Preparation
Higher Gysin Homomorphism
Adequate Equivalence Relations
On a smooth curve,
Zero Cycles on Surfaces
Investigating \(T(X)\) in different settings.
\(A_0(X)\) and \(T(X)\) are divisible.
First Example: Hyperelliptic
Let \(E\) and \(F\) be elliptic curves, with \(\eta\in F\) a point of order \(2\). Define an involution \(\sigma\) on \(F\times E\) by \(\sigma(f,g)=(-f,g+\eta)\), and let \(X=(F\times E)/\sigma\). Let \(E'=E/\{1,\eta\}\) we then conclude a diagram
therefore all the fibres of \(\rho\) should be isomorphic to \(F\). To see this, track the inverse image of the origin \(o\in E'\). \(o\) goes upward to \(\{o,\eta\}\in E\) then back to two copies of \(F\) in \(F\times E\), which are identified under quotient of the involution \(\sigma\) (reflection).
Note that the Albanese variety to \(X\) is defined as \(\mathrm{H}^0(X,\Omega_{X}^1)^*/\mathrm{H}_1(X,\mathbb{Z})\), which is the initial object in the category of abelian varieties under \(X\). That is, \(\mathrm{Alb}(X)\) must be the largest abelian variety admitting a morphism from \(X\), in the sense of dimension. Here \[ \begin{aligned} \mathrm{H}^0(X,\Omega_{X}^1)&=\mathrm{H}^0(F\times E,\Omega_{F\times E}^1)/\sigma\ (to\ be\ explained)\\ &=\left(\mathrm{H}^0(F,\Omega_{F}^1)\oplus\mathrm{H}^0(E,\Omega_{E}^1)\right)/\sigma\\ &=\mathrm{H}^0(E,\Omega_{E}^1)=\mathbb{C}. \end{aligned} \] Hence \(\mathrm{Alb}(X)\) has dimension \(1\), and the induced morphism \(\mathrm{Alb}(X)\rightarrow E'\) is quasi-finite. Since \(\rho\) has connected fibres, \(\mathrm{Alb}(X)\rightarrow E'\) is an isomorphism.
Claim \(A_0(X)\cong E'=\mathrm{Alb}(X)\), or \(T(x)=0\). If \(z\in T(X)\), we have \[ \pi^*z=\sum_i r_i\left((q_i,p_i)+(-q_i,p_i+\eta)\right) \] where \(\sum r_i=0\) (algebraically equivalent to \(0\)) and \(\sum 2r_ip_i=0\) (maps to \(0\in E'\)). According to the arithmetic of elliptic curves, \[ (p)+^{pic}(q)\sim (p+^{ellip}q)+(\infty) \] so \[ \begin{aligned} 2(q,p)&\sim 2(q,p+\eta)\\ (q,p)+(-q,p)&\sim 2(0,p) \end{aligned} \] on \(F\times E\). Therefore \[ \begin{aligned} 2\pi^*z&\sim\sum 2r_i\left((q_i,p_i)+(-q_i,p_i+\eta)\right)\\ &\sim\sum 4r_i(0,p_i)\\ &\sim [(0)\times E]\cdot[F\times\sum 4r_i(p_i)]\\ &\sim 0. \end{aligned} \] It follows that \(0\sim 2\pi_*\pi^*z=4z\) (is a cover with two branches), hence \(4T(x)=0\), which implies that \(T(X)=0\) since \(T(X)\) is divisible.
Second Example: Surface of degree \(5\)
Let \(Y\) be the subvariety of \(\mathbb{P}^3\) cut out by \(T_0^5+T_1^5+T_2^5+T_3^5=0\). Define an action of \(\mathbb{Z}/5\mathbb{Z}\) on \(Y\) by \[ (t_0,t_1,t_2,t_3)\mapsto (t_0,\omega t_1,\omega^2 t_2,\omega^3 t_3) \] where \(\omega\) is a primitive \(5\)-th root of unity. This action is fixed point free, so the quotient \(X=Y/(\mathbb{Z}/5\mathbb{Z})\) is smooth.
Claim \(A_0(X)=\{0\}\).
Let \(\pi:Y\rightarrow X\) be the projection. Since \(A_0(X)\) is divisible, it suffices to show \(5A_0(X)=0\), or \(\pi_*\pi^*A_0(X)=0\). Since \(\pi_*\) is surjective, it suffices to show \(\pi^*\pi_*:A_0(Y)\rightarrow A_0(Y)\) is the zero map.
