| Izvestiya: Mathematics |
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Algebraic de Rham theorem and Baker–Akhiezer function I. M. Kricheverab, L. A. Takhtadzhyancd a Columbia University, New York, USA b Skolkovo Institute of Science and Technology c Department of Mathematics, Stony Brook University, NY, USA d Euler International Mathematical Institute, St. Petersburg
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    § 1. IntroductionLet $X$ be a smooth algebraic variety over $\mathbb{C}$ with the classical topology of a complex manifold. According to Atiyah and Hodge [1], a closed meromorphic $p$-form $\varphi$ on $X$ is called differential of a second kind if it has zero residues on open subsets $U=X\setminus D$ for sufficiently large divisors $D$. A far-reaching generalization of Atiyah and Hodge results was given by Grothendieck [2]. The quotient groups
$$
\begin{equation*}
\frac{\{p\text{-forms of the second kind}\}}{\{\text{exact forms}\}}
\end{equation*}
\notag
$$
have a natural interpretation in terms of a spectral sequence of certain complex of sheaves of meromorphic forms on $X$ (see [3], Ch. 3, § 5). In particular,
$$
\begin{equation*}
H_{\mathrm{dR}}^1(X,\mathbb{C}) \simeq \frac{\{1\text{-forms of the second kind}\}}{\{\text{exact forms}\}}.
\end{equation*}
\notag
$$
When $X$ is a smooth algebraic curve of genus $g$, this isomorphism follows from the Riemann–Roch theorem. It turns out that in this case the space of differentials of the second kind carries a natural skew-symmetric bilinear form, which is non-degenerate when taking a quotient by the subspace of exact forms. Using this bilinear form, in Theorem 1 we give a more explicit formulation of the algebraic de Rham theorem. Specifically, we show that each non-special effective divisor $D$ of degree $g$ on the Riemann surface $X$ defines a symplectic basis of $H_{\mathrm{dR}}^1(X,\mathbb{C})$, which makes it possible to explicitly describe a complement to the Lagrangian subspace of holomorphic $1$-forms on $X$ as the subspace of differentials of the second kind with the poles in $D$. In § 4, it will be shown that every non-special effective divisor $D$ of degree $g$ on $X$ defines an explicit isomorphism between the vector space $H^{0,1}(X,\mathbb{C})$ and the Lagrangian subspace of differentials of the second kind with poles in $D$. This allows us to explicitly describe the tangent space to the Picard variety (and its “incarnations”, Albanese and Jacobian varieties) in pure algebro-geometric terms. It is quite remarkable that this formalism is connected with the theory of integrable systems. In the standard approach (see, for example, [4]), an integrable system is described by the zero curvature equation
$$
\begin{equation*}
\frac{\partial L}{\partial t} -\frac{\partial M}{\partial x}+LM-ML=0,
\end{equation*}
\notag
$$
where $L(x,t,\lambda)$ and $M(x,t,\lambda)$ are certain $r\times r$ matrix-valued rational functions of the spectral parameter $\lambda\in \mathbb{C}\mathbb{P}^1$ which also depend on extra arguments $x$ and $t$ (physical space and time variables). In [5], [6] by the first author, the zero curvature formalism was extended to the case when spectral parameter varies on an algebraic curve. Namely, it was shown in [5] that a natural framework for this generalization is provided by the Hitchin system expressed in terms of Tyurin parameters for stable holomorphic vector bundles of rank $r$ and degree $rg$ on an algebraic curve. Correspondingly, the rational functions $L$ on $\mathbb{C}\mathbb{P}^1$ become special $r\times r$ meromorphic matrix $1$-forms $L(z)\,dz$ on an algebraic curve, and the set of such matrices is parameterized by the moduli space (more precisely, by its Zariski open