Complex differential form




In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.


Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and CR structures.


Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex k-form can be decomposed uniquely into a sum of so-called (p,q)-forms: roughly, wedges of p differentials of the holomorphic coordinates with q differentials of their complex conjugates. The ensemble of (p,q)-forms becomes the primitive object of study, and determines a finer geometrical structure on the manifold than the k-forms. Even finer structures exist, for example, in cases where Hodge theory applies.




Contents






  • 1 Differential forms on a complex manifold


    • 1.1 One-forms


    • 1.2 Higher-degree forms


    • 1.3 The Dolbeault operators


    • 1.4 Holomorphic forms




  • 2 See also


  • 3 References





Differential forms on a complex manifold


Suppose that M is a complex manifold. Then there is a local coordinate system consisting of n complex-valued functions z1,...,zn such that the coordinate transitions from one patch to another are holomorphic functions of these variables. The space of complex forms carries a rich structure, depending fundamentally on the fact that these transition functions are holomorphic, rather than just smooth.



One-forms


We begin with the case of one-forms. First decompose the complex coordinates into their real and imaginary parts: zj=xj+iyj for each j. Letting


dzj=dxj+idyj,dz¯j=dxj−idyj,{displaystyle dz^{j}=dx^{j}+idy^{j},quad d{bar {z}}^{j}=dx^{j}-idy^{j},}dz^{j}=dx^{j}+idy^{j},quad d{bar {z}}^{j}=dx^{j}-idy^{j},

one sees that any differential form with complex coefficients can be written uniquely as a sum


∑j=1n(fjdzj+gjdz¯j).{displaystyle sum _{j=1}^{n}left(f_{j}dz^{j}+g_{j}d{bar {z}}^{j}right).}{displaystyle sum _{j=1}^{n}left(f_{j}dz^{j}+g_{j}d{bar {z}}^{j}right).}

Let Ω1,0 be the space of complex differential forms containing only dz{displaystyle dz}dz's and Ω0,1 be the space of forms containing only dz¯{displaystyle d{bar {z}}}d{bar {z}}'s. One can show, by the Cauchy–Riemann equations, that the spaces Ω1,0 and Ω0,1 are stable under holomorphic coordinate changes. In other words, if one makes a different choice wi of holomorphic coordinate system, then elements of Ω1,0 transform tensorially, as do elements of Ω0,1. Thus the spaces Ω0,1 and Ω1,0 determine complex vector bundles on the complex manifold.



Higher-degree forms


The wedge product of complex differential forms is defined in the same way as with real forms. Let p and q be a pair of non-negative integers ≤ n. The space Ωp,q of (p,q)-forms is defined by taking linear combinations of the wedge products of p elements from Ω1,0 and q elements from Ω0,1. Symbolically,


Ωp,q=Ω1,0∧⋯∧Ω1,0∧Ω0,1∧⋯∧Ω0,1{displaystyle Omega ^{p,q}=Omega ^{1,0}wedge dotsb wedge Omega ^{1,0}wedge Omega ^{0,1}wedge dotsb wedge Omega ^{0,1}}Omega ^{{p,q}}=Omega ^{{1,0}}wedge dotsb wedge Omega ^{{1,0}}wedge Omega ^{{0,1}}wedge dotsb wedge Omega ^{{0,1}}

where there are p factors of Ω1,0 and q factors of Ω0,1. Just as with the two spaces of 1-forms, these are stable under holomorphic changes of coordinates, and so determine vector bundles.


If Ek is the space of all complex differential forms of total degree k, then each element of Ek can be expressed in a unique way as a linear combination of elements from among the spaces Ωp,q with p+q=k. More succinctly, there is a direct sum decomposition


Ek=Ωk,0⊕Ωk−1,1⊕⋯⊕Ω1,k−1⊕Ω0,k=⨁p+q=kΩp,q.{displaystyle E^{k}=Omega ^{k,0}oplus Omega ^{k-1,1}oplus dotsb oplus Omega ^{1,k-1}oplus Omega ^{0,k}=bigoplus _{p+q=k}Omega ^{p,q}.}E^{k}=Omega ^{{k,0}}oplus Omega ^{{k-1,1}}oplus dotsb oplus Omega ^{{1,k-1}}oplus Omega ^{{0,k}}=bigoplus _{{p+q=k}}Omega ^{{p,q}}.

Because this direct sum decomposition is stable under holomorphic coordinate changes, it also determines a vector bundle decomposition.


In particular, for each k and each p and q with p+q=k, there is a canonical projection of vector bundles


πp,q:Ek→Ωp,q.{displaystyle pi ^{p,q}:E^{k}rightarrow Omega ^{p,q}.}pi ^{{p,q}}:E^{k}rightarrow Omega ^{{p,q}}.


