Blow-up compatibility of reflection vectors in quantum cohomology: convergence of the quantum cohomology of Bl(X) and the reflection-vector property for pullbacks of exceptional objects

For a smooth projective variety $X$ with convergent, semisimple big quantum cohomology, Milanov and Xia proved that the cohomology classes $\Psi_q(\mathcal{O}_E(k))$ attached to the exceptional-locus line bundles of the point blow-up $\pi\colon \mathrm{Bl}_{\mathrm{pt}}(X)\to X$ are reflection vectors for the quantum cohomology of $\mathrm{Bl}_{\mathrm{pt}}(X)$, provided the quantum cohomology of the blow-up is itself assumed convergent. Their paper leaves two questions open: (1) whether the blow-up operation preserves convergence of quantum cohomology, i.e. whether convergence of the big quantum product of $X$ implies convergence for $\mathrm{Bl}_{\mathrm{pt}}(X)$, so that the auxiliary hypothesis is redundant; and (2) whether the pullbacks $\pi^*E_i$ of the members of a full exceptional collection of $X$ correspond to reflection vectors $\Psi_q(\pi^*E_i)$ for the quantum cohomology of the blow-up. Affirmative answers, combined with the already-proven reflection vectors from the exceptional divisor and Orlov's semiorthogonal decomposition of the derived category of a blow-up, would complete the reflection-vector form of Dubrovin's conjecture (Conjecture 1.6 of Milanov-Xia) for blow-ups at finitely many points.

Background

Gromov-Witten (GW) invariants of a smooth complex projective variety $X$ are rational counts of stable maps from genus-0 curves into $X$, recorded per effective curve class $d$; the class variables $Q^d$ are called Novikov variables. The 3-point genus-0 invariants deform the cup product into the small quantum product, and the full system of genus-0 descendants defines the big quantum cohomology: a commutative, associative product on $H^*(X;\mathbb{C})$ depending formally on the Novikov variables and on coordinates $t$ on $H^*(X;\mathbb{C})$. The big quantum cohomology is called convergent if these formal structure constants have a non-empty domain of analytic convergence in $(Q,t)$, and semisimple if the resulting algebra is, at generic parameters, a product of copies of $\mathbb{C}$. On such a domain the big quantum cohomology defines a semisimple Frobenius manifold in the sense of Dubrovin: a complex manifold with a flat metric, flat unit, and Euler vector field satisfying the WDVV associativity equations. Convergence is conjectural for general $X$; it is automatic for Fano varieties, where only finitely many curve classes contribute, and was proved by Cotti under the assumption that the small quantum cohomology is semisimple at a point, together with a finiteness condition on the Novikov sums.

On a semisimple Frobenius manifold $M$ one has locally defined canonical coordinates $u_1,\dots,u_N$ in which the Frobenius product diagonalizes; the discriminant is the locus where two canonical coordinates collide. Dubrovin's structure connection and its Laplace transform, the second structure connection, are meromorphic flat connections on $M\times\mathbb{C}$ whose monodromy around the discriminant is a reflection group. Milanov and Xia formalize this via period vectors - vectors $\alpha$ in the finite-dimensional space $\mathcal{A}$ of solutions of an auxiliary linear system, equipped with a monodromy-invariant intersection pairing $(\,|\,)$ - and define a reflection vector to be a period vector $\alpha$ with $(\alpha|\alpha)=2$ whose analytic continuation along a small loop around a generic discriminant point acts as $-1$ on the corresponding solution and as the orthogonal reflection $x\mapsto x-(x|\alpha)\alpha$ on $\mathcal{A}$. These vectors generate the stable monodromy group of the Frobenius manifold and encode its Stokes data.

In parallel, an object $E$ of the bounded derived category $D^b(X)$ of coherent sheaves is exceptional if $\mathrm{Ext}^k(E,E)=\mathbb{C}$ for $k=0$ and $0$ otherwise, and a sequence $(E_1,\dots,E_N)$ of such objects is a full exceptional collection if it generates $D^b(X)$ in the sense of triangulated categories. Dubrovin's conjecture (1998), in its refined versions (the Gamma conjecture II of Galkin-Golyshev-Iritani and the refined Dubrovin conjecture of Cotti-Dubrovin-Guzzetti), predicts for Fano $X$ that semisimplicity of quantum cohomology is equivalent to the existence of a full exceptional collection, with the monodromy data (central connection matrices) given by explicit characteristic classes: the gamma class $\widehat{\Gamma}(X)=\prod_x\Gamma(1+x)$ over the Chern roots $x$ of the tangent bundle, and Chern characters $\mathrm{ch}(E)$. Bayer suggested that the Fano condition is unnecessary. For a variety with semisimple quantum cohomology, Milanov and Xia use Iritani's map $\Psi_q\colon K^0(X)\to H^*(X;\mathbb{C})$, $\Psi_q(E)=(2\pi)^{(1-n)/2}\,\widehat{\Gamma}(X)\cup e^{-\sum_i p_i\log q_i}\cup(2\pi i)^{\deg}\mathrm{ch}(E)$ (where $n=\dim_{\mathbb{C}}X$, the classes $p_i$ are first Chern classes of ample line bundles forming a $\mathbb{Z}$-basis of the torsion-free part of $H^2(X,\mathbb{Z})$, $q_i$ are the matching quantum variables, and $(2\pi i)^{\deg}$ multiplies $H^{2j}$ by $(2\pi i)^j$), and conjecture (their Conjecture 1.6): (a) if the quantum cohomology of $X$ is convergent and semisimple, then $\Psi_q(E)$ is a reflection vector for every exceptional object $E$; (b) for a full exceptional collection $(E_1,\dots,E_N)$, the reflection vectors $\Psi_q(E_i)$ generate, up to the monodromy group, all reflection vectors.

