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The quadratic Euler characteristic of a smooth projective same-degree complete intersection

Anna M. Viergever

TL;DR

The paper develops an algorithm to compute the quadratic Euler characteristic $\chi(X/k)\in\mathrm{GW}(k)$ for smooth projective complete intersections of hypersurfaces of the same degree. It accomplishes this by lifting the problem to a hypersurface $\mathcal{X}$ in $\mathbb{P}^r\times\mathbb{P}^n$, establishing a canonical isomorphism between bigraded pieces of the Jacobian ring $J$ and primitive cohomology, and then mapping cup products to Jacobian-ring multiplication. A key technical advance is the explicit trace calculation: for $A,B$ giving $AB=\lambda C$ in $J^\rho$, one has $\operatorname{Tr}(\omega_A\cup\omega_B)=(-1)^{r+1}m^{n+1}\binom{n+r}{r}\lambda$, with a Scheja–Storch-type generator providing normalization. The main application computes the quadratic Euler characteristic for the intersection of two generalized Fermat hypersurfaces of the same degree, yielding explicit hyperbolic and non-hyperbolic components depending on parity, and cross-checks via a quadratic Riemann–Hurwitz computation. This extends hypersurface results to equal-degree complete intersections and highlights a concrete, computable link between Hodge-theoretic data and Jacobian-ring structure in the motivic setting.

Abstract

We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective complete intersection of two generalized Fermat hypersurfaces. The results presented here also form a chapter in the author's thesis, which was submitted on May 30'th, 2023.

The quadratic Euler characteristic of a smooth projective same-degree complete intersection

TL;DR

The paper develops an algorithm to compute the quadratic Euler characteristic for smooth projective complete intersections of hypersurfaces of the same degree. It accomplishes this by lifting the problem to a hypersurface in , establishing a canonical isomorphism between bigraded pieces of the Jacobian ring and primitive cohomology, and then mapping cup products to Jacobian-ring multiplication. A key technical advance is the explicit trace calculation: for giving in , one has , with a Scheja–Storch-type generator providing normalization. The main application computes the quadratic Euler characteristic for the intersection of two generalized Fermat hypersurfaces of the same degree, yielding explicit hyperbolic and non-hyperbolic components depending on parity, and cross-checks via a quadratic Riemann–Hurwitz computation. This extends hypersurface results to equal-degree complete intersections and highlights a concrete, computable link between Hodge-theoretic data and Jacobian-ring structure in the motivic setting.

Abstract

We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective complete intersection of two generalized Fermat hypersurfaces. The results presented here also form a chapter in the author's thesis, which was submitted on May 30'th, 2023.
Paper Structure (24 sections, 53 theorems, 299 equations)

This paper contains 24 sections, 53 theorems, 299 equations.

Table of Contents

  1. Quadratic Euler characteristics
  2. Quadratic Euler characteristics
  3. The quadratic Euler characteristic of a smooth projective hypersurface
  4. Setup, cohomology of differential forms and primitive cohomology
  5. Two exact sequences
  6. Cohomology of differential forms
  7. Primitive cohomology
  8. Isomorphism from the Jacobian ring to primitive cohomology
  9. An exact sequence relating $\delta$ and $\pi_q$
  10. Proof of Proposition \ref{['Levine Proposition 3.2']}
  11. Comparing the two products
  12. Proof of Proposition \ref{['Proposition comparison products']}
  13. Proof of Proposition \ref{['representation of image cech']} and Corollary \ref{['corollary definition map Jrho to cohomology']}
  14. One dimensionality of $J^\rho$ and $\tilde{J}^{\rho + (r+1,n+1)}$
  15. Computing the trace map
  16. ...and 9 more sections

Key Result

Proposition 1

Consider the bidegree There is a surjective homomorphism $\phi: J^\rho\to H^{n+r}(\mathbb{P}^r\times\mathbb{P}^n, \Omega_{\mathbb{P}^r\times\mathbb{P}^n}^{n+r})\cong k$, such that the diagram \begin{tikzcd} H^q(\mathcal{X}, \Omega_{\mathcal{X}}^p)_{prim}\otimes H^p(\mathcal{X}, \Omega_{\mathcal{X}}^q)_{prim} \arrow[r,

Theorems & Definitions (131)

  • Proposition 1: See Corollary \ref{['corollary definition map Jrho to cohomology']}
  • Theorem 1: See Lemma \ref{['construction of ctilde']} and Theorem \ref{['theorem trace']}
  • Theorem 2: See Corollary \ref{['qec of X']}
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • ...and 121 more