This course develops the analytic and geometric foundations of positivity in complex geometry and follows their consequences through vanishing theorems, deformation theory, and moduli problems. It begins with holomorphic line bundles and the notion of positivity, then studies how sections and jets encode projective embeddings and linear systems. From there, Bochner identities and $L^2$ methods provide the analytic engine behind Kodaira and Nakano vanishing, which in turn feed into Riemann-Roch-type consequences and sharpened control over the geometry of complex manifolds.

The later chapters broaden the setting from line bundles to vector bundles, then shift to deformation theory of complex manifolds, submanifolds, and line bundles. Period maps and variations of Hodge structure explain how complex structures move in families, while Calabi-Yau manifolds and Ricci-flat metrics bring in the special geometry of varieties with holomorphically neutral canonical class. The final chapters connect unobstructedness, Calabi-Yau moduli, and broader compactness themes, tying together the analytic and algebro-geometric tools developed earlier into a coherent picture of how positivity governs both local and global behavior in complex geometry.

# Introduction

This opening chapter fixes the purpose and vocabulary of the course before the technical work begins. The central theme is that positivity in complex geometry has two complementary faces: differential-geometric curvature positivity and algebro-geometric abundance of holomorphic sections. The course develops the analytic tools that pass between these faces, then applies them to vanishing theorems, projective embeddings, deformations, period maps, and moduli questions for Calabi-Yau manifolds.

The preceding material in complex manifolds, holomorphic vector bundles, sheaf cohomology, Kähler geometry, and Hodge theory supplies the language. This course asks what extra structure becomes available when the line bundles or vector bundles under study have positive curvature, and how this positivity turns cohomological obstructions into effective geometric information.

## The Guiding Problem of Positivity

What does it mean for a holomorphic line bundle to be positive, and why should a curvature inequality have consequences for projective geometry? A holomorphic line bundle is locally generated by a non-vanishing holomorphic frame, while a Hermitian metric records the length of that frame. Differentiating the logarithm of the metric produces the Chern curvature form, so positivity becomes a local analytic condition on a real $(1,1)$-form.

The hypothesis is not cosmetic. On a compact connected complex manifold, the structure sheaf line bundle $\mathcal{O}_X$ with the flat metric has zero curvature, and its holomorphic sections are often too few to distinguish points. Positivity is the extra convexity condition that turns local analytic weights into global section-producing estimates.

[definition: Positive Hermitian Line Bundle] Let $X$ be a complex manifold and let $L \to X$ be a holomorphic line bundle with Hermitian metric $h$. The pair $(L,h)$ is positive if the Chern curvature form \begin{align*} \frac{i}{2\pi}F_h \end{align*} is a positive real $(1,1)$-form on $X$. [/definition]

This definition is the analytic starting point of the course. In local holomorphic coordinates $(z_1,\dots,z_n)$ and a local holomorphic frame $e$ with $|e|_h^2=e^{-\varphi}$, the local curvature representative is $i\partial\bar{\partial}\varphi$, up to the harmless $2\pi$ normalization and the sign convention used for $F_h$. Thus positivity is controlled by the complex Hessian of the local weight $\varphi$.