For blow-ups the picture is partially worked out. By a theorem of Orlov, if $(E_1,\dots,E_N)$ is a full exceptional collection of $X$, then $(\mathcal{O}_E(-n+1),\dots,\mathcal{O}_E(-1),\pi^*E_1,\dots,\pi^*E_N)$ is a full exceptional collection of the blow-up $\mathrm{Bl}_{\mathrm{pt}}(X)$, where $E\cong\mathbb{P}^{n-1}$ is the exceptional divisor. Bayer proved that blowing up a point preserves semisimplicity of the quantum cohomology, and Gathmann computed the relevant GW invariants. A new difficulty is that the line bundle of the exceptional divisor is not ample, so the GW invariants of the blow-up are in general Laurent series, not power series, in the Novikov variable $Q$ of the exceptional divisor. Milanov and Xia nevertheless proved (their Theorem 1.7) that if the quantum cohomology of $X$ is convergent and semisimple and the quantum cohomology of $\mathrm{Bl}(X)$ is convergent, then $\Psi_q(\mathcal{O}_E(k))$, $k\in\mathbb{Z}$, are reflection vectors for the quantum cohomology of $\mathrm{Bl}(X)$. Their introduction states explicitly that the convergence hypothesis on $\mathrm{Bl}(X)$ is expected to be redundant and that proving $\pi^*E_i$ to be reflection vectors is the remaining step toward Conjecture 1.6 for blow-ups at finitely many points; their attempts on the latter were unsuccessful.

Problem Statement

Let $X$ be a smooth complex projective variety whose big quantum cohomology is convergent and semisimple, let $\pi\colon \mathrm{Bl}_{\mathrm{pt}}(X)\to X$ be the blow-up of $X$ at a point, with exceptional divisor $E\cong\mathbb{P}^{n-1}$, $n=\dim_{\mathbb{C}}X$, and let $\Psi_q\colon K^0(\mathrm{Bl}(X))\to H^*(\mathrm{Bl}(X);\mathbb{C})$ be Iritani's map defined in the background. Milanov and Xia proved (Theorem 1.7) that, under the additional assumption that the quantum cohomology of $\mathrm{Bl}(X)$ is convergent, the vectors $\Psi_q(\mathcal{O}_E(k))$, $k\in\mathbb{Z}$, are reflection vectors for the quantum cohomology of $\mathrm{Bl}(X)$, where a reflection vector is a period vector $\alpha$ with $(\alpha|\alpha)=2$ whose analytic continuation along a small loop around a generic point of the discriminant acts as $-1$ on the associated solutions and as the orthogonal reflection $x\mapsto x-(x|\alpha)\alpha$ on the space of period vectors. Two questions, stated as open in the source, constitute the problem.

(1) Redundancy of the convergence hypothesis. Is it the case that the blow-up operation preserves convergence of quantum cohomology - that is, if the structure constants of the big quantum product of $X$ converge on a non-empty open domain (so that $X$ carries its Frobenius-manifold domain $M\subset H^*(X;\mathbb{C})$), then the structure constants of the big quantum product of $\mathrm{Bl}_{\mathrm{pt}}(X)$ likewise converge on a non-empty domain, so that the convergence assumption in Theorem 1.7 is automatic? The GW invariants of the blow-up are Laurent series in the Novikov variable of the exceptional divisor.

(2) Reflection vectors from pullbacks. Let $(E_1,\dots,E_N)$ be a full exceptional collection in $D^b(X)$ (conjecturally, the same should hold for every exceptional object of $X$, per part (a) of Conjecture 1.6). Is $\Psi_q(\pi^*E_i)$ a reflection vector for the quantum cohomology of $\mathrm{Bl}_{\mathrm{pt}}(X)$ for each $i$?

Affirmative answers to (1) and (2), combined with Theorem 1.7 and Orlov's full exceptional collection $(\mathcal{O}_E(-n+1),\dots,\mathcal{O}_E(-1),\pi^*E_1,\dots,\pi^*E_N)$ of $\mathrm{Bl}(X)$, would complete the proof of Conjecture 1.6 of Milanov-Xia - the reflection-vector form of Dubrovin's conjecture - for blow-ups at finitely many points, exhibiting a complete system of reflection vectors for such blow-ups generated by the exceptional-locus bundles and the pullbacks.

References

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