The spectral radius ρ(c^1)\rho(\hat c_1) of quantum multiplication by the first Chern class on the small quantum cohomology (specialized at q=1q=1) of a Fano manifold is a central numerical invariant of the Gamma-conjecture program, but its behavior under birational modifications is unknown. Hu, Ke, Li and Song proposed the conjecture that if YY is a Fano manifold and Z⊂YZ\subset Y an irreducible smooth Fano subvariety of codimension at least two such that the blow-up X=BlZYX=\mathrm{Bl}_Z Y is again Fano, then ρ(c^1(X))>ρ(c^1(Y))\rho(\hat c_1(X))>\rho(\hat c_1(Y)). The conjecture refines Galkin's lower bound conjecture — which is now known to be false in general — in the smooth blow-up setting. The conjecture is proved only for the blow-ups of the four-dimensional quadric Q4Q^4 at a point and along a plane, and numerically for del Pezzo surfaces; the case BlP1(Q4)\mathrm{Bl}_{\mathbb{P}^1}(Q^4), the blow-ups of Grassmannians along sub-Grassmannians, the general statement, and the authors' proposed removal of the smoothness hypothesis on the center all remain open. The known counterexamples to Galkin's lower bound conjecture and to Conjecture O\mathcal{O} arise from resolutions of singular weighted projective spaces and do not affect the smooth-center statement, which remains a precise, open problem.

Background

A Fano manifold is a smooth complex projective variety whose first Chern class c1(X)c_1(X) is positive, equivalently whose anticanonical line bundle is ample. Its small quantum cohomology ring QH∗(X)QH^*(X) deforms the classical cohomology ring: the product incorporates genus-zero, three-pointed Gromov–Witten invariants, which are integrals over moduli spaces of stable maps and measure counts of rational curves on XX. Multiplication by c1(X)c_1(X) in this deformed ring, evaluated at all quantum variables equal to 11, defines a linear operator c^1\hat c_1 on the even cohomology Hev(X)H^{\mathrm{ev}}(X). Its spectrum controls the asymptotics of solutions of the quantum differential equation, and its spectral radius ρ(c^1)=max⁡{∣λ∣:λ∈Spec(c^1)}\rho(\hat c_1)=\max\{|\lambda|:\lambda\in\mathrm{Spec}(\hat c_1)\} is a fundamental numerical invariant of the Fano variety.

Two conjectures organize the study of this invariant. Conjecture O\mathcal{O} of Galkin–Golyshev–Iritani asserts that ρ(c^1)\rho(\hat c_1) is a simple eigenvalue of c^1\hat c_1, and that every eigenvalue of modulus ρ(c^1)\rho(\hat c_1) is ρ(c^1)\rho(\hat c_1) times an ss-th root of unity, where ss is the Fano index, i.e. the largest integer ss with c1(X)=s cc_1(X)=s\,c for an integral class c∈H2(X,Z)c\in H^2(X,\mathbb{Z}); it underlies Gamma conjecture I, which relates the quantum connection to exceptional collections of coherent sheaves. Galkin's lower bound conjecture asserts ρ(c^1)≥dim⁡X+1\rho(\hat c_1)\ge \dim X+1, with equality only for projective space. Both statements were verified for many homogeneous spaces and complete intersections, but both are now known to fail in general: for the Picard-rank-two toric Fano manifolds PPn(O⊕O(3))\mathbb{P}_{\mathbb{P}^n}(\mathcal{O}\oplus\mathcal{O}(3)) with nn sufficiently large one has ρ(c^1)<dim⁡X+1\rho(\hat c_1)<\dim X+1, and PPn(O⊕O(n))\mathbb{P}_{\mathbb{P}^n}(\mathcal{O}\oplus\mathcal{O}(n)) gives counterexamples to Conjecture O\mathcal{O} and Gamma conjecture I. Notably, all known counterexamples are toric Fano manifolds that arise as resolutions of singular weighted projective spaces, not as blow-ups of smooth Fano manifolds along smooth centers.

Small quantum cohomology is not functorial under birational maps, so its behavior under blow-ups must be analyzed case by case. If π ⁣:X=BlZY→Y\pi\colon X=\mathrm{Bl}_Z Y\to Y is the blow-up of a smooth variety YY along a smooth subvariety ZZ (the operation replacing ZZ by the projectivized normal bundle, an exceptional divisor EE over ZZ), then blowing up along a smooth irreducible divisor returns YY itself, which is why centers of codimension at least two are the interesting ones; the condition that XX remain Fano is additionally quite restrictive. In this setting Hu, Ke, Li and Song proposed the following conjecture, formulated while studying the quantum cohomology of blow-ups of the four-dimensional quadric Q4⊂P5Q^4\subset\mathbb{P}^5.

