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Lemma 3.4. Let \( r \) be a \( {C}^{k} \) defining function for \( D, k \geq 1 \), and let \( \iota : {bD} \rightarrow {\mathbb{R}}^{n} \) be the inclusion map. Then\n\n\[ d{S}_{P} = \frac{{l}^{ * }\left( {*d{r}_{P}}\right) }{\begin{Vmatrix}d{r}_{P}\end{Vmatrix}}\;\text{ for }P \in {bD}. \] | Proof. Fix \( P \in {bD} \) and choose \( {\omega }_{2},\ldots ,{\omega }_{n} \in {T}_{P}^{ * }{\mathbb{R}}^{n} \), such that \( d{r}_{P}/\begin{Vmatrix}{d{r}_{P}}\end{Vmatrix},{\omega }_{2},\ldots \) , \( {\omega }_{n} \) is a positively oriented orthonormal basis. By (3.17), \( * \left( {d{r}_{P}/\begin{Vmatrix}{d{r}... | Yes |
Corollary 3.5. If \( D \) is open in \( {\mathbb{C}}^{n} \) with \( {C}^{k} \) defining function \( r \), then\n\n\[ \n{dS} = \frac{2}{\parallel {dr}\parallel }{\iota }^{ * }\left( {*\partial r}\right) \;\text{ on }{bD}.\n\] | Proof. We use (3.21) and compute \( * {dr} = * \left( {\partial r + \bar{\partial }r}\right) = * \partial r + \overline{*\partial r} \) . By (3.20),\n\n\[ \n* \partial r = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{1}{{2}^{n - 1}{i}^{n}}\frac{\partial r}{\partial {z}_{j}}d{z}_{j} \land \left( {\mathop{\bigwedge }\limits_... | Yes |
Lemma 3.6. If\n\n\\[ \n\\psi = \\mathop{\\sum }\\limits_{{I, J}}{\\psi }_{IJ}d{z}^{I} \\land d{\\bar{z}}^{J} \\in {C}_{p, q + 1}^{1}\\left( {\\mathbb{C}}^{n}\\right) \n\\]\n\nthen\n\n(3.26)\n\n\\[ \n{\\vartheta \\psi } = - 2{\\left( -1\\right) }^{p}\\mathop{\\sum }\\limits_{{I, K}}\\left( {\\mathop{\\sum }\\limits_{{k,... | Proof. Rather than computing \\( \\vartheta \\) from \\( - * \\partial * \\), which is messy, we carry out the integration by parts in more explicit form. Let\n\n\\[ \n\\varphi = \\mathop{\\sum }\\limits_{{I, K}}{\\varphi }_{IK}d{z}^{I} \\land d{\\bar{z}}^{K} \\in {C}_{p, q}^{1}\\left( {\\mathbb{C}}^{n}\\right) \n\\]\n... | Yes |
Lemma 3.7. Suppose \( D \subset \subset {\mathbb{C}}^{n} \) has piecewise \( {C}^{1} \) boundary and \( \varphi \in {C}_{p, q}^{1}\left( \bar{D}\right) \) , \( \psi \in {C}_{p, q + 1}^{1}\left( \bar{D}\right) \) . Then\n\n(3.29)\n\n\[ \n{\left( \bar{\partial }\varphi ,\psi \right) }_{D} = {\left( \varphi ,\vartheta \ps... | Proof. One proceeds as in (3.24), with all integrals taken over \( D \) instead of \( {\mathbb{C}}^{n} \) . Stokes’ Theorem now gives\n\n\[ \n{\int }_{D}d\left( {\varphi \land * \bar{\psi }}\right) = {\int }_{bD}\varphi \land * \bar{\psi }\n\]\n\nand (3.29) follows. Equation (3.30) follows from (3.29) by conjugation. | Yes |
Lemma 3.8. If \( f \in {C}^{2}\left( {\mathbb{C}}^{n}\right) \), then\n\n\[ \n▱f = - 2\mathop{\sum }\limits_{{j = 1}}^{n}\frac{{\partial }^{2}f}{\partial {z}_{j}\partial {\bar{z}}_{j}} = \frac{1}{2}{\Delta f}.\n\] | Proof. Since \( \vartheta = 0 \) on functions,\n\n\[ \n▱f = \vartheta \bar{\partial }f = \vartheta \left( {\mathop{\sum }\limits_{{j = 1}}^{n}\left( {\partial f/\partial {\bar{z}}_{j}}\right) d{\bar{z}}_{j}}\right)\n\]\n\nThe first equation now follows from Lemma 3.6, and the second equation then follows from\n\n\[ \n\... | No |
Theorem 1.1. The kernel \( \Gamma = {\Gamma }^{\left( n\right) } \) defined on \( {\mathbb{C}}^{n} \times {\mathbb{C}}^{n} \) by (1.3) satisfies\n\n(1.4)\n\n\[ f\left( z\right) = {\left( \bar{\partial }f,\bar{\partial }\Gamma \left( \cdot, z\right) \right) }_{{\mathbb{C}}^{n}} = \int \bar{\partial }f \land * \partial \... | The equivalence of (1.2) and (1.4) for \( f \in {C}_{0}^{2} \) involves just an integration by parts (see III.3.5), since\n\n(1.5)\n\[ \left( {▱f,\Gamma }\right) = \left( {\vartheta \bar{\partial }f,\Gamma }\right) = \left( {\bar{\partial }f,\bar{\partial }\Gamma }\right) . \] \n\nThe reader should check that (1.5) hol... | No |
Lemma 1.2. (i) \( ▱\Gamma \left( {\zeta, z}\right) = 0 \) for \( \zeta \neq z \) ; (ii) \( - {\int }_{{bB}\left( {z,\varepsilon }\right) } * \partial \bar{\Gamma }\left( {\cdot, z}\right) = 1 \) for all \( z \in {\mathbb{C}}^{n} \) and \( \varepsilon > 0 \) . | Proof. (i) is a straightforward computation which is left to the reader (use Lemma III.3.8). For (ii), it follows from (1.3) that\n\n\[ \partial \bar{\Gamma } = - {\sigma }_{{2n} - 1}^{-1}{\beta }^{-n}\partial \beta \]\n\nwhere \( \beta = {\left| \zeta - z\right| }^{2} \). Since \( \beta = {\varepsilon }^{2} \) on \( {... | No |
Corollary 1.3. Suppose \( D \subset \subset {\mathbb{C}}^{n} \) has piecewise \( {C}^{1} \) boundary. Then\n\n\[ f\left( z\right) = - {\int }_{bD}f \land * \partial \bar{\Gamma }\left( {\cdot, z}\right) + {\left( \bar{\partial }f,\bar{\partial }\Gamma \left( \cdot, z\right) \right) }_{D} \]\n\nfor \( f \in {C}^{1}\left... | Proof. In the proof of Theorem 1.1 one replaces the region of integration \( {\mathbb{C}}^{n} - B\left( {z,\varepsilon }\right) \) by \( D - B\left( {z,\varepsilon }\right) \), where \( \varepsilon \) is so small that \( B\left( {z,\varepsilon }\right) \subset \subset D \) . According to Lemma III.3.7, there is now an ... | Yes |
Lemma 1.4. The Bochner-Martinelli kernel satisfies \( {\bar{\partial }}_{\zeta }{K}_{0} = 0 \) on \( {\mathbb{C}}^{n} \times {\mathbb{C}}^{n} - \) \( \{ \zeta = z\} \) . | Proof. For \( \zeta \neq z,{\bar{\partial }}_{\zeta }{K}_{0} = - {\bar{\partial }}_{\zeta } * {\partial }_{\zeta }\bar{\Gamma } = * \bar{▱}\bar{\Gamma } = 0 \) . ∎ | No |
