—how to detect rationality when gaps of spectra cannot be used. The example we have considered there was the Artin–Mumford example. Initially it was shown that the Artin–Mumford example is not rational since it has torsion in its third cohomology group. Our conjectural interpretation in [50] is that the Artin–Mumford example is not rational since it is a conic bundle which contains the derived category of an Enriques surface in the SOD of its derived category. The derived category of an Enriques surface has 10 exceptional objects and a category $\mathcal{A}$, which does not look like a category of a curve, in its SOD. We conjecture in [50] that derived category of the Artin–Mumford example has no gap in its spectra, but it is the moving scheme which determines its nonrationality. We also exhibit the connection between category $\mathcal{A}$ and its moving scheme—see also [49].

We bring a totally new prospective to rationality questions—the parallel of spectra and gaps of categories with topological superconductors. Indeed, if we consider generators as Hamiltonians and generation times as states of matter we get a far-reaching parallel. The first application of this parallel was a prediction of existence of phantom categories which we have defined above. The phantoms are the equivalent of topological superconductors in the above parallel, which, we conjecture, allows us to compute spectra in the same way as Turaev–Viro procedure allows us to compute topological states in the Kitaev–Kong models. In fact, the parallel produces a new spectra code which can be used in quantum computing, opening new horizons for research. We outline this parallel in Table 7.

Table 7 Gaps, spectra, and topological superconductors

The discovery of phantoms came contrary to the expectations of the founding fathers of derived categories. Today it is known that phantoms are everywhere, in the same way as topological insulators and topological superconductors—a truly groundbreaking unconventional parallel. We anticipate striking applications of phantoms in the study of rationality of algebraic varieties. Let us briefly outline one of these applications. We consider another conic bundle—Sarkisov’s example; see [90] and Table 8. This example can be described as follows—we start with an irreducible singular plane curve C _sing in ${\mathbb{P}}^{2}$ of degree $d \geq 3$ that has exactly (d − 1)(d − 2) ∕ 2 − 1 ordinary double points (such curves exists for every $d \geq 3$). Then we blow up ${\mathbb{P}}^{2}$ at the singular points of C _sing . Denote by S the resulting surface and by C the proper transform of the curve C _sing . Then C is a smooth elliptic curve (easy genus count). Let $\tau : \tilde{C} \rightarrow C$ be some unramified double cover. Then it follows from [90, Theorem 5.9] there exists a smooth threefold X of Picard rank rk(Pic(S)) + 1 with a morphism π: X → S whose general fiber is ${\mathbb{P}}^{1}$, i.e., π is a conic bundle, such that C is the discriminant curve of π, and τ is induced by interchanging components of the fibers over the points of C. Moreover, it follows from [90, Theorem 4.1] that X is not rational if $d \geq 12$. On the other hand, we always have ${H}^{3}(X, \mathbb{Z}) = 0$, since C is an elliptic curve. Note that birationally, X can be obtained as a degeneration of a standard conic bundle over ${\mathbb{P}}^{2}$ whose discriminant curve is a smooth curve of degree d. On the side of Landau–Ginzburg models we can observe the following. The mirrors of conic bundles are partially understood—see [1]. The degeneration procedure on the B side amounts to conifold transitions on the A side. These conifold transitions define a moving scheme for the Landau–Ginzburg model which suggests the following conjecture.

Conjecture 4.24.

The SOD of D ^b(X) contains a phantom category—a nontrivial category with trivial K ⁰ group.

This phantom category is the reason for nonrationality of Sarkisov’s conic bundle, which we conjecture has no gaps in the spectra of its derived category—see Table 8.

Table 8 Sarkisov example

So conjecturally we have

$$\displaystyle{{D}^{b}(X) =\langle \mathcal{A},E_{ 1},\ldots,E_{112}\rangle,\ \ \ {K}^{0}(\mathcal{A}) = 0,\ \ \ \mathcal{A}\neq 0.}$$

The degeneration construction above suggests an ample opportunity of constructing phantom categories.

Conjecture 4.25.

The SOD of derived category of degeneration of a generic quadric bundles over a surface contains a phantom category.

As a result, we conjecture nonrationality of such quadric bundles. An interesting question is where this phantom categories come from. The analysis of Sarkisov’s example suggests the following. We start with a conic bundle over ${\mathbb{P}}^{2}$ with a curve of degeneration C. Such a conic bundle has a nontrivial gap in the spectra of its derived category. Via degeneration we reduce this gap in the same way as via degeneration we get rid of the intermediate Jacobian. The degeneration of the two sheeted covering of ${\mathbb{P}}^{2}$ produces a surface with a phantom category in its SOD. This observation provides us with many possibilities to construct geometric examples of phantom categories. After all, many of the classical examples of surfaces of general type are obtained from rational surfaces by taking double coverings, quotienting by group actions, degenerations, and smoothings.

So the existence of nontrivial categories in SOD with trivial Hochschild homology can be conjecturally seen in the case of classical surfaces of general type, Campedelli, Godeaux (see [12]), Burniat (see [3]), Dolgachev surfaces, product of curves ([39]), and also in the categories of quotients of product of curves and fake ${\mathbb{P}}^{2}$. These surfaces are not rational since they have nontrivial fundamental groups but also since they have conjecturally a quasi-phantom subcategory in their SOD.

We call a category a quasi-phantom if it is a nontrivial category with a trivial Hochschild homology. On the Landau–Ginzburg side these quasi-phantoms are described by the moving scheme. The deformation of Landau–Ginzburg models is determined by the moving scheme, thus the (quasi-)phantoms factor in the geometry of the Landau–Ginzburg models. On the mirror side this translates to the fact Ext’s between the (quasi-)phantom category and the rest of the SOD determine the moduli space.

