Differential forms of the first and second kind on modular algebraic varieties

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1 to J Math Soc Japan Vol 16, No 2, 1964 Differential forms the first and second kind on modular algebraic varieties By Satoshi ARIMA (Received Aug 26, 1963) (Reved Nov 27, 1963) Let $V$ be a complete non-singular variety We denote the universal domain by $K$ and its charactertic by All vector spaces and their dimensions will mean those with respect to $K$ Differential forms the first kind degree on $V$ are, as well known, elements $H^{0}(V, \Omega^{r})$ $\Omega^{r}$ where the sheaf $r$ germs holomorphic differential forms degree $r$ The dimension th space denoted by $h^{r,0}$ We define the differential forms the second kind on $V$ after Picard and Rosenlicht [9] as follows Let to be a differential form degree $r\geqq 1$ on $V$ ; we call to be the second kind at a point $P$ $V$ $\theta_{p}$ if there exts a differential form degree $r-1$ on $V$ such that $\omega-d\theta_{p}$ holomorphic at ; $P$ if the second kind at every point $V$, we call be the second kind on $V$ We denote by the space all closed $9_{2}^{(r)}(V)$ differential forms the second kind degree on $r$ $V$, and by that $9_{e}^{(r)}(V)$ all exact differential forms among them The dimension the factor space $9_{2}^{(1)}(V)/9_{e}^{(1)}(V)$ known to be: (1) or $2h^{1,0}$ $h^{1,0}$ respectively, in case $\dim V=1$, according as $p=0$ or $>1$ (Rosenlicht [9]), (2) in case $2h^{1,0}$ $K=C,$ $\dim V$ being arbitrary (Hodge-Atiyah [5]) Our purpose in \S 1 to show that the dimension the factor space 9 $2(1)(V)/9_{e}^{(1)}(V)$, whenever $p>1,$ $h^{1,0}$ $\dim V$ being arbitrary We shall prove th in making use the operator $C$ Cartier; a pro th fact in case $\dim V=1$, making also use $C$, has been given by Cartier (Tamagawa s lecture in Tokyo University 1960) and Barsotti [2, p 63], but even in th case our pro based on other property $C$ than that used by them Theorem 2 generalizes Theorem 2 [9] In \S 2, we shall give some results on closed semi-invariant differential forms on modular abelian varieties; in case charactertic zero, the corresponding results are found in Barsotti [1] In \S 3 we shall prove the following result It known that the dimension $q$ the Albanese variety a variety $V$ $\leqq h^{1,0}$ (Igusa [6]); there a famous example $V$ due to Igusa [7] for which the strict inequality $q<h^{1,0}$ holds We shall prove that if $V$ defined over a field prime charactertic and if $V$ has p-torsion divors, then the inequality $q<h^{1,0}$ holds