Lemma 1.5. With \( \beta = {\left| \zeta - z\right| }^{2} \), one has the following representations for \( {K}_{0} = \) \( - * \partial \bar{\Gamma } \) :\n\n(a)\n\[ \n{K}_{0} = \frac{\left( {n - 1}\right) !}{{\left( 2\pi i\right) }^{n}}{\beta }^{-n}\mathop{\sum }\limits_{{j = 1}}^{n}\left( {{\bar{\zeta }}_{j} - {\bar{... | Proof. By calculating \( \partial \bar{\Gamma } \), one obtains\n\n\[ \n{K}_{0} = \frac{\left( {n - 1}\right) !}{2{\pi }^{n}}{\beta }^{-n}\mathop{\sum }\limits_{{j = 1}}^{n}\left( {{\bar{\zeta }}_{j} - {\bar{z}}_{j}}\right) * d{\zeta }_{j}, \n\]\n\nand by III, (3.20), \n\n\[ \n* d{\zeta }_{j} = 2 \cdot {\left( 2i\right... | Yes |
Proposition 1.6. (The Bochner-Martinelli Integral Formula). Let \( D \subset \subset {\mathbb{C}}^{n} \) be a domain with piecewise \( {C}^{1} \) boundary. Then, for \( f \in C\left( \bar{D}\right) \cap \mathcal{O}\left( D\right) \), \[ {\int }_{bD}f\left( \zeta \right) {K}_{0}\left( {\zeta, z}\right) = \left\{ \begin{... | Proof. Assume first that \( f \in {C}^{1}\left( \bar{D}\right) \) and \( \bar{\partial }f = 0 \) on \( D \). If \( z \in D \), the result follows from Corollary 1.3. For \( z \notin \bar{D} \), note that \[ d\left( {f{K}_{0}}\right) = \bar{\partial }\left( {f{K}_{0}}\right) = f \land \bar{\partial }{K}_{0} = 0 \] on \(... | Yes |
Theorem 1.7. Let \( D \subset \subset {\mathbb{C}}^{n} \) be a domain with piecewise \( {C}^{1} \) boundary. Fix \( 0 \leq q \leq n \), and let \( f \in {C}_{0, q}^{1}\left( \bar{D}\right) \) . Then\n\n\[ f\left( z\right) = {Q}_{D}\left( {f,{\Gamma }_{q}\left( {\cdot, z}\right) }\right) - {\int }_{bD}f \land * \partial... | Proof. First we consider \( f \in {C}_{0, q}^{2}\left( \bar{D}\right) \) . Fix \( z \in D \) and write \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} \in {C}_{0, q}^{2} \) has compact support in \( D \), and \( {f}_{2} \equiv 0 \) on \( \overline{B\left( {z,\varepsilon }\right) } \subset \subset D \), for some \( \varep... | Yes |
Lemma 1.8. Let \( 0 \leq q \leq n \) . Then\n\n(a)\n\n\[{\vartheta }_{\zeta }{\Gamma }_{q} = {\partial }_{z}{\Gamma }_{q - 1}\]\n\n(b)\n\[{\partial }_{\zeta }{\bar{\Gamma }}_{q - 1} = {\vartheta }_{z}{\bar{\Gamma }}_{q}\]\non \( {\mathbb{C}}^{n} \times {\mathbb{C}}^{n} - \{ \zeta = z\} \) . | Proof. This is a straightforward computation. Since \( {\Gamma }_{-1} \equiv 0 \), it is enough to consider \( q \geq 1 \) . Using the definitions of \( {\Gamma }_{q},\vartheta \), and \( \partial \), one obtains\n\n\[{\vartheta }_{\zeta }{\Gamma }_{q} = - 2\frac{-\left( {n - 1}\right) !}{{2}^{q + 1}{\pi }^{n}}{\beta }... | Yes |
Proposition 1.9. Let \( D \subset \subset {\mathbb{C}}^{n} \) be a domain with piecewise \( {C}^{1} \) boundary. Let \( 0 \leq q \leq n \), and \( f \in {C}_{0, q}^{1}\left( \bar{D}\right) \) . Then\n\n(a)\n\n\[{\left( f,\bar{\partial }{\Gamma }_{q - 1}\left( \cdot, z\right) \right) }_{D}\text{is of class}{C}^{1}\text{... | Proof. By Lemma 1.8.,\n\n\[{\left( \mathcal{I}f,\mathcal{I}{\Gamma }_{q}\left( \cdot, z\right) \right) }_{D} = {\left( \mathcal{I}f,{\partial }_{z}{\Gamma }_{q - 1}\left( \cdot, z\right) \right) }_{D}\]\n\n(1.18)\n\n\[= {\bar{\partial }}_{z}{\left( \vartheta f,{\Gamma }_{q - 1}\left( \cdot, z\right) \right) }_{D}.\]\n\... | Yes |
Theorem 1.10. (The Bochner-Martinelli-Koppelman formula [Kop 2]). Let \( D \subset \subset {\mathbb{C}}^{n} \) be a domain with piecewise \( {C}^{1} \) boundary. Let \( 0 \leq q \leq n \) . Then every \( f \in {C}_{0, q}^{1}\left( \bar{D}\right) \) is represented on \( D \) by\n\n\[ f\left( z\right) = {\int }_{bD}f \la... | Proof. Theorem 1.10 shows in both cases that\n\n\[ u\left( z\right) = - {\int }_{D}f \land {K}_{q - 1}\left( {\cdot, z}\right) = {\left( f,\bar{\partial }{\Gamma }_{q - 1}\left( \cdot, z\right) \right) }_{D} \] | No |
Lemma 1.12. For \( 0 \leq q \leq n \) one has\n\n\[{\bar{\partial }}_{\zeta }{K}_{q} = {\left( -1\right) }^{q}{\bar{\partial }}_{z}{K}_{q - 1}\]\non \( {\mathbb{C}}^{n} \times {\mathbb{C}}^{n} - \{ \zeta = z\} \) . | Proof. By Lemma 1.8(a),\n\n\[{\bar{\partial }}_{z}{K}_{q - 1} = - { * }_{\zeta }{\bar{\partial }}_{z}{\partial }_{\zeta }{\bar{\Gamma }}_{q - 1} = - { * }_{\zeta }{\partial }_{\zeta }\overline{{\partial }_{z}{\Gamma }_{q - 1}}\]\n\n\[= - { * }_{\zeta }{\partial }_{\zeta }\overline{{\vartheta }_{\zeta }{\Gamma }_{q}} = ... | Yes |
Lemma 1.13. With \( \beta = {\left| \zeta - z\right| }^{2} \), the following representations are valid for the BMK kernel \( {K}_{q} \) for \( 0 \leq q \leq n \) :\n\n(a)\n\n\[ \n{K}_{q} = \frac{\left( {n - 1}\right) !}{{2}^{q + 1}{\pi }^{n}}{\beta }^{-n}\mathop{\sum }\limits_{\substack{{j, J} \\ {\left| L\right| = q +... | Proof. Using the definitions and calculating \( {\partial }_{\zeta }{\bar{\Gamma }}_{q} \) immediately gives (a). Next, note that by Lemma III.3.3, if \( \left| L\right| = q + 1 \), then\n\n\[ \n* d{\zeta }^{L} = \frac{{\left( -1\right) }^{\left( {q + 1}\right) q/2}}{{2}^{n - q - 1}{i}^{n}}d{\zeta }^{L} \land \mathop{\... | Yes |
Theorem 1.14. Let \( D \) be a bounded domain in \( {\mathbb{C}}^{n} \) . Then\n\n(a) for \( 0 < \alpha < 1,{\mathbf{K}}_{q}^{D} \) defines a bounded linear transformation\n\n\( {L}_{0, q + 1}^{\infty }\left( D\right) \rightarrow {\Lambda }_{\alpha ,\left( {0, q}\right) }\left( D\right) \), i.e., there is a constant \(... | Proof. Using Lemma 1.13(a), one sees that up to a constant factor, the coefficient \( {\left( {\mathbf{K}}_{q}^{D}f\right) }_{J} \) is given by\n\n\[ \n\sum {\varepsilon }_{jJ}^{L}{\int }_{D}{f}_{L}\left( \zeta \right) \frac{{\bar{\zeta }}_{j} - {\bar{z}}_{j}}{{\left| \zeta - z\right| }^{2n}}{dV}\left( \zeta \right) \n... | No |
