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Register a new userThe Logarithmic Bogomolov-Tian-Todorov Theorem
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We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed.
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Let $X$ be a complex analytic manifold, $Dsubset X$ a free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), $j: U=X-D to X$ the corresponding open inclusion, $E$ an integrable logarithmic connection with respect to $D$ and $L$ the local system of the horizontal sections of $E$ on $U$. In this paper we prove that the canonical morphisms between the logarithmic de Rham complex of $E(kD)$ and $R j_* L$ (resp. the logarithmic de Rham complex of $E(-kD)$ and $j_!L$) are isomorphisms in the derived category of sheaves of complex vector spaces for $kgg 0$ (locally on $X$)
We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langers method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint appendix with Hao Max Sun, we generalize the Bogomolov inequality formula to Simpson Higgs sheaves on tame Deligne-Mumford stacks.
We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langers estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.
In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log etale maps and use this to define derived log stacks.
We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following the same steps as in the general case, treated next.
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