2 denote be Differential forms on modular algebraic varieties 103 $\mathfrak{p}_{w}$ $NoTATIONS$ : $U$ If $W$ a subvariety $U$, then $K(U)$ denotes the field all rational functions on a variety $0_{W}$ and respectively the local ring $W$ on $U$ over $K$ and the ideal non-units $0_{W}$ A prime divor on $U$ means a simple subvariety $U$ codimension 1; a prime divor $W$ gives re to a dcrete valuation rank 1 the field $K(U)$ over $K$ which will be denoted by $v_{w}$ \S 1 Differential forms the second kind Let $U$ be a variety defined over a field prime charactertic, and ru be a closed differential form degree on $r\geqq 1$ $U$ $\cdot$ $\{x_{1}$ Let,, $x_{m}\}$ be a separating transcendence bas $K(U)$ over $K$ Then known to be written uniquely in the form to $=d\theta+\sum_{i_{1}<\cdots<i_{\gamma}}z_{\dot{b}1^{\theta}r}x_{i}dx_{i_{1}}\wedge\cdots\wedge x_{i}^{p_{r}-1}dx_{i_{\mathcal{t}}}$, $z_{i_{1}\cdots ir}\in K(U)$, and the operator $C$ defined by the formula (1) Cto $=\sum_{i_{1}<\cdots<i_{t}}z_{i_{1}\cdots i_{r}}dx_{i_{1}}\lambda\ldots\wedge dx_{i_{r}}$, which independent the choice $\{x_{1}, \cdots, x_{m}\}$ $C$ -semi-linear in the $p^{-1}$ sense $C(\omega+\omega^{\prime})=C\omega+C\omega^{\prime}$, $ C(z^{p}\omega)=zC\omega$ $(z\in K(U))$ If to a closed differential form degree 1, (1) equivalent to the formula (2) $(C\omega(D))^{p}=\omega(D^{p})-D^{p-1}(\omega(D))$, and $C\omega=0$ if and only if form $\frac{df}{f}$ with $f\in K(U)$ (Cf Cartier [4]) exact, and $ C\omega=\omega$ if and only if $\omega\dot{i}s$ the THEOREM 1 Let $V$ be a complete non-singular variety which defined over a field prime charactertic Then $\omega\rightarrow C\omega$ induces a -semilinear $p^{-1}$ bijective homomorphm 9 ; especially we have $2(1)(V)/\mathscr{D}_{e}^{(1)}(V)\rightarrow H^{0}(V, \Omega^{1})$ $p^{-1}$ $\dim 9_{2}^{(1)}(V)/9_{e}^{(1)}(V)=h^{1,0}$ We first prove the following LEMMA 1 Let $V$ be as in Theorem 1, and let a closed differential form degree on V $r\geqq 1$ If holomorphic at $P\in V$, then the second kind at P Especially if the first kind on $V$, then the second kind on $V$ PROOF Let $\{x_{1}, \cdot, x_{m}\}$ be a set uniformizing coordinates $P$ on $V$ Then if the form $\omega=d\theta+\sum_{i_{1}<\cdots<i_{\gamma}}z_{i_{1}\cdots i\gamma}^{p}x_{i}^{p_{1}-1}dx_{i_{1}}\wedge\cdots$ $x_{ir}^{p-1}dx_{i_{r}}$ A, we have $ $C\omega=\sum_{i_{1}<\cdots<i_{\gamma}}z_{i_{1}\cdots i\gamma}dx_{i_{1}}\lambda\ldots\wedge dx_{i_{\gamma}}$ If $z_{t_{1}\cdots i_{t}}\in 0_{P}$ and so $z_{i_{1}\cdots C\omega$ holomorphic at $P$, then we have i\gamma}^{p}x_{i}^{p_{1}-1}\ldots x_{i}^{p_{r}-1}\in 0_{P}$, which implies that $\omega-d\theta$ holomorphic

3 be be can 104 S ARIMA at $P$ Th completes the pro LEMMA 2 Let $V$ be as in Theorem 1, and let a closed differential form degree 1 on V If the second kind at $P\in V$, then holomorphic at P Especially if the second kind on $V$, then Cru the first kind on $V$ PROOF Let $\theta=\omega-df$ be holomorphic at $P$ Take any derivation $D$ $K(V)$ over $K$ $Do_{P}\subseteqq \mathfrak{o}_{p}$ $\theta(d)\in \mathfrak{o}_{p},$ with ; then we have $D^{p-1}\theta(D)\in 0_{P}$ and $\theta(d^{p})$ $\theta$ $\in \mathfrak{o}_{p}$, since holomorphic at $P$ It follows from th and the formula (2) that Since the $(C\theta(D))^{p}\in 0_{P}$ local ring $0_{P}$ a simple point $P$ integrally closed in $K(V)$ and since $C\theta(D)\in K(V)$ $C\theta(D)\in \mathfrak{o}_{p}$, we must have Thus $ C\omega=C\theta$ holomorphic at $P$ Theorem 1 now an immediate consequence Lemma 1, Lemma 2 and the fact that maps the $C$ set all closed differential forms on $V$ onto that all differential forms As a corollary to Lemma 1 and Lemma 2, we have COROLLARY Let $V$ be as in Theorem 1 If a closed differential form degree 1 on $V$ the second kind at every prime divor $V$, then the second kind on $V$ PROOF It follows from the assumption and Lemma 2 that holomorphic at each prime divor V then the first kind on $V$ (cf Zarki [11, p 26, Proposition 87]) therefore the second kind on $V$ by Lemma 1 THEOREM 2 Let $U$ be a variety defined over a field prime charactertic Let a closed differential form degree 1 on U If be approximated arbitrarily closely at a prime divor $W$ $e$ on $U(i$ for each natural number, there exts $n$ $f_{n}\in K(U)$ such that $v_{w}(\omega-df_{n})\geqq n)$, then ru exact PROOF Let $x=x_{1}$ be a uniformizing parameter $W$ on $Ui$ $e$ $\mathfrak{p}_{w}=0_{w}x$ $\mathfrak{p}$ (which we shall denote by below), be a set uniformizing $a_{\partial}nd\{x_{1}, x_{2}, \cdots x_{m}\}$ coordinates $W$ on $U$ Put $\partial_{i}=\overline{\partial}_{x_{i}^{-}}(1\leqq i\leqq m)$ and $\theta_{n}=\omega-df_{n}=z_{1}dx_{1}+\cdots$ $+z_{m}dx_{m},$ $z_{i}\in K(U)$ Denoting $v=v_{w}$, we have $v(\theta_{n})=_{1\leqq i}{\rm Min}_{\leqq m}\{v(z_{\dot{t}})\}\geqq n$ Since $\partial_{i}^{p}=0$, we have $(C\theta_{n}(\partial_{i}))^{p}=-\partial_{i}^{p-1}(\theta_{n}(\partial_{i}))$ $\in \mathfrak{p}^{v(zi)}\subseteqq \mathfrak{p}^{n}=0_{w}x^{n},$ by the formula (2) Since $\theta_{n}(\partial_{i})=z_{i}$ $z_{i}$ the form $z_{i}=x^{n}u_{i}$ with $u_{i}\in 0_{W}$, so that we have $\partial_{i}(\theta_{n}(\partial_{i}))=\partial_{i}(x^{n}u_{i})=nx^{n-1}\cdot\partial_{i}x\cdot u_{i}+x^{n}\cdot\partial_{i}u_{i}\in \mathfrak{v}^{n-1},$ $\in \mathfrak{p}^{n-(p-1)}(n\geqq p-1)$ We have therefore $(C\theta_{n}(\partial_{i}))^{p}\in $=C\theta_{n}(\partial_{i})\in K(U)$ (which independent $n$), $\partial_{i}^{2}(\theta_{n}(\partial_{i}))\in \mathfrak{p}^{n-2},$ $\cdots,$ $\partial_{i}^{p-1}(\theta_{n}(\partial_{i}))$ \mathfrak{p}^{n-(p-1)}$ Putting $a_{i}=c\omega(\partial_{i})$ we have $a_{i}^{p}\in\bigcap_{n=p-1}^{\infty}\mathfrak{p}^{n-(p-1)}=\{0\}$, $a_{i}^{p}=0$ and $a_{i}=0(1\leqq i\leqq m)$ Thus we have $C\omega=a_{1}dx_{1}+\cdots+a_{m}dx_{m}=0$, and that $\omega=df$ with some $f\in K(U)$, which completes the pro

4 be Differential forms on modular algebraic varieties 105 \S 2 Invariant and semi-invariant differential forms on abelian varieties Let $A$ be an abelian variety A differential form degree on $r\geqq 1$ $A$ said to be invariant if $\omega\circ T-\omega=0$ for any $a\in A$, where denotes the $T_{a}$ translation by ; $a$ said to be semi-invariant if for any $ a\in A\omega\circ T_{\alpha}-\omega$ exact (Cf Barsotti [1]) The space all closed semi-invariant differential