Lemma 1.15. Let \( D \) be a bounded open set in \( {\mathbb{R}}^{n}, n \geq 2 \) . For \( 1 \leq j \leq n \) and \( f \in {L}^{\infty }\left( D\right) \), define\n\n\[ \n{\mathbf{T}}_{j}f\left( x\right) = {\int }_{D}f\left( y\right) \frac{{y}_{j} - {x}_{j}}{{\left| y - x\right| }^{n}}{dV}\left( y\right) \n\]\n\nThen\n... | Proof. It is enough to consider \( \mathbf{T} = {\mathbf{T}}_{1} \) . Direct estimation shows \( {\left| \mathbf{T}f\right| }_{0} \lesssim {\left| f\right| }_{0} \) . \( {}^{1} \) Choose \( R < \infty \), so that \( D \subset B\left( {0, R}\right) \) . Fix \( x,{x}^{\prime } \in B\left( {0, R}\right), x \neq {x}^{\prim... | Yes |
Corollary 2.3. Let \( f \in C\left( {bD}\right) \), and suppose that\n\n(2.9)\n\n\[{\int }_{bD}f{K}_{0}\left( {\cdot, z}\right) = 0\;\text{ for }z \notin \bar{D}.\n\]\n\nThen \( {\left. {\mathbf{K}}^{bD}f\right| }_{D} \) extends to a continuous function \( F \) on the closure of \( D \) and\n\n\[F\left( z\right) = f\le... | Proof. Simply apply Proposition 2.2, observing that (2.9) implies the existence of the continuous extension \( {F}^{ - } \), and \( {F}^{ - }\left( z\right) = 0 \) for \( z \in {bD} \) . | No |
Lemma 2.4. Suppose \( 1 \leq k \leq \infty \), and let \( D \) be a bounded domain with boundary of class \( {C}^{k} \) . Let \( f \in {C}^{k}\left( {bD}\right) \) be a CR-function. There are a neighborhood \( U \) of \( {bD} \) and an extension \( \widetilde{f} \in {C}^{k - 1}\left( U\right) \) of \( f \), such that\n... | Proof. For \( p \in {bD} \), let \( {L}_{1}\left( p\right) ,\ldots ,{L}_{n - 1}\left( p\right) \) be a basis of \( {T}_{p}^{1,0}\left( {bD}\right) \) . Let \( {L}_{n}\left( p\right) \neq 0 \) be orthogonal to \( {T}_{p}^{1,0}\left( {bD}\right) \) . The extension \( \widetilde{f} \) must be chosen, so that \( {\bar{L}}_... | Yes |
Lemma 2.6. If \( n > 1 \), then\n\n\[ \n{H}_{\bar{\partial }, c}^{\frac{1}{2}}\left( {\mathbb{C}}^{n}\right) = 0 \n\] | Lemma 2.6 can also be proved directly, without using the Hartogs Extension Theorem. For example, the solution \( u = - {\int }_{{\mathbb{C}}^{n}}f \land {K}_{0} \) of \( \bar{\partial }u = f \) given by Corollary 1.11 vanishes at \( \infty \), and is holomorphic outside supp \( f \), so, if \( n > 1 \) , \( u \equiv 0 ... | Yes |
Theorem 2.7. Let \( {nbe} \geq 1 \) ; then\n\n\[ \n{H}_{\partial, c}^{q}\left( {\mathbb{C}}^{n}\right) = 0 \n\]\n\nfor \( q = 0,1,\ldots, n - 1 \) . | Proof. The case \( q = 0 \) is obvious, and the case \( q = 1 \leq n - 1 \) was discussed above. The case \( 1 < q \leq n - 1 \) is done similarly, except the extension of holomorphic functions is replaced by extension of \( \bar{\partial } \) -closed \( \left( {0, q - 1}\right) \) -forms.\n\nLet \( f \in {\mathcal{D}}... | No |
Lemma 3.1. Let \( W\left( \zeta \right) = \mathop{\sum }\limits_{{j = 1}}^{n}{w}_{j}\left( \zeta \right) d{\zeta }_{j} \) be a \( {C}_{1,0}^{2} \) form on a set \( U \subset {\mathbb{C}}^{n} \) . Suppose there is \( z \notin U \), such that\n\n\[ \langle W\left( \zeta \right) ,\zeta - z\rangle = \mathop{\sum }\limits_{... | Proof of Lemma 3.1. It is immediate that\n\n\[ {\left( 2\pi i\right) }^{n}\;d{\Omega }_{0}\left( W\right) = {\left( 2\pi i\right) }^{n}\overline{\partial }{\Omega }_{0}\left( W\right) = {\left( \overline{\partial }W\right) }^{n} \]\n\n\[ = {\left( \mathop{\sum }\limits_{{j = 1}}^{n}\overline{\partial }{w}_{j} \land d{\... | Yes |
Proposition 3.2. Let \( D \subset \subset {\mathbb{C}}^{n} \) be a domain with piecewise \( {C}^{1} \) boundary. Let \( z \in D \) and suppose that \( W \in {C}_{1,0}^{1}\left( {\bar{D}-\{ z\} }\right) \) is a generating form for \( z \) such that \( W = \partial \beta /\beta \) on a ball \( B\left( {z,\varepsilon }\ri... | The proof is left as an exercise for the reader (see E.3.3). | No |
Let \( B = B\left( {0,1}\right) \subset {\mathbb{C}}^{n} \) . For \( 1 \leq j \leq n \) and \( \left| \zeta \right| > \left| z\right| \) define\n\n\[ \n{s}_{j}\left( {\zeta, z}\right) = \frac{{\bar{\zeta }}_{j}}{\mathop{\sum }\limits_{{j = 1}}^{n}{\bar{\zeta }}_{j}\left( {{\zeta }_{j} - {z}_{j}}\right) },\;S\left( {\ze... | Let \( 0 < a < 1 \) and choose \( \chi \in {C}^{\infty }\left( {\mathbb{C}}^{n}\right) \), such that \( \chi \equiv 1 \) on \( {bB} \) and \( \chi \equiv 0 \) on \( B\left( {0, a}\right) \) . For \( z \in B\left( {0, a}\right) \) the \( \left( {1,0}\right) \) -form\n\n\[ \nW\left( {\zeta, z}\right) = \chi \left( \zeta ... | Yes |
Theorem 3.4. Let \( D \subset \subset {\mathbb{C}}^{n} \) be a convex domain with \( {C}^{2} \) defining function \( r \) . Then\n\n\[ f\left( z\right) = \frac{1}{{\left( 2\pi i\right) }^{n}}{\int }_{bD}f\left( \zeta \right) \frac{\partial r\left( \zeta \right) \land {\left( \bar{\partial }\partial r\left( \zeta \right... | Proof. As in Example 3.3, one shows that\n\n\[ f\left( z\right) = {\int }_{bD}f{\Omega }_{0}\left( {{C}^{\left( r\right) }\left( {\cdot, z}\right) }\right) \] for \( f \in {A}^{1}\left( D\right) \) . Another proof of (3.8), independent of Proposition 3.2, will follow from a more general result in the next section. Sinc... | Yes |
