Quadratic Euler Characteristic

• Quadratic Euler characteristic is a motivic invariant that refines the classical Euler–Poincaré number by encoding bilinear forms and richer arithmetic data.
• It is constructed using motivic stable homotopy theory and duality in algebraic geometry, with explicit formulas derived from Jacobian rings.
• Applications include refined enumerative geometry and singularity theory, offering insights into degenerations and quadratic conductor formulas.

The quadratic Euler characteristic is a motivic refinement of the classical Euler–Poincaré characteristic for algebraic varieties, enriching the invariant from integers to elements of the Grothendieck–Witt ring $\text{GW}(k)$. In this formalism, the quadratic Euler characteristic $\chi(X/k) \in \text{GW}(k)$ encodes not just the ranks of cohomological contributions, but the full structure of non-degenerate symmetric bilinear forms and their assembly via motivic stable homotopy, categorical traces, and Hodge-theoretic pairings. This invariant has deep connections with the theory of degenerations, singularity theory, motivic nearby cycles, and quadratic refinements of classical conductor and enumerative formulas.

1. Definition and Construction of the Quadratic Euler Characteristic

Let $k$ be a perfect field (often char $k \neq 2$), and let $SH(k)$ be the stable motivic homotopy category over $k$. Morel’s fundamental theorem identifies the ring of endomorphisms of the unit object $1_k$ in $SH(k)$ with the Grothendieck–Witt ring, i.e.,

$\mathrm{End}_{SH(k)}(1_k) \cong GW(k),$

where $GW(k)$ is the group completion of isometry classes of nondegenerate quadratic forms over $k$.

Given a smooth projective $k$-scheme $X$, the suspension spectrum $\Sigma^\infty_+ X$ is strongly dualizable in $SH(k)$, allowing the Euler trace to define the quadratic Euler characteristic as

$$\chi(X/k) := \chi_{SH(k)}(\Sigma^\infty_+ X) \in GW(k).$$

Levine–Raksit further establish that this $\chi(X/k)$ coincides with the bilinear form defined on Hodge cohomology,

$$H^\bullet(X, \Omega^\bullet_{X/k}) := \bigoplus_{p+q = \dim X} H^q(X, \Omega^p_{X/k}),$$

via cup-product into $H^n(X, \Omega^n)$ and the trace mapping $\mathrm{tr}_X : H^n(X, \Omega^n) \to k$.

In the classical setting, the Euler characteristic is $$\chi_\text{top}(X) = \sum_i (-1)^i \dim H^i(X, \mathbb{Q})$$, but passing to $GW(k)$, the quadratic refinement retains the full structure of forms, encoding additional information on the underlying algebraic and arithmetic geometry.

2. Explicit Formulas: Hypersurfaces, Weighted Cases, and Complete Intersections

For smooth projective hypersurfaces $X \subset \mathbb{P}^{n+1}_k$ given by $F = 0$ of degree $e$, the computation of $\chi(X/k)$ relies on the structure of the Jacobian ring,

$$J(F) = k[X_0, ..., X_{n+1}]/(F, \partial F/\partial X_i),$$

and specifically its top graded piece $J(F)_N$ with $N = (e-2)(n+2)$, which is 1-dimensional and canonically generated by the Scheja–Storch generator $e_F$. The nondegenerate pairing $B_J$ on $J(F)$ translates, under an explicit cochain isomorphism $\psi_q$ (residue-Koszul resolution), to the trace form on primitive Hodge cohomology: for $A, B \in J(F)_m$,

$$\mathrm{tr}_{X/k}(\psi_q(A) \cup \psi_p(B)) = -e \lambda,$$

where $A \cdot B = \lambda e_F$ in $J(F)$.

The quadratic Euler characteristic is then expressed as

$\chi(X/k) = -e \cdot [J(F), B_J] \in GW(k) \quad \text{(up to a hyperbolic factor if $n$ odd)}.$

In the weighted projective case, for $Y = \{F = 0\} \subset \mathbb{P}(a_*)$, with $F$ weighted homogeneous of degree $e$, a similar structure holds using the weighted Jacobian ring. The pairing gains a factor $\prod a_i$, and the formula becomes

$$\chi(Y/k) = -e \left( \prod a_i \right) \cdot [J(F), B_J] \in GW(k).$$

For complete intersections of $r+1$ hypersurfaces of same degree $m$, Viergever’s algorithm involves constructing a universal hypersurface $\mathcal{X}$ in $\mathbb{P}^r \times \mathbb{P}^n$, relating the quadratic Euler characteristic $\chi(X/k)$ of $X$ to $\chi(\mathcal{X}/k)$ via motivic cut-and-paste and the explicit calculation of trace forms in the primitive cohomology, normalized by the Scheja–Storch generator. The output is a closed-form expression in $GW(k)$, involving Chern classes and Jacobian data (Viergever, 2023).

3. Degeneration, Quadratic Conductor Formulas, and Nearby Cycles

The quadratic Euler characteristic encodes the behavior of families under degeneration. For a degeneration over a discrete valuation ring $\mathcal{O}$ with uniformizer $t$ to a cone over a smooth hyperplane,

$X_t \subset \mathbb{P}^{n+1}_K \leadsto X_0,$

the difference in compactly supported $A^1$–Euler characteristics,

$$\Delta_t = sp_t(\chi_c(X_t/K)) - \chi_c(X_0/k) \in GW(k)$$

admits a quadratic refinement of the classical conductor formula:

$$\Delta_t = \langle e \rangle - \langle 1 \rangle + (-\langle e \rangle)^n \cdot [J(F), B_J],$$

replacing the classical sign $(-1)^n$ with the "quadratic sign" $(-\langle e \rangle)^n$ and adding a rank-zero error form.