forms degree on $r$ $A$ $\mathscr{d}_{s^{(r)}}(a)$ will be denoted by Clearly we have $9_{e}^{(\gamma)}(A)$ $\subseteqq 9_{s^{(r)}}(A)$ PROPOSITION 1 Let be a closed differential form degree on an $\geqq 1$ abelian variety A which defined over a field prime charactertic Then semi-invariant if and only if invariant PROOF For a given $a\in A,$ exact if and only if ; $\omega\circ T_{a}-\omega$ $ C(\omega\circ T_{a})=C\omega$ and th so if and only if $(C\omega)\circ T_{a}=C\omega$, since Proposition 1 an immediate consequence th and the definition semi- $(C\omega)\circ T_{a}=C(\omega\circ T_{a})$ invariant Proposition 1 may be stated also as follows COROLLARY 1 Let $A$ $\omega\rightarrow C\omega$ be as in Proposition 1 induces a $p^{-}$ -semilinear bijective homomorphm ; especially we have $9_{s^{(r)}}(A)/9_{e}^{(r)}(A)\rightarrow H^{0}(A, \Omega^{r})$ $\dim 9_{s^{(r)}}(A)/9_{e}^{(r)}(A)=h^{r,0}(1\leqq r\leqq\dim A)$ COROLLARY 2 Let $A$ and as in Proposition 1 (1): If semiinvariant, then the second kind on A (2): If degree 1 then semi-invariant if and only if to the second kind on $A$ PROOF Note that a differential form degree on $r\geqq 1$ $A$ invariant if and only if it the first kind on A (1) follows immediately from Proposition 1 and Lemma 1 (2) follows from Lemma 2 and Proposition 1 \S 3 -torsions and the inequality $q<h^{1,0}$ 1) Let $V$ be a complete non-singular variety defined over a field prime charactertic, and $ V^{\varphi}\rightarrow$ $A$ be the Albanese variety $V$ Pic (V) and Pic $(A)$ denote the group all divor classes (with respect to the linear equivalence) on $V$ and $A$ respectively; Pic and Pic denote respectively the subgroup the elements in Pic (V) and Pic $(A)$ whose orders divide $(V)_{p}$ $(A)_{p}$ We shall $\equiv$ $\mathfrak{a}\rightarrow\varphi^{-1}(\mathfrak{a})$ denote the algebraic and linear equivalence relations between divors by and $\sim$ respectively $\mathfrak{a}$ If a divor on $A$ $\mathfrak{a}\sim 0$ with, $\varphi^{-1}(\mathfrak{a})\sim 0$ then on $V$ (if it defined); $\varphi^{*}:$ induces a homomorphm Pic $(A)\rightarrow Pic(V)$ and Pic $(A)_{p}\rightarrow Pic(V)_{p}$ (Lang [8, p 236, p 65]) A divor $X$ on $V$ said to be p-torsion if $X\not\equiv O$ but $px\equiv 0$ $\delta\varphi$ induces the homomorphm the $\omega\rightarrow\omega\circ\varphi$ space $H^{0}(A, \Omega^{1})$ into the subspace closed differential forms in $H^{0}(V, \Omega^{1})$, $\rightarrow C(V)$ and which known to be injective (Igusa [6]) denotes the additive 1) We owe to D Mumford the formulation which follows