Lemma 3.5. Let \( W \in {C}_{1,0}^{1,\infty }\left( {{bD} \times D}\right) \) be a generating form. Then, for any \( 0 \leq q \leq n, \)\n\n\[ {d}_{\zeta ,\lambda }{\Omega }_{q}\left( \widehat{W}\right) = {\bar{\partial }}_{\zeta ,\lambda }{\Omega }_{q}\left( \widehat{W}\right) = {\left( -1\right) }^{q}{\bar{\partial }... | Proof. The right side is a continuous form; so, unless \( W \) is of class \( {C}^{2,\infty } \) ,(3.12) has to be interpreted in the distribution sense (cf. the addendum to Lemma 3.1).\n\nFrom (3.9) it follows that, at a fixed point \( \left( {\zeta ,\lambda, z}\right) \in \left( {{bD} \times I}\right) \times D \), th... | Yes |
Theorem 3.6. Let \( W \in {C}_{1,0}^{1,\infty }\left( {{bD} \times D}\right) \) be a generating form for points \( z \) in the domain \( D \subset \subset {\mathbb{C}}^{n} \) with \( {C}^{1} \) boundary and set \( \widehat{W} = {\lambda W} + \left( {1 - \lambda }\right) B \) on \( \left( {{bD} \times I}\right) \times D... | Proof of 3.6. Since the first integral in (3.13) is \( {C}^{\infty } \) in \( z \), part (a) follows from Theorem 1.14. By the Remark above, it is enough to prove (b) for \( q < n \) . We use the representation given by Theorem 1.10 and modify the boundary integral as follows. By (3.11),\n\n\[ \n{\int }_{bD}f \land {K}... | Yes |
Corollary 3.8. Let \( K \) be a compact convex set in \( {\mathbb{C}}^{n} \). Then\n\n\[ \n{H}_{\bar{\partial }}^{q}\left( K\right) = 0\;\text{ for }q \geq 1.\n\] | Proof. Let \( q \geq 1 \) and suppose \( f \in {C}_{0, q}^{\infty }\left( K\right) \) is defined and \( \bar{\partial } \) -closed on the open neighborhood \( U \) of \( K \). Since \( K \) is convex, there is a smoothly bounded open convex neighborhood \( D, K \subset D \subset \subset U \), with defining function \( ... | Yes |
Corollary 3.10. If \( K = \\left\\{ {{R}_{1} \\leq \\left| z\\right| \\leq {R}_{2}}\\right\\} \\subset {\\mathbb{C}}^{n} \) with \( 0 < {R}_{1} \\leq {R}_{2} < \\infty \), then\n\n\[ \n{H}_{\\partial }^{q}\\left( K\\right) = 0\\;\\text{ for }q \\neq 0, n - 1.\n\] | The proof is immediate (see the proof of Corollary 3.8). | No |
Lemma 4.1. The Bergman kernel \( {K}_{D} \) satisfies\n\n(4.3)\n\n\[ \n{K}_{D}\left( {\zeta, z}\right) = \overline{{K}_{D}\left( {z,\zeta }\right) }\;\text{ for all }\zeta, z \in D, \]\n\nand hence \( {K}_{D}\left( {\zeta, z}\right) \) is conjugate holomorphic in \( z \) . | Proof. By definition, for fixed \( z \in D \) we have \( {K}_{D}\left( {\cdot, z}\right) \in {\mathcal{H}}^{2} \) ; so we may apply (4.1) with \( f = {K}_{D}\left( {\cdot, z}\right) \), and \( a = \zeta \in D \), giving\n\n\[ \n{K}_{D}\left( {\zeta, z}\right) = \left( {{K}_{D}\left( {\cdot, z}\right) ,{K}_{D}\left( {\c... | Yes |
Lemma 4.2. For every compact set \( K \subset D \) there is a constant \( {C}_{K} < \infty \) such that for every orthonormal basis \( \left\{ {{\varphi }_{j}, j = 1,2,\ldots }\right\} \) of \( {\mathcal{H}}^{2} \) one has\n\n\[ \mathop{\sup }\limits_{{z \in K}}\mathop{\sum }\limits_{{j = 1}}^{\infty }{\left| {\varphi ... | Proof. Let \( K \subset D \) be compact. Then \( \operatorname{dist}\left( {K,{bD}}\right) > 0 \), and it follows from (4.2) that there is a constant \( {C}_{K} \) such that\n\n\[ {\begin{Vmatrix}{K}_{D}\left( \cdot, z\right) \end{Vmatrix}}_{{L}^{2}\left( D\right) } \leq {C}_{K} \]\n\nfor all \( z \in K \) . Given an o... | Yes |
Theorem 4.3. For any orthonormal basis \( \left\{ {{\varphi }_{j}, j = 1,2,\ldots }\right\} \) for \( {\mathcal{H}}^{2}\left( D\right) \) one has the representation\n\n(4.10)\n\n\[ \n{K}_{D}\left( {\zeta, z}\right) = \mathop{\sum }\limits_{{j = 1}}^{\infty }{\varphi }_{j}\left( \zeta \right) \overline{{\varphi }_{j}\le... | Proof. The representation (4.10) follows directly from (4.7) and (4.9), since convergence in \( {\mathcal{H}}^{2} \) implies pointwise convergence. For the remaining statement, it is enough to prove uniform boundedness of the partial sums \( \mathop{\sum }\limits_{{j = 1}}^{m}\left| {{\varphi }_{j}\left( \zeta \right) ... | Yes |
Proposition 4.5. The Bergman projection \( {\mathbf{P}}_{D} : {L}^{2}\left( D\right) \rightarrow {\mathcal{H}}^{2}\left( D\right) \) satisfies\n\n\[ \left( {{\mathbf{P}}_{D}f}\right) \left( z\right) = \left( {f,{K}_{D}\left( {\cdot, z}\right) }\right) = {\int }_{D}f\left( \zeta \right) {K}_{D}\left( {z,\zeta }\right) {... | Proof. Given \( f \in {L}^{2}\left( D\right) \), we apply the reproducing property (4.1) to \( \mathbf{P}f \in {\mathcal{H}}^{2} \) , giving\n\n\[ \mathbf{P}f\left( z\right) = \left( {\mathbf{P}f,{K}_{D}\left( {\cdot, z}\right) }\right) \]\n\nSince \( \mathbf{P} \) is Hermitian and \( {K}_{D}\left( {\cdot, z}\right) \i... | Yes |
Theorem 4.6. The Bergman kernel \( {K}_{B} \) for the unit ball \( B = \left\{ {z \in {\mathbb{C}}^{n} : \left| z\right| < 1}\right\} \) is given by | \[ {K}_{B}\left( {\zeta, z}\right) = \frac{n!}{{\pi }^{n}}{\left\lbrack 1 - \left( \zeta, z\right) \right\rbrack }^{-\left( {n + 1}\right) }.\] | Yes |
Theorem 4.7. Suppose \( {D}_{i} \subset \subset {\mathbb{C}}^{{n}_{i}}, i = 1,2 \), are bounded domains with Bergman kernels \( {K}_{{D}_{1}} \) and \( {K}_{{D}_{2}} \) . Then the Bergman kernel \( {K}_{D} \) for the product domain \( D = {D}_{1} \times {D}_{2} \) is given by\n\n(4.21)\n\n\[ \n{K}_{D}\left( {\left( {{\... | Proof. Let \( G \) denote the function on the right in (4.21). It is clear that \( G\left( {\cdot ,\left( {{a}_{1},{a}_{2}}\right) }\right) \in {\mathcal{H}}^{2}\left( D\right) \) for each fixed \( \left( {{a}_{1},{a}_{2}}\right) \in D \) ; moreover, the reproducing property\n\n\[ \nf\left( {{a}_{1},{a}_{2}}\right) = {... | Yes |
