Let \(X\) be a nonsingular projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \(Y\) be a nonsingular subvariety of codimension \(p\), hence of dimension \(n-p\). From the natural map \(\Omega_X \tensor \mco_Y \to \Omega_Y\) of (II, 8.12) we deduce a map \(\Omega_X^{n-p} \to \Omega_Y^{n-p}\). This induces a map on cohomology \[ H^{n-p}(X, \Omega_X^{n-p}) \to H^{n-p}(Y, \Omega_Y^{n-p}) .\] Now \(\Omega_Y^{n-p} = \omega_Y\) is a dualizing sheaf for \(Y\), so we have the trace map \[ t_Y: H^{n-p}(Y, \Omega_Y^{n-p}) \to k .\] Composing, we obtain a linear map \(H^{n-p}(X, \Omega_X^{n-p}) \to k\). By (7.13) this corresponds to an element \(\eta(Y) \in H^p(X, \Omega_X^p)\), which we call the cohomology class of \(Y\).
For this exercise, use the standard compatible normalization of the trace maps on canonical sheaves: the trace on \(\Spec k\) is the identity, the trace on a finite reduced \(k\)-scheme is the sum of the traces of its points, and traces are compatible with the Gysin maps for nonsingular closed immersions. Equivalently, use the trace normalization from Grothendieck–Serre duality rather than an arbitrary scalar multiple of the trace supplied merely by the existence statement of Serre duality.
If \(P \in X\) is a closed point, show that \(t_X(\eta(P)) = 1\), where \(\eta(P) \in H^n(X, \Omega^n)\) and \(t_X\) is the trace map.
If \(X = \PP^n\), identify \(H^p(X, \Omega^p)\) with \(k\) by (Ex. 7.3), and show that \(\eta(Y) = (\deg Y) \cdot 1\), where \(\deg Y\) is the degree of \(Y\) as a projective variety (I, §7).
Hint: cut with a hyperplane \(H \subseteq X\), and use Bertini’s theorem (II, 8.18) to reduce to the case where \(Y\) is a finite set of points.
For any scheme \(X\) of finite type over \(k\), we define a homomorphism of sheaves of abelian groups \(d\log: \mco_X^* \to \Omega_X\) by \(d\log(f) = f^{-1} \, df\). Here \(\mco_X^*\) is a group under multiplication, and \(\Omega_X\) is a group under addition. This induces a map on cohomology \[ \Pic X = H^1(X, \mco_X^*) \to H^1(X, \Omega_X) ,\] which we denote by \(c\). See (Ex. 4.5).
Returning to the hypotheses above, suppose \(p = 1\). Show that \(\eta(Y) = c(\mcl(Y))\), where \(\mcl(Y)\) is the invertible sheaf corresponding to the divisor \(Y\).
(Trace normalization)
As written in the source, the exercise suppresses a normalization needed to make its numerical assertions canonical. In III.7.1 a dualizing object is a pair consisting of a sheaf and a trace, and replacing a trace by a nonzero scalar multiple gives another valid dualizing pair after rescaling the identification with the canonical sheaf. Corollary III.7.12 identifies the dualizing sheaf with \(\omega_X\), but the text does not there state a scalar normalization of that identification; Remark III.7.14 explicitly notes that at that stage little is known about the trace map beyond its existence. The standard trace/Gysin normalization stated in the problem is what makes \(t_P(1)=1\), transverse pullback of fundamental classes, and the divisor identity in part (d) compatible simultaneously.
Solution
Write \(j:Y\hookrightarrow X\) for the closed immersion. We use the normalized Serre pairing \[ \langle\alpha,\beta\rangle_X =t_X(\alpha\smile\beta), \] between \(H^p(X,\Omega_X^p)\) and \(H^{n-p}(X,\Omega_X^{n-p})\). With the trace normalization stated in the problem, the Gysin morphism \[ j_*:H^q(Y,\Omega_Y^q)\longrightarrow H^{q+p}(X,\Omega_X^{q+p}) \] is characterized by the adjunction identity \[ \langle j_*(a),b\rangle_X =\langle a,j^*b\rangle_Y. \] In particular, the class defined in the problem is \[ \eta(Y)=j_*(1). \]
1. For a closed point \(P\in X\),\[ \boxed{t_X(\eta(P))=1}. \]
Here \(p=n\), so the functional defining \(\eta(P)\) is the composite \[ H^0(X,\OO_X)\xrightarrow{j^*}H^0(P,\OO_P) \xrightarrow{t_P}k. \] Since \(X\) is projective and integral over the algebraically closed field \(k\), \[ H^0(X,\OO_X)=k, \] and since \(P\cong\Spec k\), the restriction map is the identity on \(k\). By the chosen normalization \(t_P=\operatorname{id}_k\), so the functional sends \(1\) to \(1\).
Under Serre duality, evaluating that functional at \(1\in H^0(X,\OO_X)\) is the same as pairing \(\eta(P)\) with \(1\): \[ 1 =\langle\eta(P),1\rangle_X =t_X(\eta(P)). \] This proves part (a).
2. Cohomology classes are compatible with transverse intersection by a nonsingular hyperplane: if \(H\subseteq X\) is nonsingular and meets \(Y\) transversely, then\[ i^*\eta_X(Y)=\eta_H(Y\cap H), \]where \(i:H\hookrightarrow X\).