subset) of stable holomorphic vector bundles of rank $r$ and degree $rg$ (see [5] for details). A similar explicit description is given for $r\times r$ meromorphic matrix-valued functions $M(z)$. It turns out that, in the simplest case $r=1$, this formalism is still non-trivial and is naturally connected with Theorem 1 (see the related discussion in § 3). Namely, as shown in § 5, the meromorphic $1$-forms $L(z)\,dz$ become differentials of the first kind on an algebraic curve $X$, while analogs of $M(z)$ are meromorphic functions $f$ defined using two non-special effective divisors $D$ and $D_0$ of degree $g$ on $X$. The varying divisors $D$ parameterize the Jacobian of $X$ with the base point $D_0$, and the vector fields describing the motion of points of $D$ are naturally expressed in terms of the meromorphic functions $f$. Remarkably, in the case when $X$ is a hyperelliptic curve, equations (5.3) for the integral curves of these vector fields coincide with the Dubrovin equations arising in the theory of finite-gap integration of the Korteweg–de Vries equation [7]. Moreover, integrating meromorphic functions $f$ along these integral curves and using the Dubrovin equations, we naturally obtain the Baker–Akhiezer function, a fundamental object in the algebro-geometric approach to integrable systems, which was introduced by the first author in [8]. The second author is grateful to the referee for constructive remarks and suggestions. § 2. Differentials of the second kindLet $X$ be a connected compact Riemann surface of genus $g$ with classical topology. We denote by $\mathcal{O}_X$ the sheaf of germs of holomorphic functions on $X$. Let $\mathcal{M}_X$ be the sheaf of germs of meromorphic functions on $X$, and let $\mathcal{M}$ be the vector space of meromorphic functions on $X$. For every divisor $D$ on $X$, let $L=\mathcal{O}(D)$ be the holomorphic line bundle associated with $D$, and let $H^0(X,L)$ be the vector space of holomorphic sections of $L$ over $X$. The isomorphism
$$
\begin{equation*}
H^0(X,L)\simeq \mathcal{L}_{D}=\{f\in \mathcal{M}\colon (f)+D\geqslant 0\}.
\end{equation*}
\notag
$$
is very useful. From the Riemann–Roch theorem together with the Kodaira–Serre duality it follows that where $h^0(L)=\dim_{\mathbb{C}} H^0(X,L)$, $\deg L$ is the degree of $L$, and $K_X$ is the canonical class of $X$ (the holomorphic cotangent bundle to $X$).Let $\mathrm{d}$ be the exterior derivative on $X$. The sheaf $\mathrm{d} \mathcal{M}_X$ is a sheaf of germs of differentials of the second kind on $X$ and $\Omega^{(\mathrm{2nd})}= H^0(X,\mathrm{d} \mathcal{M}_X)$ is the infinite-dimensional vector space of the differentials of the second kind (the meromorphic $1$-forms on $X$ with zero residues). The infinite-dimensional vector space $\Omega^{(\mathrm{2nd})}$ has the natural skew-symmetric bilinear form1
$$
\begin{equation*}
\omega_X(\theta_1,\theta_2) =\sum_{P\in X}\operatorname*{Res}_P(\mathrm{d}^{-1}\theta_1\theta_2),\qquad \theta_1,\theta_2\in \Omega^{(\mathrm{2nd})},
\end{equation*}
\notag
$$
where $\mathrm{d}^{-1}\theta_1$ denotes any locally defined function $f$ such that $\mathrm{d} f = \theta_1$ (the local antiderivative). The ambiguity in the choice of $f$ does not matter. Indeed, it is clear that the bilinear form $\omega_X$ is defined by a finite sum and the choice of an additive constant in the definition of a local antiderivative is irrelevant. The skew-symmetry of $\omega_X$ follows from the basic property
$$
\begin{equation*}
\operatorname*{Res}_P(f_1\, \mathrm{d} f_2)=-\operatorname*{Res}_P(f_2\, \mathrm{d} f_1),
\end{equation*}
\notag
$$
where the meromorphic functions $f_1$ and $f_2$ are the local antiderivatives of $\theta_1$ and $\theta_2$ in a neighbourhood of $P\in X$.