Define an action of \((\mathbb{Z}/5\mathbb{Z})^{\oplus 3}\) on \(Y\) by \[ e_i^*T_j=\omega^{\delta_{ij}}T_j. \]
This gives a representation \(\mathbb{Z}[\mathbb{Z}/5\mathbb{Z}^{\oplus 3}]\rightarrow\mathrm{End}(A_0(Y))\) sending \(x\) to \(\bar{x}\), which image is denoted by \(R\subset\mathrm{End}(A_0(Y))\). It follows immediately That \[ \pi^*\pi_*=1+\bar{e}_1\bar{e}_2^2\bar{e}_3^3+(\bar{e}_1\bar{e}_2^2\bar{e}_3^3)^2+\cdots+(\bar{e}_1\bar{e}_2^2\bar{e}_3 ^3)^4. \]
Observation The quotient of the surface \(Y\) under any of the following groups
- \(<1,e_i,\cdots,e_i^4>\)
- \(<1,e_ie_i,\cdots,e_i^4e_i^4>\)
- \(<1,e_1e_2e_3,\cdots,e_1^4e_2^4e_3^4>\)
is rational (birational to \(\mathbb{P}^2\)). To prove it…
As a result, \(A_0(Y/G)\) must be trivial when \(Y/G\) is rational, which could be seen easily using moving lemma (just to push the zero cycle away from the exceptional loci). As a result, every zero cycle of zero degree is rational. Denote by \(\pi_G\) the projection from \(Y\) to \(Y/G\), for each zero cycle \(z\in Y/G\) we have \[ \int_{g\in G} g^*z=\pi_{G*}\pi^*z \] so \[ \sum_{n=1}^4\bar{e}_i^n=\sum_{n=1}^4\bar{e}_i^n\bar{e}_j^n=\sum_{n=1}^4\bar{e}_1^n\bar{e}_2^n\bar{e}_3^n=0 \] in \(R\subset\mathrm{End}(A_0(Y))\).
Now go back to the proof of the claim.
Family of Lines on a Hypersurface
Case 1
\(X\subset\mathbb{P}^4\) a smooth hypersurface of degree \(3\) (cubic threefold). The family of lines on \(X\) forms a smooth connected surface \(S\subset\mathrm{Gr}(2,5)\) called the Fano surface (not Del Pezzo surface). (A. N. Tyurin, Five lectures on three-dimensional varieties (in Russian), Uspehi Mat. Nauk, 27 (1972), no. 5, (167) 3–50. [Translation: Russian Math. Surveys, 27 (1972), no. 5, 1–53.])
The Fano surface \(S\) of a smooth cubic threefold \(F\) into \(\mathbb{P}^4\) carries many remarkable geometric properties. The surface \(S\) is naturally embedded into the grassmannian of lines \(\mathrm{Gr}(2,5)\) of \(\mathbb{P}^4\). Let \(U\) be the restriction to \(S\) of the universal rank \(2\) bundle on \(G\). We have the:
Tangent bundle Theorem (Fano, Clemens-Griffiths, Tyurin) The tangent bundle of \(S\) is isomorphic to \(U\).
Torelli-type Theorem Let \(g'\) be the natural morphism from \(S\hookrightarrow \mathrm{Gr}(2,5)\) defined by the cotangent sheaf of \(S\) generated by its 5-dimensional space of global sections (to add some recall on morphism to grassmannians). Let \(F'\) be the union of the lines corresponding to \(g'(S)\). The threefold \(F'\) is isomorphic to \(F\).
Case 2
\(X\subset\mathbb{P}^4\) a smooth hypersurface of degree \(4\). The family of lines on \(X\) is known to form a connected (in general, smooth) curve \(F\subset\mathrm{Gr}(2,5)\). (S. Bloch and J. P. Murre, On the Chow groups of certain types of Fano threefolds, Compositio Math., 39 (1979), 47–105.)