Let \(Z=Y\cap H\) and write the transverse Cartesian square \[ \begin{array}{ccc} Z&\xrightarrow{j_H}&H\\ \downarrow&&\downarrow i\\ Y&\xrightarrow{j}&X. \end{array} \] For regular closed immersions meeting transversely, the normalized Gysin maps satisfy base change \[ i^*j_*=(j_H)_*\,i_Y^*. \] Applying this to \(1\in H^0(Y,\OO_Y)\) gives \[ i^*\eta_X(Y) =i^*j_*(1) =(j_H)_*(1) =\eta_H(Z). \] This is exactly the asserted compatibility.
3. If \(X=\PP^n\) and \(Y\) has codimension \(p\), then\[ \boxed{\eta(Y)=(\deg Y)\cdot1\in H^p(\PP^n,\Omega_{\PP^n}^p)\cong k}. \]
By Exercise III.7.3, \[ H^p(\PP^n,\Omega^p)\cong k. \] Choose a general linear subspace \[ L\cong\PP^p\subseteq\PP^n \] of complementary dimension to \(Y\). By repeated Bertini and the generic-intersection statement of Chapter I, \(L\) meets \(Y\) transversely in a reduced set \[ Z=Y\cap L=\{P_1,\ldots,P_d\}, \qquad d=\deg Y. \] Repeated application of step 2 gives \[ \eta_L(Z)=i^*\eta_{\PP^n}(Y). \]
The Gysin map is additive on a disjoint union of points, so \[ \eta_L(Z)=\sum_{a=1}^d\eta_L(P_a). \] By step 1 each point class has trace one. Since \[ H^p(L,\Omega_L^p)\cong k \] and its trace identifies the standard generator with \(1\), we obtain \[ \eta_L(Z)=d\cdot1. \]
Under the identifications of Exercise III.7.3, restriction from \(\PP^n\) to a linear \(\PP^p\) sends the standard generator of \(H^p(\PP^n,\Omega^p)\) to the standard generator of \(H^p(L,\Omega_L^p)\); this is the iterated boundary map coming from the Euler sequence, and the coordinate restriction commutes with that construction. Hence the scalar multiplying the generator upstairs is also \(d\). Thus \[ \eta(Y)=(\deg Y)\cdot1, \] proving part (b).
4. The rule\[ d\log(f)=f^{-1}df \]is a morphism of sheaves of abelian groups\[ d\log:\OO_X^*\longrightarrow\Omega_X. \]
For local units \(f,g\), \[ \begin{aligned} d\log(fg) &=(fg)^{-1}d(fg)\\ &=f^{-1}g^{-1}(f\,dg+g\,df)\\ &=f^{-1}df+g^{-1}dg\\ &=d\log(f)+d\log(g). \end{aligned} \] The construction commutes with restriction to smaller open sets because Kähler differentiation does. Therefore it is a morphism from the multiplicative sheaf \(\OO_X^*\) to the additive sheaf \(\Omega_X\). Passing to first cohomology and using \[ \Pic X\cong H^1(X,\OO_X^*) \] gives the homomorphism \[ c:\Pic X\longrightarrow H^1(X,\Omega_X) \] required in part (c).
5. If \(Y\subseteq X\) is a nonsingular divisor with local equations \(f_i=0\) on an open cover \((U_i)\), then \(c(\mcl(Y))\) is represented by the Čech cocycle\[ \left\{\frac{df_i}{f_i}-\frac{df_j}{f_j}\right\}_{ij}. \]
The invertible sheaf \(\mcl(Y)=\OO_X(Y)\) has local generator \(e_i=f_i^{-1}\) on \(U_i\). Use the Čech convention in which the transition function \(g_{ij}\) is defined by \[ e_j=g_{ij}e_i. \] Then \[ g_{ij}=\frac{f_i}{f_j}, \] and its image under \(d\log\) is \[ d\log\!\left(\frac{f_i}{f_j}\right) =\frac{df_i}{f_i}-\frac{df_j}{f_j}. \] This is the displayed cocycle.
6. For a nonsingular divisor \(Y\subseteq X\),\[ \boxed{\eta(Y)=c(\mcl(Y))\in H^1(X,\Omega_X)}. \]
Let \[ \alpha\in H^{n-1}(X,\Omega_X^{n-1}). \] By definition of \(\eta(Y)\), \[ \langle\eta(Y),\alpha\rangle_X =t_Y(j^*\alpha). \]
On a cover on which \(Y\) is given by the equations \(f_i\), the standard local description of the Gysin/residue map for a Cartier divisor sends a form \(\beta\) on \(Y\) to the Čech class represented by \[ \frac{df_i}{f_i}\wedge\widetilde\beta_i - \frac{df_j}{f_j}\wedge\widetilde\beta_j, \] where the \(\widetilde\beta_i\) are local lifts to \(X\). Equivalently, cup product with the divisor Gysin class is cup product with the Čech cocycle \[ \left\{\frac{df_i}{f_i}-\frac{df_j}{f_j}\right\}_{ij}. \] Compatibility of the normalized trace with this residue map gives \[ t_X\bigl(c(\mcl(Y))\smile\alpha\bigr) =t_Y(j^*\alpha). \] Thus \[ \langle c(\mcl(Y)),\alpha\rangle_X =\langle\eta(Y),\alpha\rangle_X \] for every \(\alpha\). The Serre pairing is nondegenerate, so \[ c(\mcl(Y))=\eta(Y). \] This proves part (d).
7. Q.E.D.
Step 1 proves part (a), steps <1>2–<1>3 prove part (b), step 4 proves part (c), and steps <1>5–<1>6 prove part (d).