Witt groups of smooth real curves and surfaces

Abstract: We study the $\mathbf{I}<sup>*$-cohomology of a smooth real algebraic curve $X$ in terms of its real locus and its geometric genus. We notably extend results of Monnier to the twisted case, which is crucial to the understanding of proper pushforwards of Witt groups. We also perform some computations related to transfers along the finite étale extension $\mathbb{C}/\mathbb{R}$. We further describe how to compute twisted Witt groups of surfaces, extending work of Sujatha, and the image of the global signature homomorphism following Monnier. As an application of the main methods of the paper, we describe the shifted and twisted Witt groups of smooth anisotropic quadrics over $\mathbb{R}$ of dimension $\leq 3$.

• The paper develops mostly topological descriptions of shifted and twisted Witt groups for smooth real varieties of dimension at most three, with controlled 2-primary corrections from algebraic cycles and Steenrod operations.
• For real curves, it gives explicit formulas involving genus, real connected components, points at infinity, and the twisting line bundle, while establishing fibre-product descriptions for important cases of Grothendieck–Witt groups.
• For surfaces, it determines torsion patterns and top Witt groups, identifies the limits of topological methods for twisted W¹, and applies the results to anisotropic quadrics, real spheres, and Enriques surfaces.

Context and objectives

This paper by Samuel Lerbet studies the shifted and twisted Witt groups $W^i(X,L)$ and Grothendieck–Witt groups $GW^i(X,L)$ of smooth algebraic varieties over $\mathbb{R}$ in small dimension, with twisting line bundle $L$ arbitrary. The guiding principle is that these groups are governed, up to controlled 2-torsion, by the topology of the real locus $X(\mathbb{R})$ together with geometric data such as the genus of the complexification. The work extends the untwisted computations of Knebusch on curves, Monnier on curves with torsion, Sujatha on surfaces, and Barge–Ojanguren on symplectic forms on affine real surfaces to the twisted setting — a generalization that matters because twisted groups are indispensable for pushforwards along proper morphisms, whose orientation sheaf $\omega_f$ introduces twists.

The technical backbone is the author's earlier analysis of $I$-cohomology via quadratic real cycle class maps $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$, combined with the Gersten–Witt spectral sequence, the Bloch–Ogus spectral sequence, and the Pardon spectral sequence whose $d_2$-differentials are identified, via Totaro's theorem and its twisted refinement due to Asok–Fasel, with the Steenrod operation $\mathrm{Sq}^2$ and its twisted variant $GW^i(X,L)$0.

The top Witt group in low dimension

The first main result gives a canonical description of $GW^i(X,L)$1 for a smooth real variety of dimension $GW^i(X,L)$2. If $GW^i(X,L)$3 is not proper or $GW^i(X,L)$4, then

$GW^i(X,L)$5

where $GW^i(X,L)$6 is generated by mod 2 fundamental classes of algebraic curves and $GW^i(X,L)$7 is the connecting homomorphism for $GW^i(X,L)$8. When $GW^i(X,L)$9 is proper with empty real locus, the answer is instead $\mathbb{R}$0. This directly generalizes Barge–Ojanguren's theorem on orientable rank-2 bundles over affine real surfaces, since $\mathbb{R}$1 classifies such bundles up to rank-zero classes in $\mathbb{R}$2.

A notable corollary of the method is a much simpler example than Totaro's of noninjectivity of $\mathbb{R}$3: for the real algebraic 4-sphere $\mathbb{R}$4, one obtains $\mathbb{R}$5 from purely topological input, without Totaro's identification of Pardon spectral sequence differentials. The paper also observes that no such counterexample exists in dimension $\mathbb{R}$6 over $\mathbb{R}$7, sharpening the known range.

Curves

For smooth real curves, the Gersten–Witt spectral sequence collapses, so all $\mathbb{R}$8 reduce to sheaf cohomology $\mathbb{R}$9. The key computation is $L$0 when $L$1 satisfies $L$2 (not proper or nonempty real locus), where $L$3 is the genus of a compactification and $L$4 counts complex points at infinity; otherwise it is $L$5. This yields:

• Twisted Witt group: if $L$6 is not a square, $L$7 under $L$8, where $L$9 is the number of connected components of $X(\mathbb{R})$0 and $X(\mathbb{R})$1 counts components on which $X(\mathbb{R})$2 restricts nontrivially; if $X(\mathbb{R})$3 is proper with empty real locus, $X(\mathbb{R})$4. Combined with Monnier's computation of the square case and the previously computed image of the signature, this gives an essentially complete description.
• Grothendieck–Witt groups: $X(\mathbb{R})$5 fits in an exact sequence $X(\mathbb{R})$6; the extension problem is left open. For $X(\mathbb{R})$7, a fibre product formula $X(\mathbb{R})$8 holds when $X(\mathbb{R})$9 is empty or connected or $\omega_f$0 is a square — extending Asok's fibre product formula, whose hypothesis on $\omega_f$1 would be vacuous for most curves.

