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The Dubrovin-Gamma conjectures for Fano varieties

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For a smooth Fano variety XX of Hodge-Tate type, the modern form of Dubrovin's conjecture predicts

QHbig∗(X) is generically semisimple⟺DbCoh⁡(X) has a full exceptional collection.QH^*_{\mathrm{big}}(X)\text{ is generically semisimple} \quad\Longleftrightarrow\quad D^b\operatorname{Coh}(X)\text{ has a full exceptional collection}.

The refined Gamma conjecture II also identifies categorical and quantum data. For an exceptional collection (E1,…,EN)(E_1,\ldots,E_N), the Stokes matrix of the quantum differential equation should match, up to conventions and mutations, the Gram matrix

(χ(Ei,Ej))i,j,\bigl(\chi(E_i,E_j)\bigr)_{i,j},

while the central connection matrix should be described by the Chern characters of the EiE_i and the Gamma-hat class of XX.

This gives a precise quantum-cohomological criterion for full exceptional collections. Gamma conjecture II is known for all del Pezzo surfaces, but the general Fano case remains open.

derived-categoriesquantum-cohomology
Posted by Vihaan Dheer Posted

Progress summary

Refreshed
Claimed progress

The conjecture remains open in general, although it has been proved for several important families of Fano varieties.

The conjecture predicts an equivalence between generic semisimplicity of big quantum cohomology and a full exceptional collection in DbCoh⁡(X)D^b\operatorname{Coh}(X), together with precise Stokes-matrix and Gamma-class identities. No general proof or counterexample has been reported.

Known results

  • Projective spaces and Grassmannians satisfy Gamma conjecture II (Galkin–Golyshev–Iritani, 2014).
  • Smooth Fano complete intersections of Fano index greater than 11 satisfy a Dubrovin-type analogue for exponential-type quantum connections (2017).
  • Gamma conjecture I was verified for all 1717 rank-one Fano threefold deformation classes (2018).
  • For Fano threefolds, small quantum cohomology is known to be generically semisimple in the cases treated, but this does not settle the general conjecture.

June 2026 status update

A 2026 paper records Gamma conjecture II as proved for complex Grassmannians, toric Fano varieties, and quadrics, while presenting the general statement as open; this is a status report, not a new proof of the full conjecture.

Current status (as of August 2026): Several major families and related analogues are settled, but the Dubrovin–Gamma conjectures for general smooth Fano varieties remain open.

Sources

Solutions 0

No solutions have been posted yet.