Big quantum cohomology of Fano complete intersections

Hu, Xiaowen

doi:10.48550/arxiv.1501.03683

Cited by 6 publications

(23 citation statements)

References 26 publications

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“…when the dimension constraint is satisfied. For fixed genus g, using the enumerativity results of [LP21], the virtual Tevelev degrees for all flag varieties in sufficiently 5 See also [ABPZ21,Hu15] for recent results on the Gromov-Witten theory of complete intersections which also confront the primitive cohomology. 6 The full definition is reviewed in Section 3.…”

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confidence: 99%

Tevelev degrees in Gromov-Witten theory

Buch¹,

Pandharipande²

2021

Preprint

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For a nonsingular projective variety X, the virtual Tevelev degree in Gromov-Witten theory is defined as the virtual degree of the morphism from Mg,n(X, d) to the product Mg,n ×X n . After proving a simple formula for the virtual Tevelev degree in the (small) quantum cohomology ring of X using the quantum Euler class, we provide several exact calculations for flag varieties and complete intersections. In the cominuscule case (including Grassmannians, Lagrangian Grassmannians, and maximal orthogonal Grassmannians), the virtual Tevelev degrees are calculated in terms of the eigenvalues of an associated self-adjoint linear endomorphism of the quantum cohomology ring. For complete intersections of low degree (compared to dimension), we prove a product formula. The calculation for complete intersections involves the primitive cohomology. Virtual Tevelev degrees are better behaved than arbitrary Gromov-Witten invariants, and, by recent results of [LP21], are much more likely to be enumerative.

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confidence: 99%

Tevelev degrees in Gromov-Witten theory

Buch¹,

Pandharipande²

2021

Preprint

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“…When QH * (X) is sufficiently well known, exact calculation is possible. For further recent progress on the Gromov-Witten of hypersurfaces, see [1,10].…”

Section: Tevelev Degreesmentioning

confidence: 99%

Enumerativity of virtual Tevelev degrees

Lian¹,

Pandharipande²

2021

Preprint

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Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus g maps of fixed complex structure in a given curve class β through n general points of a target variety X. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class β. A simple speculation is that for all Fano X, the virtual Tevelev degrees are enumerative for sufficiently large β. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension).

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“…without ψ-classes) Gromov-Witten invariants correlators, for brevity. In [Hu15] we studied the big quantum cohomology of Fano complete intersections, especially the correlators with primitive insertions. One of our basic tool is the Monodromy group of the whole family of complete intersections of a given multidegree.…”

Section: Introductionmentioning

confidence: 99%

“…We will also call them even (2, 2)-complete intersections for short. To understand our approach and its limitations, let us recall more details in [Hu15]. The basic idea is still to use WDVV equations e f…”

Section: Introductionmentioning

confidence: 99%

“…where t i are dual variables of an orthonormal basis of H n prim (X). The main novelty of [Hu15] is that we found that the coefficient of s k in F can be recursively computed by solving quadratic equations of one variable with two equal roots. For a precise conjectural algorithm and partial results on this we refer the reader to [Hu15, Section 1].…”

Section: Introductionmentioning

confidence: 99%

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Big quantum cohomology of even dimensional intersections of two quadrics

Hu¹

2021

Preprint

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For every even dimensional smooth complete intersection, of dimension at least 4, of two quadric hypersurfaces in a projective space, we compute the genus zero Gromov-Witten invariants of length 4, and then we show that, besides a special invariant, all genus zero Gromov-Witten invariants can be reconstructed from the invariants of length 4. In dimension 4, we compute the special invariant by relating it to a curve counting problem. We also show that the generating function of genus zero Gromov-Witten invariants has a positive radius of convergence, and the associated Frobenius manifold is generically semisimple.

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Big quantum cohomology of Fano complete intersections

Cited by 6 publications

References 26 publications

Tevelev degrees in Gromov-Witten theory

Tevelev degrees in Gromov-Witten theory

Enumerativity of virtual Tevelev degrees

Big quantum cohomology of even dimensional intersections of two quadrics

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