[example: Hyperplane Bundle as the Model Positive Line Bundle] On the affine chart $U_0=\{Z_0\neq 0\}\subset \mathbb{P}^N$, write $w_j=Z_j/Z_0$ for $1\leq j\leq N$. In the standard local frame of $\mathcal{O}_{\mathbb{P}^N}(1)$ over $U_0$, the Fubini-Study metric is \begin{align*} |e_0|_{h_{\mathrm{FS}}}^2=\frac{1}{1+\sum_{j=1}^N |w_j|^2}. \end{align*} Thus the local weight is \begin{align*} \varphi=\log\left(1+\sum_{j=1}^N |w_j|^2\right), \end{align*} because $|e_0|_{h_{\mathrm{FS}}}^2=e^{-\varphi}$. Set $r=1+\sum_{j=1}^N |w_j|^2$. Then \begin{align*} \frac{\partial \varphi}{\partial w_a}=\frac{\overline{w_a}}{r}. \end{align*} Differentiating this coefficient with respect to $\overline{w_b}$ gives \begin{align*} \frac{\partial^2\varphi}{\partial w_a\partial \overline{w_b}}=\frac{\delta_{ab}r-\overline{w_a}w_b}{r^2}. \end{align*} Therefore the curvature representative is \begin{align*} i\partial\overline{\partial}\varphi=i\sum_{a,b=1}^N \frac{\delta_{ab}r-\overline{w_a}w_b}{r^2}\,dw_a\wedge d\overline{w_b}. \end{align*} For a tangent vector $v=(v_1,\dots,v_N)$, the associated Hermitian form is \begin{align*} \sum_{a,b=1}^N \frac{\delta_{ab}r-\overline{w_a}w_b}{r^2}v_a\overline{v_b}=\frac{r\sum_{a=1}^N |v_a|^2-\left|\sum_{a=1}^N \overline{w_a}v_a\right|^2}{r^2}. \end{align*} Since $r=1+\sum |w_a|^2$, the [Cauchy-Schwarz inequality](/theorems/432) gives \begin{align*} \left|\sum_{a=1}^N \overline{w_a}v_a\right|^2\leq \left(\sum_{a=1}^N |w_a|^2\right)\left(\sum_{a=1}^N |v_a|^2\right)<r\sum_{a=1}^N |v_a|^2 \end{align*} whenever $v\neq 0$. Hence $i\partial\overline{\partial}\varphi$ is positive on every affine chart, so $\mathcal{O}_{\mathbb{P}^N}(1)$ with the Fubini-Study metric is the model positive line bundle. It is the model for embeddings because a Kodaira map defined by sections of $L^m$ pulls back $\mathcal{O}_{\mathbb{P}^N}(1)$ to $L^m$, transferring the positivity of the hyperplane bundle back to the line bundle that produced the projective coordinates. [/example]

The example explains why projective space is not merely a source of examples but the target of the main embedding theorems. A positive line bundle supplies sections, those sections define a map to projective space, and the curvature comparison identifies the original positivity with the pullback of the hyperplane geometry.

## Analytic Methods Behind Vanishing

Why should positivity force sheaf cohomology groups to vanish? The analytic answer is that curvature terms appear with a favourable sign in Bochner-Kodaira identities. Solving a $\bar{\partial}$-equation with estimates then converts a closed form into an exact one, which is the analytic counterpart of killing a cohomology class.

[definition: Dolbeault Cohomology with Coefficients] Let $X$ be a complex manifold and let $E \to X$ be a holomorphic vector bundle. The Dolbeault cohomology group $H^{p,q}_{\bar{\partial}}(X,E)$ is the quotient of smooth $E$-valued $(p,q)$-forms $\alpha$ satisfying $\bar{\partial}\alpha=0$ by forms of the form $\bar{\partial}\beta$, where $\beta$ is a smooth $E$-valued $(p,q-1)$-form. [/definition]

These groups are the bridge between analysis and sheaf theory. On compact complex manifolds, Dolbeault cohomology computes sheaf cohomology of holomorphic forms with coefficients, so analytic estimates become statements about coherent sheaves. The first major question is whether positivity can make the higher groups disappear, because that is the mechanism behind many later dimension counts and extension arguments.

Without positivity this disappearance fails even in basic cases. If $X$ is an elliptic curve and $L=\mathcal{O}_X$, then $K_X\cong \mathcal{O}_X$ and

\begin{align*} H^1(X,K_X\otimes L)=H^1(X,\mathcal{O}_X)\cong \mathbb{C}, \end{align*}

so the expected higher cohomology group survives. The theorem below should therefore be read as saying that curvature positivity supplies a genuine analytic reason for vanishing.

[quotetheorem:3501]