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Genus~0 Gromov-Witten theory of even dimensional complete intersections of two quadrics: the final step

Danil Gubarevich

TL;DR

The paper closes the genus-0 Gromov-Witten theory for smooth even-dimensional complete intersections X of two quadrics by computing the remaining primitive-correlator left undetermined in Hu's work. It constructs a Jun Li degeneration family with a smooth total space and a semistable central fiber, then analyzes the restriction map on cohomology and uses a degeneration formula to express the target invariant in terms of relative invariants. A careful monodromy analysis shows the primitive part is fixed, enabling the Li-based computation to proceed via a basis adapted to the central fiber. A dimension-based vanishing argument shows all relevant terms vanish for even m ≥ 4, thus yielding the required zero invariant and completing the genus-0 computation for X.

Abstract

Even dimensional complete intersections $X$ of two quadrics in projective space are exceptional from the point of view of the Gromov-Witten theory: they are (together with qubic surfaces) the only complete intersections whose Gromov-Witten theory is not invariant under the full orthogonal or symplectic group acting on the primitive cohomology. The genus~0 Gromov-Witten theory of $X$ was studied by Xiaowen Hu. He used geometric arguments and the WDVV equation to compute all genus~0 correlators except one, which cannot be determined by his methods. In this paper we compute the remaining Gromov-Witten invariant of $X$ using Jun Li's degeneration formula.

Genus~0 Gromov-Witten theory of even dimensional complete intersections of two quadrics: the final step

TL;DR

The paper closes the genus-0 Gromov-Witten theory for smooth even-dimensional complete intersections X of two quadrics by computing the remaining primitive-correlator left undetermined in Hu's work. It constructs a Jun Li degeneration family with a smooth total space and a semistable central fiber, then analyzes the restriction map on cohomology and uses a degeneration formula to express the target invariant in terms of relative invariants. A careful monodromy analysis shows the primitive part is fixed, enabling the Li-based computation to proceed via a basis adapted to the central fiber. A dimension-based vanishing argument shows all relevant terms vanish for even m ≥ 4, thus yielding the required zero invariant and completing the genus-0 computation for X.

Abstract

Even dimensional complete intersections of two quadrics in projective space are exceptional from the point of view of the Gromov-Witten theory: they are (together with qubic surfaces) the only complete intersections whose Gromov-Witten theory is not invariant under the full orthogonal or symplectic group acting on the primitive cohomology. The genus~0 Gromov-Witten theory of was studied by Xiaowen Hu. He used geometric arguments and the WDVV equation to compute all genus~0 correlators except one, which cannot be determined by his methods. In this paper we compute the remaining Gromov-Witten invariant of using Jun Li's degeneration formula.

Paper Structure

This paper contains 12 sections, 28 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The degeneration family
  4. The cohomology of $X$
  5. Cohomology of complex smooth projective varieties
  6. Cohomology of smooth complete intersections in $\mathbb C\mathbb P^N$
  7. Cohomology of complete intersection of two quadrics
  8. Monodromy of complete intersection
  9. The restriction map
  10. Degeneration formula
  11. Relative Gromov-Witten invariants
  12. Proof of the main theorem

Key Result

Theorem 1.1

Let $X$ be a smooth complete intersection of two quadrics in $\mathbb C\mathbb P^{m+2}$ of even dimension $m \geq 4$. There exists an orthonormal basis $(e_1,\dots,e_{m+3})$ of the primitive cohomology $H^m_\mathrm{prim}(X,\mathbb Z)$ of $X$ such that

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Definition 3.2
  • Theorem 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 39 more