Conjecture 1.1 (Hu–Ke–Li–Song). Let YY be a Fano manifold and Z⊂YZ\subset Y an irreducible smooth Fano subvariety with codim Z≥2\mathrm{codim}\, Z\ge 2. If the blow-up X=BlZYX=\mathrm{Bl}_Z Y is Fano, then ρ(c^1(X))>ρ(c^1(Y))\rho(\hat c_1(X))>\rho(\hat c_1(Y)).

The conjecture gives a refined lower bound for blow-ups: since ρ(c^1(Pn))=n+1\rho(\hat c_1(\mathbb{P}^n))=n+1, it reduces for Y=PnY=\mathbb{P}^n to Galkin's lower bound conjecture for the toric blow-ups BlPrPn\mathrm{Bl}_{\mathbb{P}^r}\mathbb{P}^n. Its evidence base is: numerical verification for del Pezzo surfaces (blow-ups of P2\mathbb{P}^2 at up to seven general points, with the characteristic polynomials computed from the quantum cohomology determinations of Bayer–Manin); numerical evidence from the critical values of mirror superpotentials for BlPrPn\mathrm{Bl}_{\mathbb{P}^r}\mathbb{P}^n, for which Yang proved Conjecture O\mathcal{O} and Gamma conjecture I; and a rigorous proof for the blow-ups of Q4Q^4 at a point and along a plane P2\mathbb{P}^2, the main theorem of the source paper, obtained by explicitly determining the two-point genus-zero Gromov–Witten invariants, assembling the matrix of c^1\hat c_1, and applying the Perron–Frobenius theorem (the classical theory of eigenvalues of matrices with nonnegative entries) in well-chosen bases. Since Q4Q^4 is isomorphic to the Grassmannian Gr(2,4)\mathrm{Gr}(2,4) via the Plücker embedding and ρ(c^1(Q4))=42>5=dim⁡Q4+1\rho(\hat c_1(Q^4))=4\sqrt{2}>5=\dim Q^4+1, Conjecture 1.1 implies Galkin's lower bound for every Fano blow-up of Q4Q^4.

The main structural tool available, Iritani's decomposition theorem for the quantum D-module of a blow-up, expresses the big quantum cohomology of BlZY\mathrm{Bl}_Z Y through that of YY and of ZZ; but it does not diagonalize the small quantum product at q=1q=1, which is what the operator c^1\hat c_1 sees, so the conjecture cannot currently be reduced to known decompositions. The treated cases, together with BlPrPn\mathrm{Bl}_{\mathbb{P}^r}\mathbb{P}^n, are the first members of the family of blow-ups of Grassmannians Gr(k,n)\mathrm{Gr}(k,n) along sub-Grassmannians, which the authors identify as the natural unifying setting for a systematic study.

Problem Statement

Prove or refute Conjecture 1.1 of Hu–Ke–Li–Song: for every Fano manifold YY and every irreducible smooth Fano subvariety Z⊂YZ\subset Y with codimYZ≥2\mathrm{codim}_Y Z\ge 2 such that the blow-up X=BlZYX=\mathrm{Bl}_Z Y is Fano, the spectral radius of the operator of quantum multiplication by the first Chern class on the even-part small quantum cohomology specialized at q=1q=1 strictly increases:

ρ(c^1(BlZY))  >  ρ(c^1(Y)).\rho\bigl(\hat c_1(\mathrm{Bl}_Z Y)\bigr)\;>\;\rho\bigl(\hat c_1(Y)\bigr).

Within this objective the source flags three related unresolved instances: the case X=BlP1(Q4)X=\mathrm{Bl}_{\mathbb{P}^1}(Q^4), the blow-up of the four-dimensional quadric along a line, whose two-point Gromov–Witten invariants require ad hoc arguments involving the WDVV equations (the associativity equations of genus-zero Gromov–Witten theory) and for which the source contains only an unproved remark that the conjectures hold; the systematic verification for blow-ups of Grassmannians Gr(k,n)\mathrm{Gr}(k,n) along sub-Grassmannians, the natural unifying family containing all previously treated cases; and the question, posed but left unaddressed by the authors, whether the smoothness hypothesis on ZZ can be removed — noting that the blow-up of a smooth variety along a singular center is itself singular, so such an extension must also specify in what sense quantum cohomology and c^1\hat c_1 are defined. The completion boundary is the determination of the smooth-center statement above; the singular-center refinement is the source's declared extension and is not part of the stated problem.

References

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