Even Dimensional Complete Intersections

• Even dimensional complete intersections are varieties defined by two quadratic forms in projective space, exhibiting rich symmetry, topology, and arithmetic invariants.
• Key methodologies include birational constructions and signature invariants that distinguish rational from nonrational cases in different dimensions.
• The studies integrate quantum cohomology, Gromov–Witten invariants, and explicit formulas for topological genera, which enable practical classification and computation.

Even dimensional complete intersections—especially intersections of two quadric hypersurfaces in projective space—form a pivotal class in complex and real algebraic geometry. These varieties, defined as the common zero loci of two quadratic forms in projective space with even dimension, are distinguished by their rich geometric, arithmetic, and cohomological structures. Their study involves rationality questions, cohomological invariants, Gromov–Witten theory, quantum cohomology, and the computation of topological genera.

1. Geometric and Cohomological Structure

Let $X \subset \mathbb{P}^{2n+2}_\mathbb{R}$ be a smooth complete intersection of two real quadrics, i.e., $X = \{ F_1(x) = F_2(x) = 0 \}$, where $F_1, F_2$ are quadratic forms with $n$ even, so $\dim X = 2n$. Associated to $(F_1, F_2)$ is the pencil $Q_t = \{ t_1F_1 + t_2F_2 = 0 \}$, $t \in \mathbb{P}^1$, whose discriminant locus $D \subset \mathbb{P}^1$ (singular fibers) controls much of $X$'s geometry. In the real case, $D$ has $r \leq 2n+3$ real points (odd), with the rest as complex-conjugate pairs.

The integral middle cohomology splits as $$H^{2n}(X(\mathbb{C}),\mathbb{Z}) \cong \mathbb{Z} \cdot h^n \oplus (-1)^n D_{2n+3}$$, where $h$ denotes the hyperplane class and $D_{2n+3}$ is the root lattice associated to the Dynkin diagram of type $D_{2n+3}$ (Hassett et al., 2021). Maximal linear subspaces $P \cong \mathbb{P}^n \subset X_\mathbb{C}$ correspond to half-spin weights in $D_{2n+3}$, and the action of complex conjugation is encoded by an involution in the Weyl group $W(D_{2n+3})$. The real isotopy class of $X(\mathbb{R})$ is classified via the so-called Krasnov invariant, derived from the signature jumps along the circle in the pencil.

2. Rationality and Birational Constructions

A completed rationality classification exists for real fourfolds ($n=2$) as established by Hassett, Kollár, and Tschinkel:

Theorem: $X \subset \mathbb{P}^6_\mathbb{R}$, a smooth complete intersection of two quadrics (dim $X=4$), is rational over $\mathbb{R}$ if and only if $X(\mathbb{R}) \neq \emptyset$ and $X(\mathbb{R})$ is connected.

If $X$ contains a real line, projection yields a birational parametrization. When $X$ has no real line, explicit constructions leverage the existence of real or conjugate pairs of half-dimensional linear/quadric subvarieties. Three cases arise according to the isotopy (Krasnov) invariant—(5), (1,3,3), (1,2,4)—with birational parametrizations constructed either via blowing up real points and projecting from planes or quadrics, or by using odd multisections in quadric surface bundles (Hassett et al., 2021).

In dimension six ($n=3$), rationality is unresolved for the (1,3,5) isotopy class; here, no construction has yet produced the requisite invariant subvarieties. The issue is closely linked to the Galois action on root lattices and the arithmetic of intermediate Jacobians.

3. Quantum Cohomology and Gromov–Witten Invariants

Even-dimensional intersections of two quadrics $X_n(2,2) \subset \mathbb{P}^{n+2}$, $n \geq 4$ even, have quantum cohomological structures reflecting both their ambient geometry and primitive cohomology. The genus-zero Gromov–Witten invariants are organized by the monodromy/Weyl group $W(D_{n+3})$ acting on the primitive cohomology $H^n_{\mathrm{prim}}(X) \cong \mathbb{Z}^{n+3}$. Critical structural properties include:

• All genus zero GW invariants with more than one primitive insertion are determined by the length-4 invariants

$$\langle \varepsilon_a, \varepsilon_a, \varepsilon_b, \varepsilon_b \rangle_{0,1}^X = 1 \quad (a \neq b),$$

and by a special length-$(n+3)$ correlator

$$S = \langle \varepsilon_1, \dots, \varepsilon_{n+3} \rangle_{0,2}^X.$$

• In dimension four ($n=4$), $S = 1$; geometrically, exactly one smooth conic meets each of the seven canonical planes in $X$ (Hu, 2021).

