The second Stiefel-Whitney class of \(\ell\)-adic cohomology (Q2848378)

Let \(\ell\) be a prime number, and let \(K\) be a field of characteristic distinct from \(2\) and \(\ell\). We have \(H^1(\mathrm{Gal}(\overline K/K), \mathbb Z/2)\cong K^\times/(K^\times)^2\) by Kummer's theory. For any \(a\in K^\times\), let \(\{a\}\in H^1(\mathrm{Gal}(\overline K/K),\mathbb Z/2)\) be the class of \(a\) in \(K^\times/(K^\times)^2\), and for any \(a,a'\in K^\ast\), let \(\{a,a'\}\in H^2(\mathrm{Gal}(\overline K/K),\mathbb Z/2)\) be the cup product of \(\{a\}\) and \(\{a'\}\). Giving a finite dimensional \(K\)-vector space \(D\) and a non-degenerate symmetric bilinear form \(b:D\otimes_KD\to K\), the corresponding quadratic form is equivalent to \(a_1X_1^2+\cdots +a_r X_r^2\) for some \(a_1,\ldots, a_r\in K^\times\). Define the first and the second Hasse-Witt classes of \((D,b)\) by NEWLINE\[NEWLINE \begin{aligned} &\mathrm{hw}_1(D,b)=\sum_{i=1}^r\{a_i\}\in H^1(\mathrm{Gal}(\overline K/K), \mathbb Z/2), \\ &\mathrm{hw}_2(D,b)=\sum_{1\leq i<j\leq r} \{a_i,a_j\}\in H^2(\mathrm{Gal}(\overline K/K), \mathbb Z/2).\end{aligned}NEWLINE\]NEWLINE Giving a finite dimensional \(\mathbb Q_\ell\)-vector space \(V\), a non-degenerate symmetric bilinear form \(b:V\otimes_{\mathbb Q_\ell} V\to\mathbb Q_\ell\) and an orthogonal representation \(\rho:\mathrm{Gal}(\overline K/K)\to \mathrm O(V,b)\), we define the second Stiefel-Whitney class \(\mathrm{sw}_2(V)\in H^2(\mathrm{Gal}(\overline K/K), \mathbb Z/2)\) to be the class corresponding to the central extension of \(\mathrm{Gal}(\overline K/K)\) by \(\mathbb Z/2\) obtained by pulling back by \(\rho\) the canonical central extension NEWLINE\[NEWLINE1\to \mathbb Z/2\to \widetilde {\mathbb O}(V)\to \mathbb O(V)\to 1.NEWLINE\]NEWLINE Let \(X\) be a proper smooth scheme of even dimension \(n\) over \(K\). The cup product defines a non-degenerate symmetric bilinear form on the de Rham cohomology \(H^n_{\mathrm{dR}}(X/K)\), and we can consider its second Hasse-Witt class \(\mathrm{hw}_2(H^n_{\mathrm{dR}}(X))\). The cup product defines a non-degenerate symmetric bilinear form on the \(\ell\)-adic cohomology \(H^n(X_{\overline K}, \mathbb Q_\ell)(\frac{n}{2})\), which gives an orthogonal representation of \(\mathrm{Gal}(\overline K/K)\). We can consider its second Stiefel-Whitney class \(\mathrm{sw}_2(H^n_\ell(X))\). The author propose a conjecture comparing the Hasse-Witt class and the Stiefel-Whitney class.NEWLINENEWLINEFor any integer \(q\), let \(b_{\mathrm{\'et}, q}=\mathrm{dim}\, H^q(X_{\overline K}, \mathbb Q_\ell)\) and \(b_{\mathrm{dR}, q}=\mathrm{dim}\, H^q_{\mathrm{dR}}(X/K)\) be