The generalized Mukai conjecture

1 Introduction

A Fano manifold is a smooth connected complex projective variety X whose anticanonical divisor -K_X is ample. Its Picard number \rho_X is the rank of its Néron–Severi group, or equivalently the dimension of the space of numerical divisor classes. If X has positive dimension, its pseudoindex is

The existence of rational curves on a Fano manifold makes this minimum well defined. The pseudoindex records the smallest anticanonical curve degree; it need not equal the largest integer dividing -K_X in \operatorname{Pic}(X), which is the Fano index.

The generalized Mukai conjecture asks for a sharp bound on the Picard number in terms of dimension and pseudoindex, together with a classification of the extremal manifolds. We prove the conjectured bound and classification in arbitrary dimension.

The index version originates in Mukai’s work [14]. Wiśniewski introduced the pseudoindex and proved that 2\iota_X>n+2 forces \rho_X=1 [24, Theorem A]. Bonavero, Casagrande, Debarre and Druel formulate the pseudoindex strengthening and establish it for Fano manifolds of dimension at most four, and for smooth toric Fano manifolds when n\le7 or 3\iota_X\ge n+3 [4]. Andreatta, Chierici and Occhetta extend the low-dimensional result to dimension five and prove high-pseudoindex cases subject to contraction hypotheses [1].

For varieties with additional symmetry, Casagrande settles the bound and equality classification for \mathbb Q-factorial Gorenstein toric Fano varieties in every dimension [6, Theorem 1(ii)], and Pasquier treats \mathbb Q-factorial Gorenstein horospherical Fano varieties [22, Theorem 1]. More recently, Gagliardi, Hofscheier and Pearson prove the generalized conjecture for \mathbb Q-factorial spherical Fano varieties [8].

Large pseudoindex gives a different route. Novelli and Occhetta prove both conclusions for all smooth Fano manifolds with \iota_X\ge(n+3)/3 [19], and Novelli later enlarges this range to \iota_X>n/3 [17, Theorem 5.1]. At the boundary n=3\iota_X, Novelli also proves the conjecture when X admits an extremal contraction of fiber type [18]. Theorem 1.1 imposes no restriction on the dimension, Picard number or geometry of the smooth Fano manifold.

The ordered-chain method of Bonavero, Casagrande, Debarre and Druel derives the inequality once \rho_X numerically independent proper irreducible components of the space of rational curves, joined by a nonempty chain, are available [4, Corollary 5.3]. Our argument first detects independent curve degrees in quantum correspondences. Proper covering families enter only at equality, when all the selected degrees are minimal.

The argument

Write r=\rho_X and \iota=\iota_X. If \iota=1, the inequality is strict because n>0, so assume \iota\ge2. Choose divisor classes D_1,\ldots,D_r forming a numerical basis. Small quantum multiplication by these divisors gives commuting matrices A_1(q),\ldots,A_r(q) on the even cohomology of X, with coefficients depending on r nonzero formal variables q_j. Each matrix consists of classical cup product and a finite sum of two-point curve correspondences. A correspondence of anticanonical degree d lowers cohomological codimension by d-1. Consequently, a nonzero product in r independent curve degrees would give

The main task is to produce that product without assuming that quantum cohomology is semisimple.

We first prove a lower bound for a point descendant, an intersection number on the space of one-pointed rational stable maps involving the (d-2)nd power of the marked cotangent class. Here d is the anticanonical degree. For any degree represented by a free rational map (a map whose pulled-back tangent bundle is globally generated), this number is at least (ad)^{-(d-1)}, where -aK_X is a fixed very ample divisor. To prove positivity, we contract all components away from the marking and map the evaluation fiber to a weighted projective stack. The image of its boundary has dimension too small to contribute. The pushed-forward virtual class is therefore a nonempty effective cycle, whose degree admits the required uniform bound. Section 3 gives the construction in families and keeps track of stack multiplicities.

Two-point descendant solutions of quantum differential equations provide a conceptual antecedent for the recurrence used next [9, §6]. Here the divisor and recursion identities give a finite-shift recurrence for descendant vectors on an arbitrary smooth Fano manifold. After factorial normalization it yields an exponential upper bound. If every joint eigenvalue tuple of the quantum divisors satisfied a polynomial relation, the same recurrence would force decay of order \exp(-c d\log d) along a suitable free curve degree. After the same factorial normalization, the descendant lower bound permits only exponential decay and rules this out. Thus one joint tuple has r algebraically independent coordinates (Section 4).