Theorem (2.2 on Lecture) Further assume the curve of lines \(F\) is smooth. Then \[ \Theta:A^2(X)\rightarrow J^2(X) \] is an isomorphism. (Construct the cycle map later on)
Remark (something really strange) If \(s\in F\) corredponds to the line \(\ell_s\subset X\), one can define an incidence correspondence \(\Sigma\) on \(F\) essentially by taking the closure of the set \[ \{(s,t)\in F\times F-\Delta|\ell_s\cap\ell_t\not=\varnothing\}. \] \(J^2(X)\) can therefore be identified (Tyurin, add ref later) with generalized Pyrm \[ P_\Sigma(F)=\mathrm{Image}(\Sigma_*-1:J(F)\rightarrow J(F)) \] where \(J(F)\) is the Jacobian of \(F\) amd \(\Sigma_*\) satisties a certain quadratic relation \[ \Sigma_*^2+(q-2)\Sigma_*-(q-1)=0. \] On \(q-1\) this is the standard Prym. For quadratic threefold, \(q=24\).
Now we come to build theorem 2.2.
Proposition 2.4 Let \(X\) be a smooth quartic threefold. Then \(\Theta:A^2(X)\rightarrow J^2(X)\) is an isogeny.
To build 2.4 we need several lemmas further.
Lemma 2.5 Let \(X\) be a smooth quartic threefold, and \(\pi:Y\rightarrow X\) obtained by blowing up \(X\) at a non-singular closed subvariety \(V\subset X\). Then \(\Theta_X:A^2(X)\rightarrow J^2(X)\) is an isogeny iff \(\Theta_Y\) is.
proof
- Case 1: If \(V\) is a point, \(A^2(X)=A^2(Y)\) and \(J^2(X)=J^2(Y)\). The former statement is due to the definition of blowup: Replacing a point with \(\mathbb{P}^2\) does not really change \(A^2\). The latter one (maybe cohomological purity theorem)
- Case 2: If \(V\) is a curve, (investigate later)
Lemma 2.6 Let \(X\) and \(Y\) be smooth projective threefolds, and \(\pi:Y\dashrightarrow\) X be a rational map of degree \(d>0\) (the field extension on generic point is deg \(d\) algebraic). The \(\Theta_Y\) being an isogeny implies \(\Theta_X\) being an isogeny.
proof One may resolve the indeterminacy (blowup from nonsingular centers) to make the map into a genuine morphism.
Using the lemma above the case is reduced to the map being everywhere defined. Now consider the diagram
- Now \(\pi':Y\rightarrow X\) is generically finite locally free of degree \(d\).
- On the left side \(\pi_*\pi^*\) induces multiplcation by \(d\). This is seen by pushing any cycle away (moving lemma) from the locus where \(\pi'\) failes to be finite locally free.
- On the right side, \(J^2(T)\) is just \[ \mathrm{H}^{3,0}(T)\oplus\mathrm{H}^{2,1}(T)/\mathrm{H}_3(T). \] We shall then analyse separatly on Hodge cohomologies and the lattice formed by homology groups. Take \(\textbf{1}\in\mathrm{H}^0(Y,\mathbb{C})\) then use the projection formula we know that \[ \pi_*(\pi^*(\alpha)\cup\textbf{1})=\alpha\cup\pi_*(\textbf{1})=\alpha\cup (d\cdot\textbf{1}). \] On the other hand, using the Poincaré duality of \(\mathrm{H}_3\) to \(\mathrm{H}^3(\mathbb{Z})\) and the functoriality of \(\pi_*\) and \(\pi^*\), \(\pi_*\pi^*\) also induces a multiplcation by \(d\).
- Note that \(A^2(X)\) and \(J^2(X)\) are divisible and
Lemma 2.6 Let \(X\) be a smooth quartic threefold. Then \(\Theta:A^2(X)\rightarrow J^2(X)\) is an isogeny.
Geometry of the Fano Surface
(Example 1.7) On the Fano Surface, Incidence Divisor
On a Fano surface \(S\) (of a cubic threefold \(X\)), intersection of divisors gives bilinear maps \[ \mathrm{Pic}(S)\otimes_{\mathbb{Z}}\mathrm{Pic}^0(S)\rightarrow A_0(S).\quad\tag*{(A)} \] Restrict the first variable to divisors algebraically equivalent to \(0\), we can see the morphism \[ \mathrm{Pic}^0(S)\otimes_{\mathbb{Z}}\mathrm{Pic}^0(S)\xrightarrow{\varphi} A_0(S)\rightarrow\mathrm{Alb}(S) \] is a continuous bilinear map between compact tori, hence \(\mathrm{Mor}(\mathrm{Pic}^0(S),\mathrm{Alb}(S))\) is descrete, which implies that the image of \(\varphi\) lies in \(T(S)\).
