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

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 $\le (p_g-1)$, (iv) the multiplication map $m_2:Sy{m}^2(H^0({\mathcal{O}}_X(K_X)))⟶H^0({\mathcal{O}}_X(2K_X))$ is surjective. It is shown that the Infinitesimal Torelli holds for such $X$. Our proof is based on the study of the cup-product $$H^1({\Theta }_X)⟶({\mathcal{O}}_X(K_X){)}^{\ast }\otimes H^1({\Omega }_X)$$ 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 $X$.

55 pages, the paper supercedes earlier incorrect version(s) with a similar title (arXiv:1402.0192)