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This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. For instance, you You can have a look at On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958

http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607

but there are several many other papers of by the same authors.

For a nice and compact exposition, you can look at these class notes of Christian Schnell: http://homepages.math.uic.edu/~cschnell/pdf/notes/kodaira.pdf

Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Goldman & Millson, Gerstenhaber, Stasheff, Deligne, Quillen, Kontsevich.

Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions, for $I$ and $I_t$:

$$T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t$$

and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually

$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)\right|_{T^{1,0}_t}^{-1}.$$

In local coordinates, $$\phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k},$$ and $T^{0,1}_t$ is generated (over the smooth functions) by

$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k} z_k}.$$

Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$, $X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures on $X\times \Delta$, which you can compare.

ADDENDUM I also second YangMills' suggestion to have a look at Chapter 2 of Gross-Huybrechts-Joyce. Also, you You can look at also try Chapter 1 of K. Fukaya's book "Deformation Theory, Homological algebra, and Mirror Symmetry", or at as well as the Appendix of Goldman-Millson paper to Homotopy invariance of the Kuranishi Space by Goldman and Millson (Illinois J. of Math, vol.34, No.2, 1990). In particular, you'll see how one uses formal Kuranishi theory to avoid dealing with the convergence of the power series for $\phi(t)$. For deformations of compact coplex complex manifolds, the convergence was proved by Kodaira-Nirenberg-Spencer. Fukaya says a little bit about the convergence of this series in general, i.e., for other deformation problems.

4 added 717 characters in body

This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. For instance, you can have a look at On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958

http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607

but there are several other papers of the same authors.

For a nice and compact exposition, you can look at these class notes of Christian Schnell: http://homepages.math.uic.edu/~cschnell/pdf/notes/kodaira.pdf

Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Goldman & Millson, Gerstenhaber, Stasheff, Deligne, Quillen, Kontsevich.

Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions, for $I$ and $I_t$:

$$T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t$$

and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually

$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)\right|_{T^{1,0}_t}^{-1}.$$

In local coordinates, $$\phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k},$$ and $T^{0,1}_t$ is generated (over the smooth functions) by

$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k}$$

Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$, $X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures on $X\times \Delta$, which you can compare.

ADDENDUM I also second YangMills' suggestion to have a look at Chapter 2 of Gross-Huybrechts-Joyce. Also, you can look at Chapter 1 of K. Fukaya's book "Deformation Theory, Homological algebra, and Mirror Symmetry", or at the Appendix of Goldman-Millson paper Homotopy invariance of the Kuranishi Space (Illinois J. of Math, vol.34, No.2, 1990). In particular, you'll see how one uses formal Kuranishi theory to avoid dealing with the convergence of the power series for $\phi(t)$. For deformations of compact coplex manifolds, the convergence was proved by Kodaira-Nirenberg-Spencer. Fukaya says a little bit about the convergence of this series in general, i.e., for other deformation problems.

3 added 20 characters in body

This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. For instance, you can have a look at On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958

http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607

but there are several other papers of the same authors.

For a nice and compact exposition, you can look at these class notes of Christian Schnell: http://homepages.math.uic.edu/~cschnell/pdf/notes/kodaira.pdf

Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Gerstenhaber,StasheffGoldman & Millson, Gerstenhaber, Stasheff, Deligne, Quillen, Kontsevich.

Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions, for $I$ and $I_t$:

$$T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t$$

and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually

$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)\right|_{T^{1,0}_t}^{-1}.$$

In local coordinates, $$\phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k},$$ and $T^{0,1}_t$ is generated (over the smooth functions) by

$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k}$$

Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$, $X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures on $X\times \Delta$, which you can compare.

2 edited body

This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. For instance, you can have a look at On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958

http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607

but there are several other papers of the same authors.

For a nice and compact exposition, you can look at these class notes of Christian Schnell: http://homepages.math.uic.edu/~cschnell/pdf/notes/kodaira.pdf

Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Gerstenhaber,Stasheff, Deligne, Quillen, Kontsevich.

Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions, for $I$ and $I_t$:

$$T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t$$

and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually

$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)^{-1}\right|_{T^{1,0}_t}.$$left.\left(\textrm{pr}^{0,1}\right)\right|_{T^{1,0}_t}^{-1}.

In local coordinates, $$\phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k},$$ and $T^{0,1}_t$ is generated (over the smooth functions) by

$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k}$$

Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$, $X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures on $X\times \Delta$, which you can compare.

1

This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. For instance, you can have a look at On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958

http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607

but there are several other papers of the same authors.

For a nice and compact exposition, you can look at these class notes of Christian Schnell: http://homepages.math.uic.edu/~cschnell/pdf/notes/kodaira.pdf

Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Gerstenhaber,Stasheff, Deligne, Quillen, Kontsevich.

Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions, for $I$ and $I_t$:

$$T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t$$

and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually

$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)^{-1}\right|_{T^{1,0}_t}.$$

In local coordinates, $$\phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k},$$ and $T^{0,1}_t$ is generated (over the smooth functions) by

$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k}$$

Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$, $X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures on $X\times \Delta$, which you can compare.
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