Mixed Frobenius Structure and Local Quantum Cohomology

Yukiko Konishi; Satoshi Minabe

Home/Archive/arXiv

arXiv · 1405.7476

Mixed Frobenius Structure and Local Quantum Cohomology

Yukiko Konishi·Satoshi Minabe

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is a sequel to arXiv:1209.5550 where the notion of mixed Frobenius structure (MFS) was introduced as a generalization of the structure of a Frobenius manifold. Roughly speaking, the MFS is defined by replacing a metric of the Frobenius manifold with a filtration on the tangent bundle equipped with metrics on its graded quotients. The purpose of the current paper is to construct a MFS on the cohomology of a smooth projective variety whose multiplication is the non-equivariant limit of the quantum product twisted by a concave vector bundle. We show that such a MFS is naturally obtained as the non-equivariant limit of the Frobenius structure in the equivariant setting.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yukiko Konishi, Satoshi Minabe. 2015-09-19. Mixed Frobenius Structure and Local Quantum Cohomology. https://arxiv.org/abs/1405.7476

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↗