On the Infinitesimal Torelli theorem for regular surfaces with very ample canonical divisor

TL;DR

This paper proves the infinitesimal Torelli theorem for certain regular surfaces with very ample canonical divisors, using cohomological methods and complex sheaf theory to connect geometry and deformation theory.

Contribution

It establishes the infinitesimal Torelli theorem for a class of surfaces under specific geometric and cohomological conditions, employing a novel approach via complexes of coherent sheaves.

Findings

Infinitesimal Torelli holds for the specified surfaces.

Cohomological cup-product analysis links geometry to deformation theory.

Method involves lifting cup-product data to complexes of coherent sheaves.

Abstract

Let $X$ be a smooth compact complex surface subject to the following conditions: (i) the canonical line bundle $\mathcal{O}_X(K_X) $ is very ample, (ii) the irregularity $q(X): = h^1(\mathcal{O}_X) =0$, (iii) $X$ contains no rational normal curves of degree $\leq (p_g-1)$, (iv) the multiplication map $m_2: Sym^2(H^0(\mathcal{O}_X(K_X))) \longrightarrow H^0 (\mathcal{O}_X (2K_X))$ is surjective. It is shown that the Infinitesimal Torelli holds for such $X$. Our proof is based on the study of the cup-product $$ H^1 (\Theta_X) \longrightarrow (\mathcal{O}_X(K_X))^{\ast} \otimes H^1 (\Omega_X) $$ where $\Theta_X$ (resp. $\Omega_X$) is the holomorphic tangent (resp. cotangent) bundle of $X$. Conceptually, the approach consists of lifting the data of the cohomological cup-product above to the category of complexes of coherent sheaves of $X$. This establishes connections between the geometry of the canonical map and the above cup-product by exhibiting geometrically meaningful objects in the category of (short) exact complexes of coherent sheaves on $X$.