In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
Definition
Historical motivation The Kodaira–Spencer map was originally constructed in the setting of complex manifolds. Given a complex analytic manifold M {\displaystyle M} with charts U i {\displaystyle U_{i}} and biholomorphic maps f j k {\displaystyle f_{jk}} sending z k → z j = ( z j 1 , … , z j n ) {\displaystyle z_{k}\to z_{j}=(z_{j}^{1},\ldots ,z_{j}^{n})} gluing the charts together, the idea of deformation theory is to replace these transition maps f j k ( z k ) {\displaystyle f_{jk}(z_{k})} by parametrized transition maps f j k ( z k , t 1 , … , t k ) {\displaystyle f_{jk}(z_{k},t_{1},\ldots ,t_{k})} over some base B {\displaystyle B} (which could be a real manifold) with coordinates t 1 , … , t k {\displaystyle t_{1},\ldots ,t_{k}} , such that f j k ( z k , 0 , … , 0 ) = f j k ( z k ) {\displaystyle f_{jk}(z_{k},0,\ldots ,0)=f_{jk}(z_{k})} . This means the parameters t i {\displaystyle t_{i}} deform the complex structure of the original complex manifold M {\displaystyle M} . Then, these functions must also satisfy a cocycle condition, which gives a 1-cocycle on M {\displaystyle M} with values in its tangent bundle. Since the base can be assumed to be a polydisk, this process gives a map between the tangent space of the base to H 1 ( M , T M ) {\displaystyle H^{1}(M,T_{M})} called the Kodaira–Spencer map.
Original definition More formally, the Kodaira–Spencer map is
K S : T 0 B → H 1 ( M , T M ) {\displaystyle KS:T_{0}B\to H^{1}(M,T_{M})}
where
M → B {\displaystyle {\mathcal {M}}\to B} is a smooth proper map between complex spaces (i.e., a deformation of the special fiber M = M 0 {\displaystyle M={\mathcal {M}}_{0}} .)
K S {\displaystyle KS} is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection T M | M → T 0 B ⊗ O M {\displaystyle T{\mathcal {M}}|_{M}\to T_{0}B\otimes {\mathcal {O}}_{M}} whose kernel is the tangent bundle T M {\displaystyle T_{M}} . If v {\displaystyle v} is in T 0 B {\displaystyle T_{0}B} , then its image K S ( v ) {\displaystyle KS(v)} is called the Kodaira–Spencer class of v {\displaystyle v} .
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