The Bergman kernel \( {K}_{{\Delta }^{n}} \) for the unit polydisc \( {\Delta }^{n} \) in \( {\mathbb{C}}^{n} \) is given by | \[ {K}_{{\Delta }^{n}}\left( {\zeta, z}\right) = \frac{1}{{\pi }^{n}}\mathop{\prod }\limits_{{j = 1}}^{n}\frac{1}{{\left( 1 - {\zeta }_{j}{\bar{z}}_{j}\right) }^{2}}. \] | Yes |
Theorem 4.9. Suppose \( F : {D}_{1} \rightarrow {D}_{2} \) is a biholomorphic map between bounded domains in \( {\mathbb{C}}^{n} \) . Then\n\n\[ {K}_{{D}_{1}}\left( {\zeta, z}\right) = \det {F}^{\prime }\left( \zeta \right) {K}_{{D}_{2}}\left( {F\left( \zeta \right), F\left( z\right) }\right) \overline{\det {F}^{\prime... | Proof. The substitution formula for integrals and Lemma I.2.1 imply that\n\n\[ {\int }_{{D}_{2}}{\left| f\left( w\right) \right| }^{2}{dV}\left( w\right) = {\int }_{{D}_{1}}{\left| f \circ F\left( \zeta \right) \right| }^{2}{\left| \det {F}^{\prime }\left( \zeta \right) \right| }^{2}{dV}\left( \zeta \right) .\n\nHence ... | Yes |
Corollary 4.10. If \( F : {D}_{1} \rightarrow {D}_{2} \) is biholomorphic, then\n\n(4.26)\n\n\[ \n{\mathbf{P}}_{{D}_{1}}\left( {\det {F}^{\prime }f \circ F}\right) = \det {F}^{\prime }\left( {{\mathbf{P}}_{{D}_{2}}\left( f\right) \circ F}\right) \n\]\n\nfor all \( f \in {L}^{2}\left( {D}_{2}\right) \) . | Proof. Recall the isometry \( {T}_{F} : {L}^{2}\left( {D}_{2}\right) \rightarrow {L}^{2}\left( {D}_{1}\right) \) from the proof of Theorem 4.9. The left side of (4.26) is \( {\mathbf{P}}_{{D}_{1}}\left( {{T}_{F}f}\right) \) . By Proposition 4.5, if \( z \in {D}_{1} \), then\n\n\[ \n{\mathbf{P}}_{{D}_{1}}\left( {{T}_{F}... | Yes |
Proposition 1.1. The form \( {\Omega }_{0}\left( {L}_{D}\right) \) is defined on \( {bD} \times \bar{D} - \{ \zeta = z\} \), with coefficients in \( {C}^{k,\infty } \) . Furthermore, \( {\Omega }_{0}\left( {L}_{D}\right) \) is holomorphic in \( z \) for \( z \in \bar{D} \) with \( 0 \neq \left| {z - \zeta }\right| < \v... | Proof. Since \( \chi \equiv 1 \) for \( \left| {z - \zeta }\right| \leq \varepsilon /2 \), the construction of \( {L}_{D} \) shows that its coefficients are holomorphic in \( z \) on \( \left| {z - \zeta }\right| < \varepsilon /2 \), provided \( \Phi \left( {\zeta, z}\right) \neq 0 \) . By (1.6) and (1.7), \( \Phi \lef... | Yes |
Lemma 1.2. If \( q \geq 1 \) and \( f \in {C}_{0, q}^{1}\left( \bar{D}\right) \) satisfies \( \bar{\partial }f = 0 \) on \( D \), then\n\n(1.12)\n\n\[ \bar{\partial }\left( {{\mathbf{E}}_{q}f}\right) = 0\text{ on }{\bar{D}}_{\delta } \] | Proof. By Proposition 1.1 and by Lemma IV.3.5, if \( q \geq 1,{\bar{\partial }}_{z}{\Omega }_{q}\left( {L}_{D}\right) = \) \( {\left( -1\right) }^{q + 1}{\bar{\partial }}_{\zeta }{\Omega }_{q + 1}\left( {L}_{D}\right) \) on \( {bD} \times {\bar{D}}_{\delta } \) . By continuity, \( \bar{\partial }f = 0 \) on \( {bD} \),... | Yes |
Proposition 1.3. The homomorphism\n\n\\[ \n{\\rho }_{q}^{ * } : {H}_{\\bar{\\partial }}^{q}\\left( {\\bar{D}}_{\\delta }\\right) \\rightarrow {H}_{\\bar{\\partial }}^{q}\\left( \\bar{D}\\right) \n\\]\n\ninduced by the restriction map \\( \\rho : {C}_{0, q}^{\\infty }\\left( {\\bar{D}}_{\\delta }\\right) \\rightarrow {C... | Proof. There is \\( {\\eta }_{0} > 0 \\), such that\n\n\\[ \n{\\mathbf{E}}_{q}^{\\eta }f = {\\int }_{b{D}_{\\eta }}f \\land {\\Omega }_{q}\\left( {L}_{D}\\right) \n\\]\n\nis well defined for \\( 0 \\leq \\eta \\leq {\\eta }_{0} \\) and \\( f \\in {C}_{0, q}\\left( {b{D}_{\\eta }}\\right) \\), and \\( {\\mathbf{E}}_{q}^... | Yes |
For \( q \geq 1 \), the operator \( \bar{\partial }{\mathbf{T}}_{q} \) defines a bounded linear transformation\n\n\[ \bar{\partial }{\mathbf{T}}_{q} : {Z}_{q}^{1} \rightarrow {Z}_{q}^{1} \]\n\nwhose range has finite codimension. | Proof. By (1.11),\n\n(1.13)\n\n\[ \bar{\partial }{\mathbf{T}}_{q} = {Id} - {\mathbf{E}}_{q} \]\n\non \( {Z}_{q}^{1} \), and by Lemma 1.2, the restriction of \( {\mathbf{E}}_{q} \) to \( {Z}_{q}^{1} \) maps \( {Z}_{q}^{1} \) into \( {C}_{0, q}^{\infty }\left( \bar{D}\right) \cap \) \( \ker \bar{\partial } \subset {Z}_{q... | Yes |
Theorem 1.5. Let \( D \subset \subset {\mathbb{C}}^{n} \) be a strictly pseudoconvex domain defined by a strictly plurisubharmonic function \( r \) on a neighborhood \( U \) of \( {bD} \) such that \( L = D \cup \{ z \in U : r\left( z\right) \leq 0\} \) is compact. Then \( D \) is holomorphically convex, and hence a do... | Proof. By Proposition II.3.4, it is enough to show that for each \( \zeta \in {bD} \) there is \( h = {h}_{\zeta } \in \mathcal{O}\left( D\right) \), such that\n\n\[ \mathop{\lim }\limits_{\substack{{z \rightarrow \zeta } \\ {z \in D} }}\left| {{h}_{\zeta }\left( z\right) }\right| = \infty \]\n\nFix \( \zeta \in {bD} \... | Yes |
Corollary 1.6. A compact set \( K \subset {\mathbb{C}}^{n} \) is a Stein compactum (i.e., an intersection of holomorphically convex domains) if and only if \( K \) has a neighborhood basis of strictly pseudoconvex domains with \( {C}^{2} \) boundary. In particular, the closure of a strictly pseudoconvex domain with \( ... | Proof. The \ | No |
Proposition 2.1. Let \( K \subset {\mathbb{C}}^{n} \) be a Stein compactum and set \( {K}_{1} = K \cap \left\{ {z \in {\mathbb{C}}^{n}}\right. \) : \( \left. {{z}_{1} = 0}\right\} \) . Then for every \( f \in \mathcal{O}\left( {K}_{1}\right) \) there is a holomorphic function \( F \in \mathcal{O}\left( K\right) \), suc... | Proof of 2.1. Given \( f \in \mathcal{O}\left( {K}_{1}\right) \), we choose a neighborhood \( W \) of \( K \) such that \( f \) is defined and holomorphic on \( {W}_{1} = W \cap \left\{ {z : {z}_{1} = 0}\right\} \) . Let \( \pi : {\mathbb{C}}^{n} \rightarrow {\mathbb{C}}^{n} \) be defined by \( \pi \left( {{z}_{1},{z}^... | Yes |