§ 3. Algebraic de Rham theoremIn abstract form, the algebraic de Rham theorem reads as follows:
$$
\begin{equation}
H^1_{\mathrm{dR}}(X,\mathbb{C})\simeq \Omega^{(\mathrm{2nd})}/\mathrm{d}\mathcal{M},
\end{equation}
\tag{3.1}
$$
which is easily proved using the sheaf-theoretic de Rham isomorphism
$$
\begin{equation*}
H^1_{\mathrm{dR}}(X,\mathbb{C})\simeq H^1(X,\underline{\mathbb{C}}),
\end{equation*}
\notag
$$
where $\underline{\mathbb{C}}$ is the locally constant sheaf. Indeed, consider the short exact sequence of sheaves
$$
\begin{equation*}
0 \to \underline{\mathbb{C}} \xrightarrow{i} \mathcal{M}_X \xrightarrow{\mathrm{d}} \mathrm{d} \mathcal{M}_X \to 0
\end{equation*}
\notag
$$
and the corresponding exact cohomology sequence
$$
\begin{equation*}
H^0(X,\mathcal{M}_X) \xrightarrow{\mathrm{d}} H^0(X,\mathrm{d} \mathcal{M}_X) \xrightarrow{\delta} H^1(X,\underline{\mathbb{C}})\to H^1(X, \mathcal{M}_X).
\end{equation*}
\notag
$$
By the Riemann–Roch theorem, if $\deg D>2g-2$, which implies (see, for example, [9], Ch. 2, § 17.7) which proves (3.1). Using the bilinear form $\omega_X$, we can make isomorphism (3.1) more concrete. Namely, we have the following result (see [10], Ch. 6, § 8, [11], Ch. III, § 5.3, § 5.4, and [12], Theorem 4). Theorem 1. The following assertions hold. (i) The restriction of the bilinear form $\omega_X$ to $\Omega^{(\mathrm{2nd})}/\mathrm{d}\mathcal{M}$ is non-degenerate and
$$
\begin{equation*}
\dim_{\mathbb{C}}\Omega^{(\mathrm{2nd})}/\mathrm{d}\mathcal{M} =2g.
\end{equation*}
\notag
$$
(ii) Each non-special effective divisor $D$ of degree $g$ on $X$ defines the isomorphism
$$
\begin{equation*}
\Omega^{(\mathrm{2nd})}/\mathrm{d}\mathcal{M}\simeq\Omega^{(\mathrm{2nd})} \cap H^0(X, K_X+2D).
\end{equation*}
\notag
$$
(iii) Let $D=P_1 + \dots +P_g$ be a non-special divisor of degree $g$ with distinct points. For every choice of local coordinates in the neighbourhoods of $P_i$, the vector space $\Omega^{(\mathrm{2nd})}\cap H^0(X, K_X+2D)$ has the basis $\{\vartheta_i,\tau_i\}_{i=1}^g$ symplectic with respect to the bilinear from $\omega_X$, This basis consists of differentials of the first kind $\vartheta_i$ and differentials of the second kind $\tau_i $ uniquely characterized by the conditions where $z_j=z(P_j)$ for a local coordinate $z$ at $P_j$, and $i,j=1,\dots,g$. Proof. Let $(\theta)_{\infty}=\sum_{i=1}^{l}n_iQ_i$ be the polar divisor of $\theta\in\Omega^{(\mathrm{2nd})}$, $n_i\geqslant 2$. Since $D$ is non-special, $h^0(K_X-D)=0$, and by Riemann–Roch formula we have $h^0(D+nQ_i)=n+1$ for $n\geqslant 0$. Thus if $Q_i$ is not a point of $D$, there exists a meromorphic function $f_i\in\mathcal{L}_{D+(n_i-1)Q_i}$ such that If $Q_i$ is a point of $D$, there is a function $f_i\in\mathcal{L}_{D+(n_i-1)Q_i} $ such that (In this case, $h^0(D)=1$, and hence one can not adjust the principle part of $df_i$ at $Q_i$ to cancel possible second order pole of $\theta$.) So, for $f=\sum_{i=1}^{l}f_i$, we have which proves assertion (ii).