Section 5 converts this algebraic independence into the required curve correspondences. An alternating trace of matrix-valued differentials is invariant even under changes of basis depending on q. Computing it in a common triangular basis detects the independent eigenvalues; computing it in the fixed cohomology basis detects a nonzero product in independent curve degrees. This argument also applies to nonsemisimple commuting matrices.

At equality all these degrees are minimal. Their nonzero product implies the existence of a chain of actual rational curves; minimality makes the full irreducible families containing them proper. The ordered-chain bound of Bonavero, Casagrande, Debarre and Druel applies to these components, as required by its erratum [3]. A cyclic reordering makes every family covering, and Occhetta’s characterization of products of projective spaces gives the asserted rigidity [20]. Section 6 verifies each of these family hypotheses.

2 Quantum divisors and point descendants

We set up a finite collection of commuting matrices and a sequence of vectors on which they act. Their grading records anticanonical degrees; the vectors will connect their joint eigenvalues to rational curves. Throughout the proof, X is a smooth projective complex Fano manifold, n=\dim X>0, r=\rho_X, and \iota=\iota_X\ge2. The case \iota=1 will require no argument beyond n>0.

Choose an integral basis D_1,\ldots,D_r of the Néron–Severi group modulo torsion. A numerical curve degree is represented by

Here and below we use the same symbol for a numerical degree and its coordinate vector. Write

A positive degree means the degree of a nonconstant genus-zero stable map. Every such degree has d_\beta\ge\iota. There are only finitely many positive numerical degrees with bounded d_\beta: for a sufficiently large integer a, both a(-K_X)+D_j and a(-K_X)-D_j are ample, so |\beta_j|\le a d_\beta on effective degrees. When several curve classes have the same numerical degree, we sum their invariants. Proper finite-type spaces of stable maps of bounded projective degree make these sums finite.

We use ordinary genus-zero Gromov–Witten invariants, defined by the virtual fundamental classes of stable-map stacks. An insertion \tau_\ell(a) means \psi^\ell\ev^*a at the indicated marking: \ev evaluates the stable map there, and \psi is the first Chern class of its cotangent line. The divisor, string and topological recursion equations, together with associativity, are the standard identities we use [2, 12, 13, 21]. The discussion below specifies all unstable conventions needed here. Set

and put H_k=0 outside this range. The pairing on H is nondegenerate. We work only with even cohomology; invariants involving a single odd intermediate class and otherwise even insertions vanish by parity.

For a positive numerical degree \beta, define S_\beta\in\operatorname{End}(H) by

The virtual dimension of the two-pointed space is n+d_\beta-1. Consequently

In particular S_\beta=0 for d_\beta>n+1. For nonzero formal variables q_1,\ldots,q_r, write q^\beta=\prod_jq_j^{\beta_j} and define

Thus A_{j,0}=D_j\cup(-), A_{j,\beta}=\beta_jS_\beta for positive degrees, and all other coefficients vanish. The grading bound and finiteness of bounded degrees make these Laurent polynomial matrices. The divisor equation identifies them with small quantum multiplication by D_j, so associativity implies that they commute. If R_t acts by t^k on H_k, then

Indeed the classical term raises the grading by one, whereas the \beta term changes it by 1-d_\beta.

2.1 The descendant recurrence

Two-point descendant generating series also occur in Givental’s quantum differential framework [9, §6]. The recurrence below is derived directly from the virtual genus-zero identities cited above, so it does not require a convexity assumption on X.

Let \pt\in H_n be the class of a point, normalized by \int_X\pt=1. Define v_0=\pt and, for positive degrees, define v_\beta\in H by

The dimension constraint makes each sum finite. Set all other v_\beta equal to zero. In particular there is no degree-zero unstable two-point invariant in this definition.

The next section gives a positive lower bound for this pairing whenever \beta has a free rational representative. It is a statement about a pushforward of the virtual evaluation fiber; it will not require the entire virtual class to be effective.

3 A lower bound for point descendants

The spectral argument will require point descendants in a sequence of large curve degrees. Their virtual definition does not give positivity directly. We prove the required lower bound by a contraction that forgets all components away from the marked component. Its boundary image is too small to contribute to the integral.

Throughout this section, X is a smooth projective complex Fano manifold of dimension n>0 and pseudoindex at least two. Fix an embedding

where a is a positive integer. A morphism f:\mathbb P^1\to X is free if f^*T_X is globally generated. For a numerical curve degree \beta, set d_\beta=-K_X\cdot\beta. The notation \overline{\mathcal M}_{0,1}(X,\beta) includes the union over actual curve classes with numerical degree \beta. This is a finite union: maps of fixed projective degree belong to a finite-type proper moduli stack, and numerical degree is locally constant.