The Dolbeault operators


The usual exterior derivative defines a mapping of sections d:Ωr→Ωr+1{displaystyle d:Omega ^{r}to Omega ^{r+1}}{displaystyle d:Omega ^{r}to Omega ^{r+1}} via


d(Ωp,q)⊆⨁r+s=p+q+1Ωr,s{displaystyle d(Omega ^{p,q})subseteq bigoplus _{r+s=p+q+1}Omega ^{r,s}}{displaystyle d(Omega ^{p,q})subseteq bigoplus _{r+s=p+q+1}Omega ^{r,s}}

The exterior derivative does not in itself reflect the more rigid complex structure of the manifold.


Using d and the projections defined in the previous subsection, it is possible to define the Dolbeault operators:


∂=πp+1,q∘d:Ωp,q→Ωp+1,q,∂¯=πp,q+1∘d:Ωp,q→Ωp,q+1{displaystyle partial =pi ^{p+1,q}circ d:Omega ^{p,q}rightarrow Omega ^{p+1,q},quad {bar {partial }}=pi ^{p,q+1}circ d:Omega ^{p,q}rightarrow Omega ^{p,q+1}}partial =pi ^{{p+1,q}}circ d:Omega ^{{p,q}}rightarrow Omega ^{{p+1,q}},quad {bar {partial }}=pi ^{{p,q+1}}circ d:Omega ^{{p,q}}rightarrow Omega ^{{p,q+1}}

To describe these operators in local coordinates, let


α=∑|I|=p,|J|=q fIJdzI∧dz¯J∈Ωp,q{displaystyle alpha =sum _{|I|=p,|J|=q} f_{IJ},dz^{I}wedge d{bar {z}}^{J}in Omega ^{p,q}}alpha =sum _{{|I|=p,|J|=q}} f_{{IJ}},dz^{I}wedge d{bar {z}}^{J}in Omega ^{{p,q}}

where I and J are multi-indices. Then



∂α=∑|I|,|J|∑ℓ∂fIJ∂zℓdzℓ∧dzI∧dz¯J{displaystyle partial alpha =sum _{|I|,|J|}sum _{ell }{frac {partial f_{IJ}}{partial z^{ell }}},dz^{ell }wedge dz^{I}wedge d{bar {z}}^{J}}partial alpha =sum _{{|I|,|J|}}sum _{ell }{frac {partial f_{{IJ}}}{partial z^{ell }}},dz^{ell }wedge dz^{I}wedge d{bar {z}}^{J}

∂¯α=∑|I|,|J|∑ℓ∂fIJ∂z¯ℓdz¯ℓ∧dzI∧dz¯J.{displaystyle {bar {partial }}alpha =sum _{|I|,|J|}sum _{ell }{frac {partial f_{IJ}}{partial {bar {z}}^{ell }}}d{bar {z}}^{ell }wedge dz^{I}wedge d{bar {z}}^{J}.}{bar {partial }}alpha =sum _{{|I|,|J|}}sum _{ell }{frac {partial f_{{IJ}}}{partial {bar {z}}^{ell }}}d{bar {z}}^{ell }wedge dz^{I}wedge d{bar {z}}^{J}.


The following properties are seen to hold:



d=∂+∂¯{displaystyle d=partial +{bar {partial }}}d=partial +{bar {partial }}

∂2=∂¯2=∂∂¯+∂¯∂=0.{displaystyle partial ^{2}={bar {partial }}^{2}=partial {bar {partial }}+{bar {partial }}partial =0.}partial ^{2}={bar {partial }}^{2}=partial {bar {partial }}+{bar {partial }}partial =0.


These operators and their properties form the basis for Dolbeault cohomology and many aspects of Hodge theory.



Holomorphic forms


For each p, a holomorphic p-form is a holomorphic section of the bundle Ωp,0. In local coordinates, then, a holomorphic p-form can be written in the form


α=∑|I|=pfIdzI{displaystyle alpha =sum _{|I|=p}f_{I},dz^{I}}alpha =sum _{{|I|=p}}f_{I},dz^{I}

where the fI{displaystyle f_{I}} f_I are holomorphic functions. Equivalently, the (p,0)-form α is holomorphic if and only if


∂¯α=0.{displaystyle {bar {partial }}alpha =0.}{bar {partial }}alpha =0.

The sheaf of holomorphic p-forms is often written Ωp, although this can sometimes lead to confusion so many authors tend to adopt an alternative notation.



See also



  • Dolbeault complex

  • Frölicher spectral sequence

  • Differential of the first kind



References




  • P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 23-25. ISBN 0-471-05059-8..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Wells, R. O. (1973). Differential analysis on complex manifolds. Springer-Verlag. ISBN 0-387-90419-0.


  • Voisin, Claire (2008). Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press. ISBN 0521718015.




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