Claim Both the morphism (A) and \[ \mathrm{Pic}^0(S)\otimes_{\mathbb{Z}}\mathrm{Pic}^0(S)\rightarrow T(S)\quad\tag*{(B)} \] are surjective.
proof (fact? for zero cycles, degree zero implies algebraic to zero) Let \(r,s\in S\) be general points (moving lemma), then it will suffice (divisibility) to show that \[ 2(r)-2(s)\in\mathrm{Image}(\mathrm{Pic}(S)\otimes_{\mathbb{Z}}\mathrm{Pic}^0(S)\rightarrow A_0(S)). \] The line on \(X\) corresponding to \(s\in S\) will be denoted \(\ell_s\), and denote by \(D_s\) the divisor \[ \overline{\{t\in S| \ell_t\cap\ell_s\not=\emptyset,t\not=s\}} \] (the incidence divisor, see C-G p.284 p.290).
In general, (needs further illustration)
- \(s\not\in D_s\),
- \(D_s\cdot D_t\) has degree \(5\).
Then suppose \(r\) and \(s\) are essentially general, there should be a general \(t\) (explain?) s.t. \(\ell_r\cap\ell_t\not=\emptyset\) and \(\ell_s\cap\ell_t\not=\emptyset\).
Since \(2((r)-(s))=2((r)-(t))+2((t)-(s))\), we may assume \(\ell_r\cap\ell_s\not=\emptyset\) (that means the case is reduced to \(2(r)-2(t)\)).
Now let \(L\subset\mathbb{P}^4\) be the plane spanned by \(\ell_r\) and \(\ell_s\). Let \(\ell_u\) be the third line in \(L\cap X=\ell_r\cup\ell_s\cup\ell_u\) (degree \(1\), codim \(2\) meets degree \(3\), dim \(3\) in general position).
Through a general point of \(X\) there pass \(6\) lines (interesting), and we denote by \(\theta:\mathrm{CH}_0(X)\rightarrow\mathrm{CH}_0(S)\) the correspondence thus defined, or to say \[ \theta(p)=([\ell_1])+\cdots+([\ell_6]). \] (Can this homomorphism be realized as a correspondence?)
In our case, we thus have (We may complete other parts first)
\[ \begin{aligned} \theta(a)&=(r)+(t)+(s_1)+\cdots+(s_4)\\ &\sim \end{aligned} \]
Intermediate Jacobian
Main example: smooth hypersurface \(X\subset\mathbb{P}^4\) of degree \(4\), which witnesses which \(A^2\) concides with \(J^2\). Here \(J^2\) is the intermediate jacobian.
TBS: the \(r\)-th Intermediate jacobian on \(X\) is defined as \[ J^r(X)=(F^{n-r+1}\mathrm{H}^{2n-2r+1}(X,\mathbb{C}))^*/\mathrm{H}_{2n-2r+1}(X,\mathbb{Z}) \] where \(F^\bullet\) is the Hodge filtration. According to the Hodge decomposition \[ \mathrm{H}^k(X,\mathbb{C})=\mathrm{H}^{k,0}\oplus\cdots\oplus\mathrm{H}^{0,k} \] the \(r\)-th filtration \(F^r\mathrm{H}^k(X,\mathbb{C})\) is \[ F^r\mathrm{H}^k(X,\mathbb{C})=\mathrm{H}^{k-r,r}\oplus\cdots\oplus\mathrm{H}^{k,0}. \]
In the case of \(r=n\) the intermediate jacobian \(J^n(X)\) is just \[ (F^1\mathrm{H}^1(X,\mathbb{C}))^*/\mathrm{H}_{1}(X,\mathbb{Z})=\mathrm{H}^0(X,\Omega_{X}^1)^*/\mathrm{H}_{1}(X,\mathbb{Z})=\mathrm{Alb}(X) \] which coincides with the ordinary Jacobian.
Intermediate Jacobian is a Complex Torus
An alternative definition of the intermediate jacobian is \[ J^r(X)=\mathrm{H}^{2r-1}(X,\mathbb{C})/(F^r) \]
To see such a construction is indeed a complex torus, one first translate using Poincaré duality, where