We use rational Chow groups of Deligne–Mumford stacks. In particular, degrees of zero-dimensional stack cycles include their stabilizer denominators. The marked cotangent line has first Chern class \psi.

We will express this descendant as the degree of a nonempty effective cycle in a weighted projective stack whose weights are at most b. The main task is to obtain that cycle from the virtual evaluation fiber: irreducible maps will contribute with stack degree one, while the boundary image will have too small a dimension to contribute. After proving the bound, we show that free degrees are Zariski dense in the numerical curve space.

3.1 A free evaluation fiber

There is a very general point x\in X such that every nonconstant morphism \mathbb P^1\to X whose image contains x is free. Here is the argument, including the quantifier over special maps. Stratify the nonfree locus in each morphism scheme into smooth irreducible locally closed reduced strata. There are countably many such strata. If the evaluation from \mathbb P^1\times T, for one stratum T, were dominant, generic smoothness would give a point (p,f) where its differential is surjective. Its image is contained in the image of

Indeed, moving p gives directions also obtained from reparametrizations of the source. Splitting f^*T_X=\bigoplus_j\mathcal O(a_j), surjectivity at one point implies a_j\ge0 for every j, contrary to the choice of T. Thus each such evaluation image has proper closure, and their complement gives the assertion.

Evaluation is smooth at a free map: the obstruction groups H^1(\mathbb P^1,f^*T_X) and H^1(\mathbb P^1,f^*T_X(-p)) vanish. Consequently the degree \beta in Proposition 3.1 has maps through an open subset of X. Choose x in this open subset and outside all the exceptional closed subsets just considered.

Let

The fiber carries the refined virtual class

Let U\subset F be the open substack with irreducible source. It is nonempty, smooth of dimension h, and

Only this open part is asserted to be unobstructed.

To use this equality of classes on U, we will construct a proper morphism that preserves the marked cotangent class, has generic stack degree one on U, and sends F\setminus U to a locus of dimension less than h. These properties will make the pushforward of [F]^{\mathrm{vir}} effective.

3.2 Contracting away from the marking

Compose maps with \jmath. Their projective degree is b, and this composition changes no stable domain, since it contracts no nonconstant component. At a geometric point, retain the component carrying the marking and replace each attached tree by a common zero of the map sections at its attachment point. More precisely, multiply the original sections restricted to the marked component by a common factor whose vanishing order at each attachment is the projective degree of the removed tree. The resulting tuple has total degree b and no common zero at the marking. This description also allows the marked component to be constant.

We now construct these data in families on \overline{\mathcal M}_{0,1}(\mathbb P^N,b) and track the marked cotangent line. The contraction also follows from the one-marked intermediate-space construction of Mustaţă–Mustaţă [16, Definition 1.2 and Proposition 1.3], with the fine-stack construction of [15, Proposition 1.7].

Write \pi:\mathcal C\to\overline{\mathcal M}_{0,1}(\mathbb P^N,b) for the universal curve, s for its marking, and \mathcal L for the pullback of \mathcal O_{\mathbb P^N}(1). The moduli stack is smooth and its boundary has the usual independent node-smoothing parameters: on a genus-zero tree, H^1(C,f^*T_{\mathbb P^N})=0. Thus the universal curve is regular, with local nodal models uv=t.

For each boundary branch separating an unmarked tail of total degree e, take in a local node-smoothing chart the Cartier divisor consisting of its unmarked side. The marked side and degree e are intrinsic, so summing these local divisors over all such branches is invariant under permutations of equal-degree branches and glues in families. Let \Delta_e be this sum, counting repeated branches once each. Only 1\le e\le b-1 occurs. A tail of degree b would leave a degree-zero marked side with only one external attachment; a finite tree of contracted components with one marking and one external attachment cannot satisfy stability.

Set

On a nodal fiber, twisting by an unmarked-side divisor transfers one unit of degree across that node toward the marked side. To compute the resulting multidegree, root the dual tree at the marked component. For a nonroot component v, let b_v be its original degree and E_v the total degree of the subtree rooted at v. If w runs over its children, then

On the root, of original degree b_0, the degree is b_0+\sum_wE_w=b. Thus \mathcal L^+ has degree b on the component containing s, and degree zero elsewhere.

The line bundle \mathcal O_{\mathcal C}(s) has degree one on that marked component and degree zero elsewhere. On each fiber it is globally generated with h^0=2 and h^1=0. Cohomology and base change consequently give a rank-two bundle \pi_*\mathcal O_{\mathcal C}(s), and its evaluation map contracts the universal curve onto a \mathbb P^1-bundle. This contraction is an isomorphism on the marked component and near the section.

