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We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with deformations. We show that a projective hypersurface is smooth if and only if the classical HKR decomposition holds for this group. In general, the first Hodge component describing scheme deformations has an interesting inner structure corresponding to the various ways in which first order deformations can be realized: deforming local multiplications, deforming restriction maps, or deforming both. We make our computations precise in the case of quartic hypersurfaces, and compute explicit dimensions in many examples.
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We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevichs formality theorem, for the cup product, does not hold for Deligne--Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruans cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature.
The Kuznetsov component $mathcal{K}u(X)$ of a Gushel--Mukai (GM) threefold has two numerical $(-1)$-classes with respect to the Euler form. We describe the Bridgeland moduli spaces for stability conditions on Kuznetsov components with respect to each of the $(-1)$-classes and prove refined and birational categorical Torelli theorems in terms of $mathcal{K}u(X)$. We also prove a categorical Torelli theorem for special GM threefolds. We study the smoothness and singularities on Bridgeland moduli spaces for all smooth GM threefolds and use this to prove a conjecture of Kuznetsov--Perry in dimension three under a mild assumption. Finally, we use our moduli spaces to restate a conjecture of Debarre--Iliev--Manivel regarding fibers of the period map for ordinary GM threefolds. We also prove the restatement of this conjecture infinitesimally using Hochschild (co)homology.
In this paper we construct a graded Lie algebra on the space of cochains on a $mathbbZ_2$-graded vector space that are skew-symmetric in the odd variables. The Lie bracket is obtained from the classical Gerstenhaber bracket by (partial) skew-symmetrization; the coboundary operator is a skew-symmetrized version of the Hochschild differential. We show that an order-one element $m$ satisfying the zero-square condition $[m,m]=0$ defines an algebraic structure called Lie antialgebra. The cohomology (and deformation) theory of these algebras is then defined. We present two examples of non-trivial cohomology classes which are similar to the celebrated Gelfand-Fuchs and Godbillon-Vey classes.
In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to injections between map-graded categories, we obtain Hochschild complexes with support. We revisit Kellers arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.
Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevichs conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology complex of an n-algebra is an (n+1)-algebra. This generalizes a conjecture by Deligne for n=1, now proven by several authors.
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