The dimension formula in (i) easily follows from assertion (ii) since
$$
\begin{equation*}
\begin{aligned} \, \!\dim_{\mathbb{C}}\Omega^{(\mathrm{2nd})}\,{\cap}\,H^0(X, K_X+2D) &=h^0(X, K_X+2D)\,{-}\,h^0(X, K_X+D)\,{+}\,h^0(X, K_X) \\ &=(3g-1)-(2g-1)+g=2g. \end{aligned}
\end{equation*}
\notag
$$
To prove assertion (iii) and the remaining part in assertion (i), consider the linear map
$$
\begin{equation*}
L\colon \Omega^{(\mathrm{2nd})}\cap H^0(X, K_X+2D)\to\mathbb{C}^{2g},
\end{equation*}
\notag
$$
defined as follows. For each $\theta\in \Omega^{(\mathrm{2nd})}\cap H^0(X, K_X+2D)$, let $\alpha_i(\theta), \beta_i(\theta)\in\mathbb{C}$ be such that
$$
\begin{equation*}
\frac{\theta}{\mathrm{d} z}-\alpha_i(\theta) -\frac{\beta_i(\theta)}{(z-z_i)^{2}} =O(z-z_i)
\end{equation*}
\notag
$$
near $P_i$. We also set
$$
\begin{equation*}
L(\theta)=\bigl(\alpha_1(\theta), \beta_1(\theta),\dots, \alpha_g(\theta), \beta_g(\theta)\bigr).
\end{equation*}
\notag
$$
The divisor $D$ is non-special, and hence the map $L$ is injective, and therefore, is an isomorphism, and we define $\vartheta_i$ and $\tau_i$ to have only non-zero components of $L$ to be, respectively, $\alpha_i=1$ and $\beta_i=1$. This proves Theorem 1. Remark 1. The choice of a non-special effective divisor $D$ on $X$ with $g$ distinct points $P_i$ and local coordinates is as an algebraic analogue of the choice of $a$-cycles on a Riemann surface. Correspondingly, the differentials $\tau_i$ are analogues of differentials of the second kind with second-order poles, zero $a$-periods and normalized $b$-periods. The symplectic property of the basis $\{\vartheta_i,\tau_i\}_{i=1}^g$ is an analogue of the reciprocity laws for differentials of the first kind and the second kind (see [13], Ch. 5, § 1, and [14], Ch. VI, § 3). Remark 2. Let $\Omega^{(\mathrm{2nd})}(2D)$ be the subspace in $\Omega^{(\mathrm{2nd})}$ spanned by $\tau_i$, Then $\Omega^{(\mathrm{2nd})}(2D)$ and $H^0(X,K_X)$ are Lagrangian subspaces in $\Omega^{(\mathrm{2nd})}/\mathrm{d}\mathcal{M}$ dual with respect to the pairing given by the symplectic form $\omega_X$. § 4. Tangent space to the Picard varietyWe have the decomposition
$$
\begin{equation}
H^1_{\mathrm{dR}}(X,\mathbb{C}) =H^{1,0}(X,\mathbb{C})\oplus H^{0,1}(X,\mathbb{C})
\end{equation}
\tag{4.1}
$$
with natural pairing
$$
\begin{equation*}
H^{1,0}(X,\mathbb{C})\otimes H^{0,1}(X,\mathbb{C}) \ni\alpha \otimes\beta\mapsto(\alpha,\beta)=\int_X\alpha\wedge\beta\in\mathbb{C}.
\end{equation*}
\notag
$$
The period map
$$
\begin{equation*}
H^{1,0}(X,\mathbb{C})\ni\vartheta\mapsto\int_c\vartheta\in\mathbb{C},
\end{equation*}
\notag
$$
where $c\in H_1(X,\mathbb{Z}$), defines the canonical inclusion of the lattice $H_1(X,\mathbb{Z})$ into the vector space $H^{1,0}(X,\mathbb{C})^{\vee}$, which is the dual space of $H^{1,0}(X,\mathbb{C})$, and defines the Albanese variety
$$
\begin{equation*}
\mathrm{Alb}(X)=H^{1,0}(X,\mathbb{C})^{\vee}/H_1(X,\mathbb{Z}).
\end{equation*}
\notag
$$
Using the Dolbeault isomorphism and the exponential exact sequence of sheaves on $X$, we have, for the Picard variety of line bundles of degree $0$ on $X$,
$$
\begin{equation*}
\mathrm{Pic}^0(X)=H^{0,1}(X,\mathbb{C})/H^1(X,\mathbb{Z}).