The line bundle \mathcal L^+(-bs) has multidegree zero on every fiber. Its fiberwise h^0=1, h^1=0, and evaluation map show that it is pulled back from the base. Hence \mathcal L^+ descends to a relative degree-b line bundle on the contracted \mathbb P^1-bundle. Fiberwise pullback identifies its sections with those of \mathcal L^+: a section is constant on every contracted tail and is determined by its value on the marked component. The corresponding H^0 bundles agree by base change, so the inclusion \mathcal L\to\mathcal L^+ sends the original map sections to sections of the descended bundle in families. Restricted to the marked component, this inclusion multiplies them by a section vanishing to order e at the attachment of a tree of degree e. The original map sections have no common zero, so the new common zero has exactly that multiplicity. These constructions commute with base change and produce the retained data in families. Since the contraction is an isomorphism near the marking, it preserves the marked cotangent line.

3.3 The weighted projective fiber

Choose target coordinates with x=[1:0:\cdots:0]. Locally on the base, identify the contracted bundle with a product \mathbb P^1-bundle and the descended line bundle with \mathcal O_{\mathbb P^1}(b). Put the mark at infinity and normalize the degree-b leading coefficient vector of the polynomial sections to (1,0,\ldots,0). Translation of the affine coordinate uniquely eliminates the coefficient of degree b-1 in the first polynomial. After these normalizations, two local choices differ only by scaling the affine coordinate. Consequently the coefficient tuples glue as points of the quotient stack for this residual scaling action.

The remaining coefficients have weights

They are not all zero. Otherwise the polynomial tuple would be (z^b,0,\ldots,0), a constant map with all degree concentrated at one basepoint. A constant marked component has at least two distinct attachments by stability, so cannot produce that tuple; a nonconstant marked component cannot produce it either.

Let w_1,\ldots,w_M be the displayed list of weights and put

This is the weighted projective stack in question. We have constructed a proper morphism

The map is proper because F is proper and \mathcal W is separated. For the reparametrization z\mapsto\lambda z, the normalized coefficient of z^{b-j} has weight -j, and the cotangent d(1/z) has weight -1. Replacing \lambda by \lambda^{-1} gives the displayed positive weights and identifies the cotangent line with the pullback of \mathcal O_{\mathcal W}(1), as in the one-marked weighted construction of [16, proof of Lemma 3.3]. Thus, with \xi=c_1(\mathcal O_{\mathcal W}(1)),

On the open substack U, the tuple has no basepoints and recovers the entire pointed map. Its image is the locus of basepoint-free tuples whose maps factor through \jmath(X) and have numerical degree \beta there. A residual scaling fixes the tuple exactly when the associated reparametrization preserves the pointed map, so the stabilizers are the automorphism groups of the pointed maps. We therefore have an identification of stacks onto this locus, with generic stack degree one on every irreducible component of U.

3.4 The boundary does not contribute

Consider a boundary stratum of F. Let d_0 be the anticanonical degree of the marked component and let u\ge1 be the number of trees directly attached to it. Its image in \mathcal W is determined by the marked map, the mark and attachment points, and the degrees of those trees. It does not retain the maps on the trees: a tree of anticanonical degree d_i is recorded only by its common-zero multiplicity ad_i.

If d_0>0, the marked map is free, since it passes through x. The space of its parametrized maps has dimension n+d_0. Adding u+1 marked points, imposing evaluation x, and dividing by source reparametrization gives dimension

If d_0=0, the map is the constant x, stability gives u\ge2, and the remaining data have dimension \dim\mathcal M_{0,u+1}=u-2, the same formula.

Every attached tree contains a nonconstant rational component. The pseudoindex assumption implies

There are finitely many discrete boundary types. We conclude that

This estimate concerns the image only; it makes no smoothness or dimension assertion about the spaces of tail maps.

Write Z_\alpha for the closures in F of the irreducible components of U. Chow localization and [F]^{\mathrm{vir}}|_U=[U] express

as a class supported on F\setminus U. Its pushforward to \mathcal W vanishes, since every h-dimensional boundary cycle has image of smaller dimension. The generic degree-one statement on U therefore gives

This is a nonempty effective sum of h-dimensional integral stack cycles. Virtual effectiveness has been established only after pushforward; no effectiveness of [F]^{\mathrm{vir}} itself was assumed.

3.5 Degrees in a weighted projective stack

3.6 Enough free degrees

The next lemma lets us apply the descendant bound in a direction avoiding any specified nonzero polynomial relation. Write N_1(X)_{\mathbb C} for the complex vector space of numerical curve classes.

