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Register a new userDeformation-obstruction theory for complexes via Atiyah and Kodaira--Spencer classes
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We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira--Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.
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Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (new results obtained with Ingrid Bauer) projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance the BCDH surfaces, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.
We construct a $k[[Q]]$-linear predifferential graded Lie algebra $L^*_{X/S}$ associated to a log smooth and saturated morphism $f: X rightarrow S$ and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction by Chan-Leung-Ma whereof $L^*_{X/S}$ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields and interestingly does not need to keep track of the log structure. The method of using Gerstenhaber algebras is closely related to recent developments in mirror symmetry.
In this note, we apply the semi-ampleness criterion in Lemma 3.1 to prove many classical results in the study of abundance conjecture. As a corollary, we prove abundance for large Kodaira dimension depending only on [BCHM10].
Let $f:,X to mathbb{P}^1$ be a non-isotrivial semi-stable family of varieties of dimension $m$ over $mathbb{P}^1$ with $s$ singular fibers. Assume that the smooth fibers $F$ are minimal, i.e., their canonical line bundles are semiample. Then $kappa(X)leq kappa(F)+1$. If $kappa(X)=kappa(F)+1$, then $s>frac{4}m+2$. If $kappa(X)geq 0$, then $sgeqfrac{4}m+2$. In particular, if $m=1$, $s=6$ and $kappa(X)=0$, then the family $f$ is Teichmuller.
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fultons intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by Deglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.
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