5 $\hat{v}$ denotes : induces 106 S ARIMA $\delta\varphi$ group differential forms in $H^{0}(V, \Omega^{1})$ which are invariant by $C;X(A_{0}^{\backslash }$ denotes the similar group on A an injective homomorphm oi $X(A)$ into $X(V)$, since $ C(\omega\circ\varphi)=(C\omega)\circ\varphi$ for $\omega\in\angle(a)$ We use the following result Cartier [3, Theorem 5] (3) Notations being as above, the group Pic $(V)_{p}$ canonically omorphic to $-\mathcal{l}(v)$ The pro th fact not publhed However, a pro th fact in case $\dim V=1$ given in Serre [10, p 28, Proposition 10], and it can be applied to the case $\dim V\geqq 1$ In fact, note that a differential form on $V$ the first kind on $V$ if and only if it holomorphic at every primc divor $V$ (cf [11, p 26, Proposition 87]) Then, as in the pro [10 Proposition 10], it can be seen that $Cl(X)^{2)}\rightarrow\underline{d}\underline{f}f$ injective homomorphm $\theta$ with $px=(f)$ gives an Pic In order to see that $(V)_{p}\rightarrow_{e}\mathcal{L}(V)$ $\theta$ surjective, we have only to show that, if a differential form $\frac{df}{f}(f\in K(V))$ holomorphic at a prime divor $W$ on $V$ $t$, then $v_{w}(f)\equiv 0(mod p)$ Let be a uniformizing parameter $W$ on $V$, and let $f=t^{e}u$ with a unit $u$ $\mathfrak{o}_{w}$ in and $e=v_{w}(j^{\cdot})$ We have $\frac{df}{f}=e\frac{dt}{t}+\frac{du}{u}$ The differential form $\frac{dt}{t}$ not holomorphic at $W$, since the derivation $D=\frac{\partial}{\partial t}$ $K(V)/K$ holomorphic at $W$ and $\frac{dt}{t}(d)=\frac{1}{t}\not\in 0_{W}$ ; th implies $e\equiv 0(mod p)$ since $e\frac{dt}{t}=\frac{dfdu}{fu}$ $\hat{a}\rightarrow\hat{v}$ (Cf holomorphic at $W$ $\theta$ Thus we see that bijective $\varphi^{*}:$ PROPOSITION 2 Notations being as above, Pic suriective if and only if $V$ has no p-torsion divor $(A)_{p}\rightarrow Pic(V)_{p}$ PROOF the subgroup Pic (V) classes which are represented $i$ by divors algebraically equivalent to zero ( $e$ the Picard variety $V$ ) $;\hat{a}$ denotes the similar group on $A$ If $D$ a Poincar\ e divor for $A$ $ D\circ\varphi$, then a Poincar\ e divor for gives therefore a canonical bijective homo- $V;\varphi^{*}$ morphm Lang [8, p 148, Proposition 1 and Theorem 1]) If $V$ has no -torsion divor, then Pic $(V)_{p}$ $\hat{v};\varphi^{*}$ a subgroup maps therefore Pic $(A)_{p}$ onto Pic $(V)_{p}$ omorphically, since the abelian variety $A$ has no -torsion divor Conversely, if $V$ has a divor $X$ such that $X\not\equiv O$ but $px\equiv 0$, then we can find a divor on $Y$ $V$ such that $Y\equiv X$ and $py\sim O$ (Lang [8, p 101, Corollary 4 to Theorem 4]) Then $y=cl(y)\in Pic(V)_{p}$ can not be an image from $\varphi^{*}:$ $Pic(A)_{p}$, which shows that $Pic(A)_{p}\rightarrow Pic(V)_{p}$ not surjective Proposition 2 thereby proved 2) $Cl(X)$ denotes the linear equivalence class containing the divor $X$

6 $\theta$ denotes $\theta$ $\varphi^{*}$ Differential forms on modular algebraic varieties 107 the canonical omorphm $Pic(V)_{p}\rightarrow\rightarrow C(V)$ in (3) PROPOSITION 3 Notations being as above, the following diagram commutative Pic $(V)_{p}-$ Pic $(A)_{p}$ Zll $\delta\varphi$ $X(V)-X(A)$ PROOF Let $a=cl(ti)\in Pic(A)_{p}$ with $p\mathfrak{a}=(\alpha),$ $\alpha\in K(A)$ Then we have $\theta(a)=\frac{d\alpha}{\alpha}\in X(A)$ and $\varphi^{*}(a)=cl(\varphi^{-1}(0))\in Pic(V)_{p}$ It will not be difficult to see here that $\alpha\circ\varphi$ $ l \theta$ defined and not constant Th being so, we have $0$ It follows from th that $\theta(\varphi^{*}(a))$ $p\varphi^{*}(a)=cl(p\varphi^{-1}(\mathfrak{a}))=cl(\varphi^{-1}(\alpha))=cl((\alpha\circ\varphi))$ $=\frac{d(\alpha\circ\varphi)}{\alpha\circ\varphi}=\delta\varphi(\theta(a))$, which implies the commutativity the diagram In view the commutative diagram in Proposition 3, Proposition 2 equivalent to the following $a$ field THEOREM 3 Let $V$ be a complete non-singular variety which defined over prime charactertic, and $ V^{\varphi}\rightarrow$ $A$ be its Albanese variety Denote by $X(V)$ the additive group differential forms $\omega\in H^{0}(V, \Omega^{1})$ such that $ C\omega=\omega$, and the similar group on A Then $by\rightarrow C(A)$ $\delta\varphi:\mathcal{l}(a)\rightarrow\rightarrow C(V)$ surjective if and only if $V$ has no p-torsion divor COROLLARY Let $V,$ and $A$ be as in Theorem 3, and $q$ the dimension A If $V$ has a p-torsion divor, then we have $q<h^{1,0}$ PROOF If $V$ has a -torsion divor, then we may find by Theorem 3 a differential form $\frac{d}{f}f_{-}\in 1i(V)$ which not an image from $\Leftrightarrow a moment that $\underline{d}f_{-}f=\omega\circ\varphi$ for $A$ We have $ C(\omega\circ\varphi)=(C\omega)\circ\varphi$, since C(A)$ Assume for some differential form to the first kind on closed (Cf [1, p 93, 21]) Since $(C\omega-\omega)\circ\varphi=C(\omega\circ\varphi)-\omega\circ\varphi=C(\frac{df}{f})-\frac{df}{f}=0$, and since $ C\omega-\omega$ the first kind by Lemma 2, we would have $C\omega-\omega=0$ because the injectivity $\delta\varphi$ would be therefore the form $\omega=\frac{d\alpha}{\alpha}$ with $\alpha\in K(A)$ Th would imply that $\frac{df}{f}=\omega\circ\varphi$ an image from $\rightarrow C(A)$, which a contradiction Musashi Institute Technology, Tokyo

7 108 S ARIMA References [1] I Barsotti, Factor sets and differentials on abelian varieties, Trans Amer Math Soc, 84 (1957), [2] I Barsotti, Repartitions on abelian varieties, Illino J Math, 2 (1958), [3] P Cartier, Une nouveller op\ eration sur les formes diff\ erentielles, C R Acad Sci, 244 (1957), [4] P Cartier, Questions de rationalit\ e des diveurs en g\ eom\ etrie alg\ ebrique, Bull Soc Math France, 86 (1958), [5] V W D Hodge and M F Atiyah, Integrals the second kind on an algebraic variety, Ann Math, 62 (1955), [6] J Igusa, A fundamental inequality in the theory Picard varieties, Proc Nat Acad Sci U S A, 41 (1955), [7] J Igusa, On some problems in abstract algebraic geometry, Proc Nat Acad Sci U S A, 41 (1955), [8] S Lang, Abelian varicties, New York, Interscience Publhers, 1958 [9] M Rosenlicht, Differentials the second kind for algebraic function fields one variable, Ann Math, 57 (1953), [10] J-P Serre, Sur la topologie des vari\ et\ es alg\ ebriques en caract\ ertique, Symposium Algebraic Topology, Mexico, 1956 [11] O Zarki, An introduction to the theory algebraic surfaces, lecture notes, Harvard University,