Lemma 2.3. Let \( K \subset {\mathbb{C}}^{n} \) be a Stein compactum. Let \( 1 \leq k \leq n \) and define\n\n\[ \n{K}_{k} = \left\{ {z \in K : {z}_{1} = \cdots = {z}_{k} = 0}\right\} .\n\]\n\nSuppose \( f \in \mathcal{O}\left( K\right) \) satisfies \( f\left( z\right) = 0 \) for all \( z \) in a neighborhood of \( K \... | Proof of 2.3. We use induction. Lemma 2.3 is certainly true for \( k = 1 \) and arbitrary \( n \geq 1 \) : simply take \( {g}_{1} = f/{z}_{1} \) . Suppose the Lemma has been proved for \( k - 1 \) and all \( n \geq k \) . Let \( {K}^{\# } = K \cap \left\{ {z \in {\mathbb{C}}^{n} : {z}_{1} = 0}\right\} \) . Then \( {K}^... | Yes |
Theorem 2.5. Let \( G \) be a neighborhood of the Stein compactum \( K \subset {\mathbb{C}}^{n} \). Then there are neighborhoods \( {V}_{0} \subset \subset V \subset \subset G \) of \( K \) and linear operators \[ {\mathbf{T}}_{q}^{V,{V}_{0}} : {C}_{0, q}\left( \bar{V}\right) \rightarrow {C}_{0, q - 1}\left( {V}_{0}\ri... | Proof. Without loss of generality we may assume that \( G \) is holomorphically convex. Apply Theorem IV.3.6 with \( D = V \) to the generating form \( W \in {C}_{1,0}^{1,\infty }\left( {{bV} \times {V}_{0}}\right) \) given by Proposition 2.4, restricting \( z \) to \( {V}_{0} \subset \subset V \). The operator \( {\ma... | Yes |
Theorem 2.7. Let \( D \subset \subset {\mathbb{C}}^{n} \) be strictly pseudoconvex with \( {C}^{2} \) boundary. For \( 1 \leq q \leq n \) there are linear operators\n\n\[ \n{\mathbf{S}}_{q} = {\mathbf{S}}_{q}^{\left( D\right) } : {C}_{0, q}\left( \bar{D}\right) \rightarrow {C}_{0, q - 1}\left( D\right) \n\]\n\nwith the... | Proof of Theorem 2.7. The construction in \( §{1.1} \) gives the parametrix \( {\mathbf{T}}_{q} = \) \( {\mathbf{T}}_{q}^{{L}_{D}} \) and the extension operator \( {\mathbf{E}}_{q} : {C}_{0, q}\left( \bar{D}\right) \rightarrow {C}_{0, q}^{\infty }\left( {\bar{D}}_{\delta }\right) \), such that (1.11) holds. Since \( K ... | No |
Lemma 2.8. Suppose \( u \in {\Lambda }_{\alpha }\left( {D}^{m}\right) \) satisfies \( \bar{\partial }u = g \) on \( {D}^{m} \) . Then \( \alpha \leq 1/{2m} \) . | Proof. For \( 0 < d < 1/2 \), the integral\n\n(2.13)\n\n\[ I\left( d\right) = {\int }_{\left| {z}_{2}\right| = {d}^{1/{2m}}}\left\lbrack {u\left( {1 - d,{z}_{2}}\right) - u\left( {1 - {2d},{z}_{2}}\right) }\right\rbrack d{z}_{2} \]\nis well defined, and, if \( u \in {\Lambda }_{\alpha }\left( {D}^{m}\right) \), one obt... | Yes |
Lemma 3.2. For \( 0 \leq q \leq n - 2 \) and any \( f \in {C}_{0, q + 1}\left( {bD}\right) \) one has\n\n\[{\int }_{{bD} \times I}f \land {\Omega }_{q}\left( {\widehat{L}}_{D}\right) = {\int }_{bD}f \land {A}_{q}\left( {{L}_{D}, B}\right)\] | Proof. The integral on the left in (3.11) equals \( {\int }_{{bD} \times I}f \land {\Omega }_{q}^{\left( 1\right) } \land {d\lambda } \) . Expanding (3.9) by multilinearity, (3.10) implies\n\n\[{\Omega }_{q}^{\left( 1\right) } = \mathop{\sum }\limits_{{j = 0}}^{{n - q - 2}}\mathop{\sum }\limits_{{k = 0}}^{q}{b}_{q}^{j,... | Yes |
Proposition 3.3. For any \( 0 \leq q \leq n - 2 \) the double form \( {A}_{q}\left( {{L}_{D}, B}\right) = \) \( \mathop{\sum }\limits_{{\left| J\right| = q}}{A}_{q, J}d{\bar{z}}^{J} \) defined in Lemma 3.2 satisfies\n\n(3.14)\n\n\[{\int }_{bD}\left| {{d}_{z}{A}_{q, J}\left( {\cdot, z}\right) }\right| \lesssim {\delta }... | We now prove Proposition 3.3. Because of (3.12) it is enough to prove (3.14) for \( {A}_{q, J}^{j, k} \) instead of \( {A}_{q, J} \) . Differentiation of (3.13) with respect to \( z \) gives\n\n\[{d}_{z}{A}_{q, J}^{j, k} = \frac{{N}_{1}}{{\Phi }^{j + k + 1}{\beta }^{n - \left( {j + k + 1}\right) }} + \frac{{d}_{z}\Phi ... | Yes |
Lemma 3.4. There are positive constants \( M, a \), and \( \eta \leq \varepsilon /2 \), and, for each \( z \) with \( {\delta }_{D}\left( z\right) \leq a \), there is a \( {C}^{1} \) coordinate system \( \left( {{t}_{1},\ldots ,{t}_{2n}}\right) = t = t\left( {\zeta, z}\right) \) on \( B\left( {z,\eta }\right) \) , such... | Proof. Fix \( z \in {bD} \) . For \( \left| {\zeta - z}\right| < \varepsilon /2,\Phi = {F}^{\# } \), by (1.5). From (1.3) one obtains\n\n(3.23)\n\n\[ \n{d}_{\zeta }\Phi \left( {z, z}\right) = {d}_{\zeta }{F}^{\# }\left( {z, z}\right) = {\partial }_{\zeta }r\left( z\right)\n\]\n\nand therefore, at the point \( \zeta = z... | Yes |
Lemma 3.5. Let \( J\left( \delta \right) \) denote the integral on the right side in (3.26). Then\n\n\[ J\left( \delta \right) \lesssim {\delta }^{-1/2}\;\text{ for }\delta > 0. \] | Proof. Integrating in \( {t}_{2} \) one obtains\n\n\[ J\left( \delta \right) = {\int }_{\left| {t}^{\prime }\right| < 1}\left\lbrack {\frac{1}{\delta + {\left| {t}^{\prime }\right| }^{2}} - \frac{1}{1 + \delta + {\left| {t}^{\prime }\right| }^{2}}}\right\rbrack \frac{d{t}^{\prime }}{{\left| {t}^{\prime }\right| }^{{2n}... | Yes |
