Gamma conjecture II via global Gamma-I

Abstract: For a Fano manifold $X$, Gamma conjecture II aims to use $\mathcal{D}_{\rm{coh}}^b(X)$ to describe the asymptotic behavior of its Dubrovin connection via $\widehatΓ$-integral structure. It was proposed by Galkin, Golyshev and Iritani, and can be regarded as a quantitative refinement of Dubrovin's conjecture on Fano manifolds with semisimple big quantum cohomology. As a step toward Gamma conjecture II, we define the Gamma-I property at points satisfying the (SR) condition, arising from the original Gamma conjecture I. We prove that the property holds globally in the following sense: if it holds at one such point, then it holds throughout the connected component of the (SR)-region containing that point. Based on this global Gamma-I property, we establish a strategy-type theorem relating Gamma conjecture II to the Gamma-I property at a possibly non-semisimple point, together with an analysis of small quantum cohomology. We further apply this theorem to prove Gamma conjecture II for del Pezzo surfaces; the proof combines Iritani's Galois action with addtional elementary operations on exceptional collections, and its most technically involved step consists in verifying the required global Gamma-I property.

• The paper establishes a global Gamma-I property, proving that if Gamma-I holds at one point in a connected SR-region, it holds throughout, thereby linking flat section asymptotics with derived categories.
• The work introduces the Strategy-Theorem, which connects non-semisimple and semisimple points in quantum cohomology through analytic continuation and combinatorial synthesis of exceptional collections.
• The paper conclusively proves Gamma Conjecture II for all del Pezzo surfaces, using detailed analyses of Perron-Frobenius theory and Gromov-Witten invariants to extend results beyond toric cases.

Gamma Conjecture II via Global Gamma-I: An Expert Overview

Introduction and Background

The paper "Gamma conjecture II via global Gamma-I" (2606.07418) addresses a significant refinement in the interplay between the asymptotics of quantum connections and the structure of derived categories of coherent sheaves on Fano manifolds. This work situates itself within the landscape of Dubrovin’s conjecture and its further refinements by Galkin-Golyshev-Iritani (GGI), focusing on the so-called Gamma conjecture II, which posits a deep quantitative correspondence between the exceptional collections in $\mathcal{D}^b_{coh}(X)$ and asymptotic data of solutions to the Dubrovin connection for big quantum cohomology, described through Iritani’s $\widehat{\Gamma}$-integral structure.

The paper’s primary contributions are (a) a proof of a “global Gamma-I property”, which allows transfer of the Gamma-I property across connected components of the $(\mathrm{SR})$-region in $H^2(X)$, (b) the formulation and proof of a strategy-type result connecting Gamma conjecture II at semisimple or even non-semisimple points and small quantum cohomology, and (c) a complete proof of Gamma conjecture II for all del Pezzo surfaces, including the non-toric cases that had remained open.

Gamma Conjecture II and the Gamma-I Property

Recall that for a Fano manifold $X$, the Dubrovin connection on the (big) quantum cohomology ring $QH_{big}(X)$ encodes isomonodromic deformation data, with the points in $H^*(X)$ supporting a semisimple quantum product parameterizing regions where the Stokes and monodromy structures can be canonically analyzed. Iritani’s $\widehat{\Gamma}$-integral structure provides a mapping from the $K$-group of $X$ to the space of flat sections of the Dubrovin connection, taking into account modified Chern classes via the Gamma class. Gamma conjecture II asserts the existence of a full exceptional collection whose associated $\widehat{\Gamma}$0-flat sections exhibit controlled asymptotics (respecting canonical normalized idempotents) in alignment with the monodromic data near $\widehat{\Gamma}$1.

Gamma conjecture I, in contrast, focuses on the existence of a unique flat section asymptotic to the Gamma class of the structure sheaf at $\widehat{\Gamma}$2 in the small quantum cohomology context at the large radius point, under a simple eigenvalue assumption for the quantum multiplication operator $\widehat{\Gamma}$3 (“SR” condition). Notably, recent counterexamples in the toric Fano case show failure of the original Gamma-I formulation at the origin, but admit validity at other points in the Kähler moduli.

A central innovation of this paper is the identification of a global property: if property Gamma-I holds at any point in a connected component of the $\widehat{\Gamma}$4-region, it holds everywhere throughout that component. This is established by analytic continuation arguments for the space of flat sections and the invariance of the asymptotic subspace $\widehat{\Gamma}$5.