The quantum product in the small quantum ring fails to be semisimple due to nilpotent elements; specifically, $h^{n-1} \star h = h^n + q$, $h \star q = 0$, showing that $QH^*(X)$ contains nilpotents. However, the big quantum cohomology, as encoded by the genus-zero GW potential $F(t)$, endows $X$ with a Frobenius manifold structure generically tame semisimple. For generic parameters, the spectrum of multiplication by the Euler vector field is simple (Hu, 2021).

Property	Description	Reference
Big quantum cohomology	Generically tame semisimple Frobenius manifold structure	(Hu, 2021)
Small quantum cohomology	Non-semisimple; nilpotent elements present	(Hu, 2021)
GW invariant generator	All genus-zero invariants from length-4 and special S	(Hu, 2021)

4. Topological Invariants: –Genus and Hirzebruch Genera

Hirzebruch genera of even-dimensional complete intersections, including the –genus and the $\alpha$-invariant, are given by polynomials in the degrees and dimension. For a smooth complex $n$-dimensional complete intersection $X_n(d_1, \ldots, d_r)$ in $\mathbb{CP}^{n+r}$:

• The –genus is computed as

$$\widehat A(X_{2n}(d_1,\dots,d_r)) = \sum_{j=0}^r (-1)^{r-j} \sum_{1 \leq k_1 < \cdots < k_j \leq r} \binom{c_1-1 + d_{k_1} + \cdots + d_{k_j}}{2n + r}$$

where $c_1 = n + r + 1 - \sum_{i=1}^r d_i$ is the first Chern number (Wang et al., 2020).

• For intersections of two quadrics, e.g., $X_4(2,2) \subset \mathbb{CP}^5$, $c_1 = 1$, yielding $\widehat{A}(X_4(2,2)) = 0$.
• The vanishing criterion for spin complete intersections is: Â vanishes if and only if $c_1 > 0$, i.e., the total degree is less than $2n + r + 1$.
• The $\alpha$-invariant is determined by the parity of $n$ in relation to $\widehat{A}$:
  • $n \equiv 0 \bmod 4$: $\alpha = +\widehat{A}$
  • $n \equiv 2 \bmod 4$: $\alpha = -\widehat{A}$

The formulas provide computational access to various genera and allow explicit determination for concrete examples.

5. Obstructions, Galois Action, and Open Problems

For nonclosed fields, the Galois action on $D_{2n+3}$ and the structure of the intermediate Jacobian yield obstructions to rationality. In higher even dimensions ($n > 2$), Clemens–Griffiths-type obstructions and failures of the $\Pi^1$ vanishing condition on cycle-class groups may prevent rationality splits—particularly when the residue Galois cohomology class of the intermediate Jacobian is nontrivial (Hassett et al., 2021).

For real sixfolds ($n=3$), the rationality is completely resolved except in the isotopy class $(1,3,5)$, where no known method produces a rational parametrization. This reflects deeper issues regarding the existence of suitable subvarieties and the types of real forms available for the $D_9$ root lattice.

6. Birational and Topological Methods

Explicit birational maps in key cases use coordinate constructions, such as:

• Projection from a carefully chosen real point,
• The residual intersection (or “third-point”) map arising from conjugate linear spaces or quadrics,
• Bundle structures (conic/quadric surface bundles) with odd-degree multisections, leveraging Springer's theorem.

Topologically, isotopy classification via signature data (Krasnov invariant) provides a complete invariant of $X(\mathbb{R})$, and Witt’s theorem gives an explicit local-global principle for fiber signatures in the pencil of quadrics. These topological and birational techniques collaborate closely in both proofs of rationality and classification.

7. Summary and Significance

Even-dimensional complete intersections of two quadrics occupy a central position in the study of algebraic varieties due to the confluence of projective, cohomological, quantum, and arithmetic properties. Their geometry is governed by rich symmetry (Weyl group action), topology (root lattices), and arithmetic structure (intermediate Jacobians, Galois action). For dimensions four and below, rationality is completely characterized, with explicit birational constructions available for all isotopy classes. Quantum cohomology exhibits a dichotomy—non-semisimplicity in the small quantum ring, but generic semisimplicity in the big ring—showcasing the subtlety of even-dimensional intersections. The topological invariants, especially the –genus and the Hirzebruch genera, admit closed and computable formulas, directly linking geometry and topology. The field remains active due to open rationality cases in higher dimensions and connections to monodromy, arithmetic, and moduli theory (Hassett et al., 2021, Hu, 2021, Wang et al., 2020).