Chern character, semi-regularity map and obstructions

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology, which allows us to reconstruct the semi-regularity map and the infinitesimal Abel-Jacobi map. Combining that construction of the semi-regularity map with obstruction theory of functors of Artin rings, we verify the semiregularity conjecture: the semi-regularity map annihilates every obstructions to embedded deformations of a closed subvariety which is locally complete intersection.

[1]  C. Weibel,et al.  Infinitesimal cohomology and the Chern character to negative cyclic homology , 2007, math/0703133.

[2]  C. Weibel.  The Hodge ?ltration and cyclic homology , 1997 .

[3]  M. Schlessinger.  Functors of Artin rings , 1968 .

[4]  B. Fantechi,et al.  Obstruction Calculus for Functors of Artin Rings, I , 1998 .

[5]  D. Spencer,et al.  A theorem of completeness of characteristic systems of complete continuous systems , 1959 .

[6]  Edoardo Sernesi,et al.  Deformations of algebraic schemes , 2006 .

[7]  M. Manetti.  DIFFERENTIAL GRADED LIE ALGEBRAS AND FORMAL DEFORMATION THEORY , 2006 .

[8]  R. Thomason,et al.  Higher Algebraic K-Theory of Schemes and of Derived Categories , 1990 .

[9]  James D. Lewis.  Lectures on algebraic cycles , 2001 .

[10]  F. Severi.  Sul teorema fondamentale dei sistemi continui di curve sopra una superficie algebrica , 1944 .

[11]  Cyclic homology, cdh-cohomology and negative K-theory , 2005, math/0502255.

[12]  C. Soulé,et al.  Intersection theory using Adams operations , 1987 .

[13]  P. Griffiths,et al.  On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. , 2005 .

[14]  S. Bloch.  Semi-regularity and de Rham cohomology , 1972 .

[15]  J. Stienstra.  On the formal completion of the chow group CH2(X) for a smooth projective surface in characteristic 0 , 1983 .

[16]  Z. Ran.  Hodge theory and the Hilbert scheme , 1993 .

[17]  Donatella Iacono,et al.  Semiregularity and obstructions of complete intersections , 2011, 1112.0425.

[18]  Sen Yang.  K-theory, local cohomology and tangent spaces to Hilbert schemes , 2016, Annals of K-Theory.

[19]  M. Lejeune-jalabert,et al.  Calcul différentiel et classes caractéristiques en géométrie algébrique , 1989 .

[20]  Lie description of higher obstructions to deforming submanifolds , 2005, math/0507287.

[21]  J. Pridham.  Semiregularity as a consequence of Goodwillie's theorem , 2012, 1208.3111.

[22]  R. Buchweitz,et al.  A Semiregularity Map for Modules and Applications to Deformations , 1999, Compositio Mathematica.

[23]  Daniel Lowengrub,et al.  Deformation in Theory , 2014 .

[24]  H. Clemens.  Geometry of formal Kuranishi theory , 1999 .

[25]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[26]  Z. Ran.  Semiregularity, obstructions and deformations of Hodge classes , 1999 .

[27]  George A. Elliott,et al.  K-theory , 1999 .