Betti numbers. If \(\mathrm{char}\,K=0\), these two Betti nubmers are equal, and we denote them by \(b_q\). Let \(\chi_\ell:\mathrm{Gal}(\overline K/K)\to \mathbb Q_\ell^\times\) be the \(\ell\)-adic cyclotomic character. Denote by \(c_\ell\in H^2(\mathrm{Gal}(\overline K/K),\mathbb Z/2)\) the class of the pulling back by \(\chi_\ell\) of the central extension NEWLINE\[NEWLINE1\to\mathbb Z/2\to \mathbb G_m\overset{x\mapsto x^2}{\mathbb G}{_m} 1.NEWLINE\]NEWLINE If \(\mathrm{char}\, K\not =0\), we have \(c_\ell=0\). The character \(\Big(\mathrm{det}(H^q(X_{\overline K}, \mathbb Q_\ell))\Big)\cdot \chi_\ell^{\frac{qb_{\mathrm{\'et},q}}{2}}\) of \(\mathrm{Gal}(\overline K/K)\) takes value in \(\{\pm 1\}\), and hence defines a homomorphism \(e_q:\mathrm{Gal}(\overline K/K)\to\{\pm 1\}\). We regard \(e_q\) as an element in \(H^1(\mathrm{Gal}(\overline K/K),\mathbb Z/2)\cong \mathrm{Hom}(\mathrm{Gal}(\overline K/K), \{\pm 1\})\). Put NEWLINE\[NEWLINEe=\sum_{q< n} e_q, \quad r=\sum_{q<n}(-1)^q b_{\mathrm{dR},q},\quad \eta=\sum_{q<\frac{n}{2}} (-1)^q\Big(\frac{n}{2}-q\Big)\chi(X,\Omega^q_{X/K}).NEWLINE\]NEWLINE If \(\mathrm{char}\, K=0\), we put NEWLINE\[NEWLINE\beta=\frac{1}{2} \sum_{q<n}(-1)^q(n-q)b_q.NEWLINE\]NEWLINE Finally, let \(d_X=\mathrm{hw}_1(H^n_{\mathrm{dR}}(X/K))\in H^1(\mathrm{Gal}(\overline K/K),\mathbb Z/2)\). The author conjectures that in \(H^2(\mathrm{Gal}(\overline K/K),\mathbb Z/2)\), we have an equality NEWLINE\[NEWLINE\begin{aligned} &\mathrm{sw}_2(H^n_\ell(X))+ \{e,-1\}+\beta \cdot c_\ell\\ = &\mathrm{hw}_2(H^n_{\mathrm{dR}}(X)) +\{2,d_X\}+\eta\cdot (c_\ell-c_2) \end{aligned}NEWLINE\]NEWLINE NEWLINE\[NEWLINE +\begin{cases} r\{d_X,-1\}+{r\choose 2}\{-1,-1\} \text{if } n\equiv 0\mod 4,\\ (r+b_{\mathrm{dR},n}-1)\{d_X,-1\}+{r+b_{\mathrm{dR},n}\choose 2}\{-1,-1\}&\text{if }n\equiv 2\mod 4.\end{cases} NEWLINE\]NEWLINE The author verifies the above conjecture in the following cases, each case being verified by a different technique:NEWLINENEWLINE(1) \(K\) is a finite extension of \(\mathbb Q_p\), \(p\not =2,\ell\), and there exists a projective regular flat model \(X_{\mathcal O_K}\) over the integer ring \(\mathcal O_K\) such that the closed fiber has at most ordinary double points as singularities.NEWLINENEWLINE(2) \(K\) is a finite extension of \(\mathbb Q_p\), \(p=\ell>n+1\), and there exists a proper smooth model \(X_{\mathcal O_K}\) over \(\mathcal O_K\).NEWLINENEWLINE(3) \(K=R\) and \(X\) is projective.NEWLINENEWLINE(4) \(K\supset \overline {\mathbb Q}\).NEWLINENEWLINE(5) \(X\) is a smooth hypersurface in \(\mathbb P_K^{n+1}\) and \(\ell>n+1\).