Finding phantoms—nontrivial categories with trivial K ⁰ groups—is a quantum leap more difficult than finding quasi-phantoms. We conjecture that derived categories of the Barlow surface (see [28]) and rational blow-downs contain phantoms in their SOD.

Applying the quadric bundles construction described above one conjecturally can produce many examples of phantom categories. There are two main parallels we build our quadric bundles construction on:

• 1. Degenerations of Hodge structures applied to intermediate Jacobians. This construction goes back to Clemens and Griffiths and later to Alexeev. They degenerate intermediate Jacobians to Prym varieties or completely to algebraic tori. The important information to remember are the data of degeneration. For us the algebraic tori is analogous to the phantom and degeneration data to the gap in the spectrum. The data to analyze is how the phantom fits in SOD. This determines the gap and the geometry of the Landau–Ginzburg moduli space. It is directly connected with the geometry of the moving scheme and the monodromy of LG models.

• 2. The Candelas idea to study rigid Calabi–Yau manifolds by including them in Fano varieties, e.g., four-, seven-, and ten-dimensional cubics. This gives him the freedom to deform. Similarly, by including the phantom in the quadric bundles we get the opportunity to deform and degenerate. The rich SOD of the quadric bundle allows us to study the phantom. This of course is a manifestation of the geometry of the moving scheme and the monodromy at infinity of the moduli space of LG models.

We turn to the A side and pose the following question.

Question 4.26.

Do A side phantoms provide examples of nonsymplectomorphic symplectic manifolds with the same Gromov–Witten invariants?

Remark 4.27.

An initial application of the conic bundles above to classical Horikawa surfaces (see [48]) seems to suggest that after deformation we get a phantom in the Fukaya category for one of them and not for the other. It would be interesting to see if a Hodge type argument would lead to the fact that these two types of Horikawa surfaces have different gaps of spectra of their Fukaya categories and as a result are not symplectomorphic. It will be analogous to degenerating Hodge structures to nonisomorphic ones for the benefit of geometric consequences.

In what follows we move to finding quantitative statements for gaps and phantoms. We have already emphasized the importance of quadric bundles. In what follows we concentrate on moduli space of a stability conditions of local CY obtained as quadric bundles.

We move to a second type of invariants we have mentioned. We take the point of view from [45] that for special type of Fukaya and Fukaya wrapped categories locally stability conditions are described by differentials with coefficients irrational or exponential functions. The main idea in [45] is that for such categories we can tilt the t-structure in a way that the heart of it becomes an Artinian category. Such a simple t-structure allows description of stability condition in terms of geometry of Lefschetz theory and as a result in terms of the moduli space of Landau–Ginzburg models.

As it is suggested in [4] there is a connection between monodromy of Landau–Ginzburg models and the gaps of a spectrum. We record our observations in Tables 9 and 10.

• 1. In the case of A _n category the stability conditions are just exponential differentials as we have demonstrated in Example 4.2. In this case the simple objects for the t-structures are given by the intervals connecting singular points of the function given by the central charge.

• 2. Similarly for one-dimensional Fukaya wrapped categories the simple objects for the t-structures are given by the intervals connecting zero sets of the differentials. This procedure allows us to take categories with quivers.

• 3. For more complicated Fukaya–Seidel categories obtained as a superposition of one-dimensional Fukaya wrapped—see [71]—we describe the stability conditions by intertwining the stability conditions for one-dimensional Fukaya wrapped—look at the last line of Table 10. At the end we obtain a number d ∕ k, where d is the degree of some of the polynomials p(z) involved in the formula and k is the root we take out of it. Such a number can be associated with Fukaya–Seidel category associated with a local Calabi–Yau manifold obtained as a quadric bundle.

Table 9 Landau–Ginzburg models and stability conditions

Table 10 Conjectural duality

The following conjecture suggests local invariants:

Conjecture 4.28.

The geometry of the moving set determines the number d ∕ k and the gap of the spectra of the corresponding category.

This conjecture suggests the following question.

Question 4.29.

Is the number d ∕ k a birational invariant?

In Table 10 we give examples of categories and their stability conditions. These categories serve as building blocks for more involved categories of Fano manifolds. In case the question above has a positive answer we get a way of comparing the moving schemes of the divisors at infinity. We will also get a way of deciding if the corresponding moduli spaces of Landau–Ginzburg models can be deformed to each other which according to [29] is a way of deciding if we can build a Sarkisov link between them.

It is clear that the numbers d ∕ k fit well in the landscape of quadric bundles. We expect particularly interesting behavior from the numbers d ∕ k coming from the phantom categories the existence of which was conjectured earlier.

Question 4.30.

Can we read off the existence of phantoms and gaps in terms of the numbers d ∕ k?

It is also clear that the geometry of the moving scheme in general has a deep connection with the geometry of the Fano manifold. In fact, following [93], we associate a complex of singularities with this moving scheme. So it is natural to expect that we can read many geometrical properties of Fano manifolds from this complex of singularities. For example, it has been conjectured by S.-T. Yau, G. Tian, and S. Donaldson that some kind of stability of Fano manifolds is a necessary and sufficient condition for the existence of Kähler–Einstein metrics on them. This conjecture has been verified in the two-dimensional case (see [95]) and in the toric case (see [97]). Moreover, one direction of this conjecture is now almost proved by Donaldson, who showed that the existence of the Kähler–Einstein metric implies the so-called K-semistability (see [31, 32]). Recall that Tian (see [96]) defined the notion of K-stability, arising from certain degenerations of the manifold or, as he called them, test configurations. Proving Yau–Tian–Donaldson conjecture is currently a major research program in Differential Geometry (see [30]). We finish with the following question:

Question 4.31.

Can we read off the existence of a Kähler–Einstein metric on the Fano manifold from this complex of singularities?