The appendix computes the transfer along $\omega_f$2: the kernel of the normalized transfer on $\omega_f$3 has dimension $\omega_f$4 (or $\omega_f$5 in the proper empty-real-locus case), and the cokernel of pullback onto that kernel is $\omega_f$6. In particular the transfer vanishes entirely when $\omega_f$7, e.g. for $\omega_f$8 or an elliptic curve with disconnected real locus.

Surfaces

For surfaces, three results are established. First, a twisted version of Sujatha's exact sequence: for any line bundle $\omega_f$9,

$I$0

which reduces the computation of $I$1 to étale cohomology. Second, the torsion subgroup $I$2 is explicitly $I$3 with $I$4 and $I$5, where $I$6 is the dimension of $I$7 and $I$8, $I$9 are dimensions determined by the real-complex exact sequence. Notably, when $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$0 is not a square, the torsion is 4-torsion even if $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$1, unlike the untwisted case — a structural difference caused by the absence of a rank-one form with coefficients in a nontrivial bundle. Third, $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$2 sits in $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$3, and the torsion subgroup of $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$4 is always 4-torsion.

The image of the global signature $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$5 is computed in terms of the image of $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$6 on $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$7; applications include Enriques surfaces, where the image is described via the halves decomposition of Degtyarev–Kharlamov, recovering and extending Monnier's results.

An instructive limitation appears here: unlike $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$8, the group $$\gamma_t^c: H^c(X,I^t(L)) \to H^c(X(\mathbb{R}),\mathbb{Z}(L))$$9 admits no analogue of the topological description, because the injectivity defect of $d_2$0 cannot be detected topologically — the paper constructs an explicit abelian surface $d_2$1 and line bundle $d_2$2 for which $d_2$3 is nonzero, ruling out the hoped-for exact sequence.

Application: anisotropic quadrics

As a demonstration of the methods, the paper computes $d_2$4 for the anisotropic quadrics $d_2$5 of dimension $d_2$6 and all twists, recovering Xie's untwisted results by largely topological means. Highlights include: $d_2$7 while $d_2$8; $d_2$9 but $\mathrm{Sq}^2$0; and $\mathrm{Sq}^2$1 for every twist. These computations rest on delicate identifications such as $\mathrm{Sq}^2$2 for the torsion class $\mathrm{Sq}^2$3 and $\mathrm{Sq}^2$4, proved using Benoist's equivariant Steenrod operations and Wu relations. A byproduct is a short proof that $\mathrm{Sq}^2$5, the key step in Fasel's theorem that all vector bundles on $\mathrm{Sq}^2$6 are trivial; the author argues this proof is considerably simpler than the original.

Limitations and open questions

Several points are conceded explicitly. The extension problem for $\mathrm{Sq}^2$7 on curves is not solved in general. For surfaces, no manageable description of $\mathrm{Sq}^2$8 in $\mathrm{Sq}^2$9 is available in general, and the failure is witnessed by the elliptic-curve product example; a topological description does hold when $GW^i(X,L)$00. The fibre product formula for $GW^i(X,L)$01 requires hypotheses ($GW^i(X,L)$02 empty or connected, or $GW^i(X,L)$03 a square) whose necessity is not addressed. Finally, the paper leaves open whether the methods extend beyond dimension 3, where the Gersten–Witt spectral sequence no longer collapses.

Conclusion

The paper provides complete, mostly topological descriptions of twisted and shifted Witt groups of smooth real varieties up to dimension 3, unifying and extending classical results of Knebusch, Monnier, Sujatha, and Barge–Ojanguren to arbitrary twists, and illustrates the machinery through explicit computations on anisotropic quadrics and real spheres. The persistent theme is that topology of $GW^i(X,L)$04 controls these groups except for geometrically determined 2-primary torsion, with the twisted $GW^i(X,L)$05 of surfaces identified as the case resisting a purely topological treatment.