Abstract. We investigate the relation between the Hodge theory of a smooth subcanonical n-dimensional projective variety X and the deformation theory of the affine cone AX over X. We start by identifying H n−1,1 prim (X) as a distinguished graded component of the module of first order deformations of AX, and later on we show how to identify the whole primitive cohomology of X as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over X. In the particular case of a projective smooth hypersurface X we recover Griffiths' isomorphism between the primitive cohomology of X and certain distinguished graded components of the Milnor algebra of a polynomial defining X. The main result of the article can be effectively exploited to compute Hodge numbers of smooth subcanonical projective varieties. We provide a few example computation, as well a SINGULAR code, for Fano and Calabi-Yau threefolds.

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In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some commutative ring R and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we construct derived versions of Grassmannians and flag varieties.Recent developments in Derived Algebraic Geometry have lead many mathematicians to revise their approach to Moduli Theory: in particular one of the most striking results in this area is certainly Lurie Representability Theorem -proved by Lurie in 2004 as the main result of his PhD thesis [21] -which provides us with an explicit criterion to check whether a simplicial presheaf over some ∞-category of derived algebras gives rise to a derived geometric stack. Unfortunately the conditions in Lurie's result are often quite complicated to verify in concrete This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/1]. arXiv:1407.5900v3 [math.AG] 15 Sep 2015 1 Homotopy Theory of Filtered StructuresThis chapter is devoted to the construction of a good homotopy theory for filtered cochain complexes; for this reason we will first recall the standard projective model structure on cochain complexes and then use it to define a suitable one for filtered objects. At last, we will also study the Rees functor from a homotopy-theoretic viewpoint and see that it liaises coherently dg structures with filtered cochain ones.2 1 In the language of [13] compact objects are called ℵ0-small. 2 The definition of cofibrantly generated model category as found in [13] Section 2.1 is slightly more general than the one provided by Definition 1.1, as it involves small objects rather than compact ones; anyway the proper definition requires some non-trivial Set Theory and moreover all examples we consider in this paper fit into the weaker notion determined by Definition 1.1, so we will stick to this.1. the domains of the maps in I are compact relative to I-cell;2. the domains of the maps in J are compact relative to J-cell;such that H n (K) = 0 ∀n ∈ Z. Take any w ∈ M n such that p n (w) = x; an immediate computation shows that dw−y ∈ Z n+1 (K) and, as K is acyclic, ∃v ∈ K n such that dv = dw−y. Now define z := w − v and the result follows. The next step is describing cofibrations and trivial cofibrations generated by the sets (1.1), but we need to understand cofibrant objects in order to do that; in the following for any Rmodule P call D R (n, P ) the cochain complex defined by D R (n, P ) :=    P if k = n, n + 1 0 otherwise and in which the only non-trivial connecting map is the identity.Proposition 1.12. If A ∈ dgMod R is cofibrant, then A n is projec...

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