Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1411.0896

Notes on the proof of the KKV conjecture

R. Pandharipande·R. P. Thomas
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Katz-Klemm-Vafa conjecture expresses the Gromov-Witten theory of K3 surfaces (and K3-fibred 3-folds in fibre classes) in terms of modular forms. Its recent proof gives the first non-toric geometry in dimension greater than 1 where Gromov-Witten theory is exactly solved in all genera. We survey the various steps in the proof. The MNOP correspondence and a new Pairs/Noether-Lefschetz correspondence for K3-fibred 3-folds transform the Gromov-Witten problem into a calculation of the full stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.

Explore related subjects

math.AGhep-th

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

R. Pandharipande, R. P. Thomas. 2016-05-09. Notes on the proof of the KKV conjecture. https://doi.org/10.4310/sdg.2016.v21.n1.a7

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Brauer-Manin obstruction for stacky curves

We show that the Brauer-Manin obstruction is the only obstruction to strong approximation for all stacky curves over global fields with finite abelian fundamental groups. This includes all stacky curves of genus $g = \frac{1}{2}$, thus explaining a recent counterexample to the Hasse principle of Bhargava-Poonen. We will furthermore show that the elementary obstruction is the only obstruction to the integral Hasse principle for smooth proper integral models of stacky curves of genus $g < 1$. We then compute the Brauer-Manin obstruction for smooth proper integral models of stacky curves of genus $\frac{1}{2}$.

math.AG↗
arXiv2026

Tropicalization of super Gromov-Witten invariants

We show that genus-0, $n$ Neveu-Schwarz marked, super Gromov-Witten invariants of a convex, toric variety $X$ can be defined and computed using tropical geometry. When $X$ is a point, the tropical, super Gromov-Witten invariants of $X$ are descendant invariants on the moduli space of tropical curves. When $X$ is a general convex, toric variety, we define a procedure that computes the tropical, inverse Euler class of the SUSY normal bundle $\overline{N}_{n, β} \rightarrow \overline{\mathcal{M}}_{0,n}(X, β)$, under the assumption that $\overline{N}_{n, β}$ is in some sense locally tropicalizable. We define the tropical, genus-0, $n$ Neveu-Schwarz marked, super Gromov-Witten invariants of $X$, and show that the definition recovers the tropical, super Gromov-Witten invariants of a point. We compute a tropical, super Gromov-Witten invariant of $\mathbb{P}^1$.

math.AG↗
arXiv2026

Optimal bounds for local volumes of threefold singularities

We establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least $9$ is either a hypersurface singularity or a quotient singularity. As applications, we obtain new restrictions on the singularities of members in K-moduli spaces of Fano threefolds, and we establish a sharp inequality between local volumes and minimal log discrepancies for threefold singularities.

math.AG↗