Theorem 1.1. Let \( K \subset {\mathbb{C}}^{n} \) be a Stein compactum and suppose \( {h}_{1},\ldots ,{h}_{l} \in \mathcal{O}\left( K\right) \) . Define\n\n\[ \n{K}_{l} = \left\{ {z \in K : \left| {{h}_{j}\left( z\right) }\right| \leq 1,1 \leq j \leq l}\right\} .\n\]\n\nThen \( {K}_{l} \) is Stein and \( \mathcal{O}\le... | Proof. \( {K}_{l} \) is Stein by Lemma II.3.22. For the main part of the theorem we may assume that \( l = 1 \) ; the general case then follows by induction on \( l \) . So set \( {h}_{1} = h \) and choose an open neighborhood \( U \) such that \( h \in \mathcal{O}\left( U\right) \) . Define the\n\n---\n\nOka map \( {\... | Yes |
Corollary 1.2. Let \( K \) be a Stein compactum and suppose \( L \subset K \) is a compact subset with\n\n\[{\widehat{L}}_{\mathcal{O}\left( \mathbf{K}\right) } = L\]\n\nThen \( L \) is Stein and \( \mathcal{O}\left( K\right) \) is dense in \( \mathcal{O}\left( L\right) \) in the supremum norm over \( L \) . | Proof. Suppose \( f \in \mathcal{O}\left( L\right) \) and choose an open neighborhood \( U \) of \( L \), so that \( f \in \mathcal{O}\left( U\right) \) . By Proposition II.3.23, \( L \) is Stein and one can find a compact analytic polyhedron\n\n\[{K}_{l} = \left\{ {z \in K : \left| {h}_{j}\right| \leq 1,1 \leq j \leq ... | Yes |
Theorem 1.4. Let \( D \subset {\mathbb{C}}^{n} \) be a Stein domain. Then \[ {H}_{\bar{\partial }}^{q}\left( D\right) = 0\;\text{ for }q \geq 1. \] | Proof. By Lemma II.3.2 there is a normal exhaustion \( \left\{ {K}_{j}\right\} \) of \( D \) by Stein compacta, such that \( {\left( {\widehat{K}}_{j}\right) }_{\mathcal{O}\left( D\right) } = {K}_{j} \) . In particular, \( {\left( {\widehat{K}}_{j}\right) }_{\mathcal{O}\left( {K}_{j + 1}\right) } = {K}_{j} \) for \( j ... | No |
Theorem 1.10. Let \( D \subset {\mathbb{C}}^{n} \) be a Stein domain. Suppose \( K \subset D \) is compact and \( {\widehat{K}}_{\mathcal{O}\left( D\right) } = K \) . Then \( \mathcal{O}\left( D\right) \) is dense in \( \mathcal{O}\left( K\right) \) . | Proof. Notice that Theorem 1.5 followed directly from Corollary 1.2 by using the Taylor expansion. This tool is not available now, so the approximating functions must be obtained by an inductive procedure. Let \( {K}_{1} \subset {K}_{2} \subset {K}_{3}\ldots \) be a normal exhaustion of \( D \) with \( {K}_{1} = K \) a... | Yes |
Theorem 1.11. The following statements are equivalent for two Stein regions \( {D}_{1} \subset {D}_{2} \) in \( {\mathbb{C}}^{n} \) :\n\n(i) \( \mathcal{O}\left( {D}_{2}\right) \) is dense in \( \mathcal{O}\left( {D}_{1}\right) \), i.e., \( {D}_{1} \) is Runge in \( {D}_{2} \).\n\n(ii) For every compact set \( K \subse... | Proof. It is obvious that (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) (note that \( {\widehat{K}}_{\mathcal{O}\left( {D}_{1}\right) } \subset \subset {D}_{1} \) since \( {D}_{1} \) is Stein). Assuming (iii), notice that the function \( f \) defined by 0 on \( {K}^{\prime } = \) \( {\widehat{K}}_{\mathcal{C}\left... | Yes |
Theorem 1.13. Let \( K \subset {\mathbb{C}}^{n} \) be a Stein compactum, and let \( \zeta \in K \) be a strictly pseudoconvex boundary point, i.e., there are a neighborhood \( U \) of \( \zeta \) and a \( {C}^{2} \) strictly plurisubharmonic function \( r : U \rightarrow \mathbb{R} \), such that \( r\left( \zeta \right... | Proof. Let \( u\left( z\right) = 1/\Phi \left( {\zeta, z}\right) \), where \( \Phi \left( {\zeta, z}\right) \) is the function constructed in V.§1.1 from the Levi polynomial of \( r \) . We have Re \( u > 0 \) on \( K - \{ \zeta \} \), and \( \bar{\partial }u = 0 \) for \( \left| {z - \zeta }\right| < \varepsilon /2 \)... | Yes |
Corollary 1.14. Let \( D \subset \subset {\mathbb{C}}^{n} \) be strictly pseudoconvex with \( {C}^{2} \) boundary. Then for every \( \zeta \in {bD} \) there is a peaking function \( {h}_{\zeta } \in \mathcal{O}\left( \bar{D}\right) \) for \( \zeta \) on \( \bar{D} \) . | Proof. If \( D = \{ \varphi < 0\} \), where \( \varphi \) is strictly plurisubharmonic near \( {bD} \), then \( \bar{D} = \{ \varphi \leq 0\} \) is a Stein compactum, and \( \zeta \in {bD} \) satisfies \( \varphi \left( \zeta \right) = 0 \) . Now apply Theorem 1.13. | Yes |
Proposition 1.15. Let \( D \subset {\mathbb{C}}^{n} \) be open and let \( \varphi : D \rightarrow \mathbb{R} \) be a \( {C}^{2} \) function such that for some real numbers \( {b}^{\# } < {c}^{\# }\varphi \) is strictly plurisubharmonic on \( {\varphi }^{-1}\left( \left( {{b}^{\# },{c}^{\# }}\right) \right) \), and such... | Proof. By Theorem V.1.5, \( {D}_{c} \) is holomorphically convex. Therefore \( K = \) \( {\left( {\widehat{L}}_{b}\right) }_{\mathcal{O}\left( {D}_{c}\right) } \subset {D}_{c} \) is compact. Let \( \eta = \max \{ \varphi \left( z\right) : z \in K\} \) ; then \( b \leq \eta < c \), and we must show \( b = \eta .{L}_{\et... | Yes |
Theorem 1.16. Let \( \varphi \) and \( D \) satisfy the hypotheses of Proposition 1.15. Then, for any \( b, c \in \mathbb{R} \) with \( {b}^{\# } < b < c < {c}^{\# } \), the region \( {D}_{b} \) is Runge in \( {D}_{c} \) . | Proof. Since both \( {D}_{b} \) and \( {D}_{c} \) are Stein, it is enough to verify condition (iv) in Theorem 1.11. Let \( K \) be a compact set in \( {D}_{b} \) . Choose \( a \in \mathbb{R} \) such that \( {b}^{\# } < a < b \) and \( K \subset {L}_{a} \) . By Proposition 1.15, \( {\widehat{K}}_{\mathcal{O}\left( {D}_{... | Yes |
Theorem 1.17. Let \( D \subset {\mathbb{C}}^{n} \) be a pseudoconvex domain. Then \( D \) is Stein. | Proof. By definition (cf. II.§2.10) there is a \( {C}^{2} \) strictly plurisubharmonic exhaustion function \( \varphi : D \rightarrow \mathbb{R} \) . So each open set \( {D}_{c} = \{ z \in D : \varphi \left( z\right) < c\} \) is Stein and, by Theorem 1.16, \( {D}_{c} \) is Runge in \( {D}_{{c}^{\prime }} \) for \( c < ... | Yes |