The Strategy-Theorem: Bridging Gamma-I and Gamma-II

A major contribution is the “Strategy-Theorem”, a reconstruction statement that shows that Gamma conjecture II for a Fano manifold $\widehat{\Gamma}$6 can be deduced under four key hypotheses:

1. Convergence of big quantum cohomology near the large radius limit.
2. Existence of a Gamma-I point $\widehat{\Gamma}$7 in the $\widehat{\Gamma}$8-region.
3. A domain $\widehat{\Gamma}$9 in the $(\mathrm{SR})$0-region containing a tame semisimple point $(\mathrm{SR})$1.
4. Combinatorial synthesis of exceptional objects obtained by Galois actions and elementary operations at $(\mathrm{SR})$2 to realize a full exceptional collection whose associated $(\mathrm{SR})$3-sections respect the canonical asymptotics.

The theorem leverages analytic control over small quantum cohomology, the action of the fundamental group on flat sections via Iritani’s Galois symmetries, and the flexibility of mutations and elementary modifications in exceptional collections. Importantly, the domain $(\mathrm{SR})$4 need not be entirely contained in the domain of convergence for big quantum cohomology, allowing non-semisimple and boundary points to be treated.

Notably, this strategy replaces the locality of previous approaches (which typically worked only at semisimple points within convergence domains) with a global analytic/categorical linking argument. This is essential for cases exhibiting non-semisimplicity at key points in the Kähler moduli.

Proof of Gamma Conjecture II for Del Pezzo Surfaces

A focal application is the proof of Gamma conjecture II for all del Pezzo surfaces $(\mathrm{SR})$5 ($(\mathrm{SR})$6), settling a longstanding problem. This is accomplished as follows:

• Gamma-I at the origin: The original Gamma-I was previously established at $(\mathrm{SR})$7 for del Pezzo surfaces [HKLY21].
• Establishment of an $(\mathrm{SR})$8-region: Detailed analysis via Perron-Frobenius theory and explicit control over the quantum multiplication matrices, using Gromov-Witten theory and precise Puiseux expansions, guarantees existence and connectivity of $(\mathrm{SR})$9-domains containing both the origin and required tame semisimple points.
• Generation of a full exceptional collection: Galois actions and mutations, informed by geometric considerations of the exceptional divisors and their cohomological behavior near boundary regions of the Kähler moduli, are used to construct a full exceptional collection at a semisimple point, whose flat sections meet the required asymptotics.

Strong quantitative control over the eigenvalues of $H^2(X)$0, through a combination of majorization estimates and positivity conditions, is achieved with minimal explicit input from Gromov-Witten invariants. The step verifying the global Gamma-I property in a high-dimensional $H^2(X)$1-domain is technically intricate and relies crucially on matrix analysis and degeneration arguments.

As a consequence, Gamma conjecture II is also extended to all rational surfaces that are blow-ups of $H^2(X)$2 at up to eight points, by deformation invariance arguments.

Theoretical and Practical Implications

The findings clarify the precise connection between the Gamma-I and Gamma-II conjectures, showing that rather than being independent, Gamma-I can be leveraged, via global analytic continuation and Galois symmetries, to furnish full exceptional collections in derived categories compatible with quantum Stokes data. In particular, the categorical structure of $H^2(X)$3—even at non-semisimple points—can be used to reconstruct the asymptotic data of flat sections, and vice versa.

Beyond the proof of the remaining cases of Gamma-II for del Pezzo surfaces, the strategy is anticipated to be widely applicable. Notably, the argument is not reliant on $H^2(X)$4 being generated by line bundles, but suggests a framework that could be generalized to Fano varieties with more complicated derived categories and quantum cohomology—potentially including cases with non-semisimple quantum products.

Practically, this expands the toolkit for researchers aiming to link quantum cohomology, Stokes phenomena, and derived categories. The technique also interfaces naturally with homological mirror symmetry, noncommutative Hodge-theoretic structures, and Dubrovin’s categorification program.

Conclusion

This paper achieves a significant unification of the Gamma conjectures for Fano manifolds by establishing a mechanism to propagate the Gamma-I property globally and synthesizing quantum/exceptional data at both semisimple and non-semisimple points. The resolution of Gamma conjecture II for all del Pezzo surfaces provides an essential milestone and paves the way for generalized approaches to Dubrovin-type correspondences in broader classes of varieties. Future directions include the application of the global Gamma-I machinery to further classes of Fano (and potentially non-Fano) manifolds, extensions to the non-semisimple field, and refinements in light of recent developments in homological mirror symmetry and quantum birational geometry.