\end{equation*}
\notag
$$
Thus, the holomorphic tangent space to $\mathrm{Pic}^0(X)$ can be identified with the vector space $H^{0,1}(X,\mathbb{C})$. However, Theorem 1 allows us to describe the tangent space to the Picard variety in purely algebro-geometric terms. Namely, the following simple result holds. Proposition 1. Each non-special effective divisor $D$ of degree $g$ on a Riemann surface $X$ defines an isomorphism Proof. It follows from assertion (iii) of Theorem 1 that the mapping satisfies for any $\vartheta\in H^0(X,K_X)$, and is an isomorphism. This proves the proposition. By identifying $H^{1,0}(X,\mathbb{C})^{\vee}$ with $\Omega^{(\mathrm{2nd})}(2D)$, we get an inclusion of the lattice $H_1(X,\mathbb{Z})$ into the vector space $\Omega^{(\mathrm{2nd})}(2D)$, which is defined as follows. Let $\theta_c$ be the $(0,1)$-component of the Poincaré dual of a cycle $c\in H_1(X,\mathbb{Z})$, so that
$$
\begin{equation*}
\int_c\vartheta=\int_X\vartheta\wedge\theta_c=(\vartheta,\theta_c)\quad \text{for all}\quad\vartheta\in H^{1,0}(X,\mathbb{C}).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
H_1(X,\mathbb{Z})\ni c\mapsto\tau_c=\psi(\theta_c) =\sum_{i=1}^g\int_c\vartheta_i\cdot \tau_i\in \Omega^{(\mathrm{2nd})}(2D),
\end{equation}
\tag{4.2}
$$
and, therefore,
$$
\begin{equation}
\mathrm{Alb}(X)=\Omega^{(\mathrm{2nd})}(2D)/H_1(X,\mathbb{Z}).
\end{equation}
\tag{4.3}
$$
Thus, by choosing a non-special effective divisor $D$ of degree $g$ one can identify the holomorphic tangent spaces to the manifolds
$$
\begin{equation*}
\mathrm{Alb}(X)\simeq \mathrm{Pic}^0(X)\simeq\mathrm{Jac}(X)
\end{equation*}
\notag
$$
with vector space $\Omega^{(\mathrm{2nd})}(2D)$ of the differentials of the second kind with poles in $D$. Correspondingly, the holomorphic cotangent space is naturally identified with the vector space of $H^{1,0}(X, \mathbb{C})$ of differentials of the first kind, and the pairing with $\Omega^{(\mathrm{2nd})}(2D)$ is given by the symplectic form $\omega_X$.
§ 5. The Baker–Akhiezer functionGiven a fixed non-special effective divisor $D_0=Q_1 \,{+}\,{\cdots}\,{+}\, Q_g$ of degree $g$ on $X$, let $\{\vartheta_i\}_{i=1}^g$ be the basis of $H^0(X, K_X)$ from Theorem 1 specialized to the divisor $D_0$. Consider the Abel–Jacobi map where $\mu^{(g)}$ is the Abel sum: for the varying divisor $D=P_1 + \dots + P_g$,
$$
\begin{equation}
\mu^{(g)}(D)=\biggl(\sum_{i=1}^g\int_{Q_i}^{P_i}\vartheta_1,\dots, \sum_{i=1}^g\int_{Q_i}^{P_i}\vartheta_g\biggr).