4 An independent joint eigenvalue tuple

We now combine the descendant lower bound with the recurrence to show that the quantum divisors vary in all r numerical directions. Let

The commuting matrices A_1,\ldots,A_r can be simultaneously put in upper triangular form over L. A joint diagonal tuple is the r-tuple of diagonal entries at one position in such a form.

We first normalize the descendant recurrence to obtain a uniform exponential upper bound. We then show that a polynomial relation among every joint tuple would force superexponential decay along a free degree, contradicting Proposition 3.1.

4.1 Factorial normalization

For d_\beta\ge0 and 0\le k\le n, put

where v_{\beta,k} is the H_k component. Put w_\beta=0 when d_\beta<0. For a sequence u indexed by \mathbb Z^r, let (E^\gamma u)_\beta=u_{\beta-\gamma} and set A_j(E)=\sum_\gamma A_{j,\gamma}E^\gamma. These are finite sums of shifts, and they commute.

If A_{j,\gamma} sends an input component H_l to H_k, then l=k-1+d_\gamma. Thus

Multiplying the H_k component of (4) by (d_\beta+k-1)! gives, for d_\beta>0,

Every nonzero input term has a nonnegative factorial argument. We do not use this formula at degree zero.

Fix a norm on H. There is a constant B>1 such that

To prove this, let C_*=(\sum_jc_jD_j)\cup(-), and let T_d multiply H_k by 1+k/d for positive integers d. Taking the linear combination of (6) with coefficients c_j yields

Since T_dC_* raises grading, the inverse (I-T_dC_*)^{-1} is the finite sum \sum_{a=0}^n(T_dC_*)^a. Its norm, and that of T_d, are uniformly bounded for d\ge1. The sum uses finitely many positive shifts, each of which strictly decreases d. Induction on the nonnegative integer d proves (7), taking B large enough for the finite set of coefficients and the initial value w_0=n!\pt. All other degree-zero coefficients are zero.

4.2 A relation would force too much decay

5 From divisor eigenvalues to independent curve degrees

We now turn algebraic independence of a joint eigenvalue tuple into a nonzero product of curve correspondences in independent degrees. The point requiring care is that a common triangular basis can depend on the variables q_1,\ldots,q_r. The following lemma accounts for the derivatives of that basis and applies without a semisimplicity assumption.

For the Hochschild generalized-trace and inner-conjugation viewpoint, see Khalkhali [10, Section 3.5, Examples 3.5.5–3.5.6 and Lemma 3.5.1]. We give an elementary proof of the specific differential-form statement needed here.

We apply these lemmas over L=\overline{\mathbb C(q_1,\ldots,q_r)} to the quantum divisor operators. Common triangularization is available for any finite commuting collection over an algebraically closed field. The differential extends to L, since the algebraic extension is separable.

Proposition 4.1 supplies the algebraically independent joint tuple required in Proposition 5.3. This proves the inequality in Theorem 1.1; the equality case is considered next.

6 The equality case

We show that equality in the grading bound forces the projective-space product. The nonzero product from Proposition 5.3 first gives a chain of actual minimum-degree rational curves. A dimension estimate then turns the corresponding families into covering families.

We use \operatorname{RatCurves}^{\mathrm n}(X) for the normalized parameter space of rational curves, obtained from maps \mathbb P^1\to X birational onto their images by reparametrization. In this section a family V is an irreducible component of that space. It is unsplit if it is proper, and covering if its universal evaluation map dominates X. Its anticanonical degree is -K_X\cdot V, the common degree of its members. For unsplit families V^1,\ldots,V^k and a point x\in X, write \operatorname{Locus}(V^1,\ldots,V^k)_x for the set of endpoints y of chains C_1,\ldots,C_k with C_j\in V^j, x\in C_1, C_j\cap C_{j+1}\ne\varnothing, and y\in C_k.

We will use the following two established results, with their family hypotheses stated explicitly.

This is the theorem of Bonavero–Casagrande–Debarre–Druel [4, Theorem 5.2], with the irreducible-component hypothesis specified in their erratum [3]. Taking arbitrary proper subfamilies would not suffice.

We use Theorem 6.2 in the form of [20, Theorem 1.1]. The next lemma verifies that the curve families needed for these results are supplied by our correspondences.

Conversely, (\mathbb P^{\iota-1})^r has dimension r(\iota-1) and Picard number r. Its anticanonical divisor has degree \iota on a line in one factor, and degree at least \iota on every irreducible rational curve. Its pseudoindex is therefore \iota, and equality holds.