The generalization proposed by Levine, Pépin-Lehalleur, and Srinivas (Levine et al., 2021) and realized via motivic nearby cycles $\Psi^t$ links local and global quadratic conductor formulas. The formula for a degeneration with isolated quasi-homogeneous hypersurface singularities is

$$\Delta_t(X/\mathcal{O}) = \sum_i sp_{k(p_i)/k}\left[ \langle e_i \prod a_i^{(i)} \rangle - \langle 1 \rangle + \left(-\langle e_i \rangle \right)^{\dim X} e_{p_i}(\Omega_{X/k}, dt) \right]$$

where $e_{p_i}(\Omega, dt)$ is the local quadratic Euler class, coinciding with the Scheja–Storch form on the Jacobian ring of the singularity. This global-local compatibility is formalized via Ayoub’s nearby and vanishing cycles functors, providing an abstract motivic interpretation.

4. Symmetric Powers and Power Structures

The quadratic $A^1$-Euler characteristic $\chi_c(-)$ extends from smooth projectives to arbitrary quasi-projective schemes and is compatible with symmetric powers via "power structures" on $GW(k)$. The central conjecture, supported by computations for curves and low symmetric powers, asserts

$\chi_c(\mathrm{Sym}^n X) = a_n(\chi_c(X)),$

where $a_n$ is the uniquely defined power structure (Pajwani–Pál) on $GW(k)$:

$$a_n(\langle \alpha \rangle) = \langle \alpha^n \rangle + \frac{n(n-1)}{2}\left( \langle 2 \rangle + \langle \alpha \rangle - \langle 1 \rangle - \langle 2\alpha \rangle \right).$$

Bröring–Viergever and Taelman–Pepin-Lehalleur verify this formula for all smooth projective curves, leveraging the motivic Gauss–Bonnet theorem to obtain explicit closed forms for $\chi_q(\mathrm{Sym}^n C)$ (Bröring et al., 25 Apr 2024):

• For $n$ odd, $$\chi_q(\mathrm{Sym}^n C) = -\frac12 \binom{2g-2}{n} H$$, with $H = \langle 1 \rangle + \langle -1 \rangle$,
• For $n$ even, $\chi_q(\mathrm{Sym}^{2m} C)$ involves a combinatorial sum with alternating $\langle 1 \rangle$ and $\langle -1 \rangle$.

This compatibility underpins refined enumerative formulas such as the quadratic Göttsche formula for Hilbert schemes of points on surfaces. The computational toolkit allows calculation of $A^1$–Euler characteristics for symmetric powers from knowledge of the $GW(k)$-valued base class.

5. Applications in Enumerative Geometry and Motivic Homotopy

Quadratic Euler characteristics appear in refined enumerative geometry, notably the quadratic count of lines on a general quintic threefold,

$$e^{A^1}(E) = 1445\, \langle 1 \rangle + 1430\, \langle -1 \rangle \in GW(k),$$

a refinement encoding both complex and real counts (Pauli, 2020). The dynamic Euler number formalism extends to situations with excess intersection and positive-dimensional zero loci via deformation, with the refined count corresponding to the sum of local $GW(k)$-valued indices.

In singularity theory, conductor formulas relate differences of $A^1$–Euler characteristics under degeneration to quadratic Milnor numbers and special forms on the Jacobian ring. These invariants simultaneously refine the classical Milnor number and genus-difference formulas, providing new insights into arithmetic and motivic enumerative geometry (Azouri, 2021).

The quadratic Euler characteristic also provides invariants for parameter spaces in dynamical systems, such as the moduli of quadratic rational maps with periodic critical points, enabling explicit counts of topological and combinatorial phenomena (Kiwi, 23 Jan 2025).

6. Computational Aspects, Generalization, and Open Problems

Algorithms for computing the quadratic Euler characteristic in higher-dimensional or more general settings (same-degree complete intersections, intersections with weighted projective spaces, etc.) typically proceed via:

1. Reduction to universal hypersurfaces,
2. Calculation of trace pairings via Jacobian rings,
3. Use of the Scheja–Storch generator,
4. Chern class computations for hyperbolic contributions,
5. Assembly via motivic Gauss–Bonnet and cut-and-paste formulas.

Procedural complexity resides in determinant and minor computations for the Jacobian matrix and linear systems for Scheja–Storch normalization; for fixed dimension, this is polynomial-time.

Extensions to the case of hypersurfaces with variable degrees or more general singularity loci are expected, possibly via toric Jacobian rings or mirror symmetry techniques. Coordinate-free constructions and realization of the quadratic conductor formula in positive characteristic, potentially involving wild Swan terms, remain conjectural. The arithmetic interaction of the quadratic Milnor number with point-counting over varying fields (e.g., real vs. complex, finite fields) is largely open.

7. Significance and Outlook

The quadratic Euler characteristic acts as a unifying invariant linking motivic homotopy theory, enumerative geometry, and singularity theory. It encodes enriched information inaccessible to classical ranks, such as discriminant data and arithmetic subtleties in real and quadratic refinements. Its compatibility with motivic nearby cycles and conductor formulas expands its reach to the paper of degenerations, singularities, and arithmetic geometry. Current evidence supports the central conjectures and explicit calculations in low-dimensional, cellular, and toric settings, with broad potential for further application in arithmetic, motivic, and quadratic enumerative geometry.