An analytic approach

Lefschetz (1,1) theorem

A homology class $\gamma \in H_2(X, \mathbf{Z})$ is in $\operatorname{Pic} \overline{X}$ if and only if $\int_\gamma \omega_X = 0$, where $\omega_X$ is the nonzero holomorphic 2-form $\omega_X$ on $X$, unique up to scaling.

$$H_2(X, \mathbf{Z}) \longrightarrow \mathbf{C}, \qquad \gamma \mapsto \int_\gamma \omega_X.$$

Hence, if $\Pi \in \mathbf{C}^{22}$ represents this map, then we are reduced to finding a (saturated) lattice $\Lambda \subset H_2(X, \mathbf{Z})$ of solutions

$$\Pi R = 0, \qquad R \in H_2(X, \mathbf{Z}) \simeq \mathbf{Z}^{22}.$$

SOURCE: L:L524-554

Say the name out loud here. Lambda is introduced on this slide and nowhere else; the Jacobian period lattice on slide 5 is a different object and carries a subscript.

What it does. Converts geometry into numerical linear algebra in one line. This is the counterpart of Lecture 1's Pic = H^2 intersect H^{1,1}, now made computational.

Spoken: So with the period vector $\Pi = [\int_{\gamma} \omega_X]_{\gamma} \in \mathbf{C}^{22}$, finding Picard classes means finding the integer solutions of $\Pi R = 0$.

Spoken: With exact periods that kernel already is $\operatorname{Pic}\overline{X}$.

Spoken: What one has is $\Pi$ to finite precision: write $\Lambda$ for the saturated lattice produced from those approximate periods, satisfying $\Pi R \approx 0$ to the working precision.

Spoken: Whether its vectors are Picard classes is what the rest of the lecture establishes.

Spoken: Every $\Lambda$ later in this lecture is this object.

TRANSCRIBED FROM: five-nomial-quartics/slides/mukai_leiden.tex:524-554 (items 1 to 3 of the closing itemize; the Lairez-Sertoz item is slide 3)