\end{equation}
\tag{5.1}
$$
We choose local coordinates at the points $P_i$ and put $z_i=z(P_i)$. It follows from (5.1) that $1$-forms $dz_i$ on $\mathrm{Jac}(X)$ at the base point $\mu^g(D_0)$ correspond to the differentials $\vartheta_i$, and the vector fields $\partial/\partial z_i$ correspond to the differentials of the second kind $\tau_i$ from Theorem 1. If divisor $D$ is also non-special, it follows from the group law on the Jacobian and Theorem 1 that $\mathrm{d} z_i$ and $\partial/\partial z_i$ at a point $\mu^{(g)}(D)\in \mathrm{Jac}(X)$ are given by the symplectic basis of $\Omega^{(\mathrm{2nd})}\cap H^0(X, K_X+2D)$ from Theorem 1. Equivalently, these vector fields on $\mathrm{Jac}(X)$ can be described using the formalism of Lax equations on algebraic curves, developed by the first author in [5] and [6]. The main ingredients in [5] and [6] are stable vector bundles of rank $r$ and degree $rg$, Lax operators, certain meromorphic $r\times r$ matrix-valued $1$-forms $L(z)\,dz$ on a Riemann surface $X$, and $r\times r$ meromorphic matrix-valued functions $M(z)$. Specialization to the Jacobian corresponds to the case $r=1$ and substantially simplifies construction in [5] and [6]. Namely, the meromorphic $1$-forms $L(z)\, \mathrm{d} z$ become differentials of the first kind $\vartheta\in H^0(X,K_X)$, while analogs of the meromorphic functions $M(z)$ are defined as follows. Consider the vector space
$$
\begin{equation*}
\mathcal{L}_{D+D_0}=\{f\in\mathcal{M} \colon (f)+D+D_0\geqslant 0\}.
\end{equation*}
\notag
$$
It follows from the Riemann–Roch theorem that $\dim_{\mathbb{C}}\mathcal{L}_{D+D_0}=g+1$. Thus for any fixed choice of principal parts of $f$ at the points of $D_0$, not all of them are equal to zero, there is a unique, up to an inessential additive constant, function $f\in\mathcal{L}_{D+D_0}$ satisfying
$$
\begin{equation}
f(z)=\frac{\alpha_i}{z-z_i} +O(1),\qquad z_i=z(P_i),
\end{equation}
\tag{5.2}
$$
at all points of the divisor $D=P_1+\dots+P_g$. Functions $f$, parametrized by their fixed principal parts at $D_0$, play the role of meromorphic functions $M(z)$ in case $r=1$; coefficients $\alpha_i$ depend on the principal parts at $D_0$. We have a unique decomposition where $\tau\in\Omega^{(\mathrm{2nd})}(2D)$ (see Remark 2) and $(\tau_0)+2D_0\geqslant 0$. By the residue theorem,
$$
\begin{equation*}
-\sum_{i=1}^g\operatorname*{Res}_{P_i}(f\vartheta) =\omega_X(\vartheta,\tau) =\omega_X(\vartheta,\tau_0),\qquad \vartheta\in H^0(X,K_X),
\end{equation*}
\notag
$$
and so the pairing (2.22) in [5], as given by the Krichever–Phong form, coincides with the pairing given by the symplectic form $\omega_X$. A choice of a symplectic basis of establishes the correspondence
$$
\begin{equation*}
f\mapsto \mathscr{L}_{f} = - \sum_{i=1}^g\alpha_i\,\frac{\partial}{\partial z_i}
\end{equation*}
\notag
$$
between the rational functions $f\in\mathcal{L}_{D+D_0}$ and the vector fields on $\mathrm{Jac}(X)$. Along the integral curve $D(t)=P_1(t)+\dots +P_g(t)$ of $\mathscr{L}_{f}$, where $D(0)=D$, we have where the dot stands for the $t$ derivative. If $X$ is a hyperelliptic curve, equations (5.3) are the classical Dubrovin equations arising in the theory of finite-gap integration for the Korteweg–de Vries equation [7], written in terms of the Abel transform. Using the Dubrovin equations, we see that, along the integral curve, equations (5.2) take the form
$$
\begin{equation}
f_t(z)=-\frac{\dot{z}_i(t)}{z-z_i(t)} +O(1),\qquad i=1,\dots,g.
\end{equation}
\tag{5.4}
$$
Thus integrating and introducing we see from (5.4) that $\Psi$ is a meromorphic function on $X\setminus D_0$ with simple poles only at $D$, simple zeros only at $D(T)$, and essential singularities at $D_0$. The function $\Psi$ is nothing but the celebrated Baker–Akhiezer function, which was introduced by the first author in [8]. We leave it to the interested reader to describe by explicit formulas this connection between algebraic de Rham theorem and the integrable systems. 
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\Bibitem{KriTak24} |
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