Theorem 1.18. Let \( D \subset {\mathbb{C}}^{n} \) be PS-convex and suppose that \( K \subset D \) is compact. Then \[ {\widehat{K}}_{{PS}\left( D\right) } = {\widehat{K}}_{\mathcal{O}\left( D\right) } \] | Proof. Obviously \( {\widehat{K}}_{{PS}\left( D\right) } \subset {\widehat{K}}_{\mathcal{O}\left( D\right) } \) . To prove the reverse inclusion, suppose \( w \in D - {\widehat{K}}_{{PS}\left( D\right) } \) ; we will show that \( w \notin {\widehat{K}}_{\mathcal{O}\left( D\right) } \) . We know that \( D \) is pseudoco... | Yes |
Corollary 1.19. Let \( D \subset {\mathbb{C}}^{n} \) be PS-convex and suppose \( \varphi \) is an arbitrary pluri-subharmonic function on \( D \) . For \( c \in \mathbb{R} \) set \( {D}_{c} = \{ z \in D : \varphi \left( z\right) < c\} \) . Then \( {D}_{c} \) is Runge in \( {D}_{{c}^{\prime }} \) for all \( c < {c}^{\pr... | Proof. It is clear that each \( {D}_{c} \) is \( {PS} \) -convex and hence Stein by Theorem 1.18. Suppose \( c < {c}^{\prime } \) and let \( K \subset {D}_{c} \) be compact. By Theorem 1.11 it suffices to show that \( {\widehat{K}}_{\mathcal{O}\left( {D}_{{c}^{\prime }}\right) } \subset {D}_{c} \) . But this is obvious... | Yes |
Theorem 2.1. Let \( D \subset {\mathbb{C}}^{n} \) be pseudoconvex. Then\n\n\[ \n{H}_{\bar{\partial }}^{q}\left( D\right) = 0\;\text{ for }q \geq 1 \n\] | Proof. Let \( \varphi \) be a \( {C}^{2} \) strictly plurisubharmonic exhaustion function for \( D \) . If \( {L}_{j} = \left\{ {z \in D : \varphi \left( z\right) \leq j}\right\} \), then \( {H}_{\partial }^{q}\left( {L}_{j}\right) = 0 \) for \( q \geq 1 \) and \( j = 1,2,\ldots \) (see Chapter V, §2). Furthermore, by ... | Yes |
Lemma 2.2. Let \( D \) be open in \( {\mathbb{C}}^{n} \) and set \( {D}_{1} = \left\{ {z \in D : {z}_{1} = 0}\right\} \) . Denote by \( \iota \) the inclusion \( {D}_{1} \rightarrow D \) . If \( q \geq 0 \) and \( {H}_{\bar{\partial }}^{q + 1}\left( D\right) = 0 \), then for every \( \bar{\partial } \) -closed \( {C}_{... | Proof. Denote by \( \pi \) the projection \( \pi \left( z\right) = \left( {0,{z}_{2},\ldots {z}_{n}}\right) .{D}_{1} \) and \( D - {\pi }^{-1}\left( {D}_{1}\right) \) are two disjoint closed subsets of \( D \), so there is \( \chi \in {C}^{\infty }\left( D\right) \) such that \( \chi \equiv 1 \) in a neighborhood of \(... | Yes |
Corollary 2.3. With \( D \) and \( {D}_{1} \) as in Lemma 2.2, assume that \( {H}_{\bar{\partial }}^{q}\left( D\right) = {H}_{\bar{\partial }}^{q + 1}\left( D\right) = 0 \) for some \( q \geq 1 \) . Then \( {H}_{\bar{\partial }}^{q}\left( {D}_{1}\right) = 0 \) . | Proof. If \( f \in {C}_{0, q}^{\infty }\left( {D}_{1}\right) \) and \( \bar{\partial }f = 0 \), choose \( F \) as in Lemma 2.2. Since \( {H}_{\bar{\partial }}^{q}\left( D\right) = 0 \) , there is \( u \in {C}_{0, q - 1}^{\infty }\left( D\right) \) with \( \bar{\partial }u = F \) . Then \( {\iota }^{ * }u \in {C}_{0, q ... | Yes |
Theorem 2.4. The following statements are equivalent for a domain \( D \subset {\mathbb{C}}^{n} \) :\n\n(i) \( D \) is pseudoconvex;\n\n(ii) \( {H}_{\bar{\partial }}^{q}\left( D\right) = 0 \) for \( q = 1,2,\ldots, n \) ;\n\n(iii) \( D \) is holomorphically convex, i.e., \( D \) is Stein;\n\n(iv) \( D \) is locally Ste... | Proof. (i) \( \Rightarrow \) (ii) by Theorem 2.1,(iii) \( \Rightarrow \) (iv) is trivial, and (iv) implies that \( D \) is locally pseudoconvex, hence (i) holds. We now prove (ii) \( \Rightarrow \) (iii) by induction on the dimension \( n \) .\n\nFor \( n = 1 \) the implication is true since every \( D \subset \mathbb{... | Yes |
Lemma 3.1. Suppose \( {H}_{\partial }^{q}\left( D\right) = 0 \) for \( 1 \leq q \leq r \) . Then \( f \in {C}_{r}^{\infty }\left( D\right) \) is d-cohomologous to a holomorphic \( r \) -form if and only if \( {df} \in {\Omega }^{r + 1}\left( D\right) \) . | Proof. If \( f - {dg} \in {\Omega }^{r}\left( D\right) \) for some \( g \in {C}_{r - 1}^{\infty }\left( D\right) \), then clearly \( {df} \in {\Omega }^{r + 1} \) . To prove the reverse implication we must show\n\n(3.2)\n\nIf \( {df} \in {\Omega }^{r + 1}\left( D\right) \), there is \( g \in {C}_{r - 1}^{\infty }\left(... | Yes |
Theorem 3.2. Let \( D \subset {\mathbb{C}}^{n} \) be a Stein domain. Then \[ {H}_{d}^{r}\left( D\right) \simeq \left\{ {f \in {\Omega }^{r}\left( D\right) : {df} = 0}\right\} /d{\Omega }^{r - 1} \] for \( r \geq 0 \) . | Proof. The case \( r = 0 \) is trivial. If \( r > 0 \), consider the homomorphism \[ {\alpha }_{r} : \left\{ {f \in {\Omega }^{r} : {df} = 0}\right\} \rightarrow {H}_{d}^{r}\left( D\right) \] where \( {\alpha }_{r}\left( f\right) \) is the \( d \) -cohomology class of \( f \) in \( {H}_{d}^{r}\left( D\right) \) . Lemma... | Yes |
Let \( D = \left\{ {z \in {\mathbb{C}}^{n} : 0 < \left| {z}_{j}\right| < 2,1 \leq j \leq n}\right\} \) . \( D \) is Stein, since it is a product of planar domains. For \( 1 \leq r \leq n \) define the \( r \) -cycle \( {\gamma }_{r} \) in \( D \) by \( {\gamma }_{r}\left( {{t}_{1},\ldots ,{t}_{r}}\right) = \left( {{e}^... | \[ {f}_{r} = \frac{d{z}_{1} \land \ldots \land d{z}_{r}}{{z}_{1}\ldots {z}_{r}} \] on \( D \) is \( d \) -closed. Since \[ {\int }_{{\gamma }_{r}}{f}_{r} = {\left( 2\pi i\right) }^{r} \neq 0 \] \( {f}_{r} \) is not the differential of a holomorphic \( \left( {r - 1}\right) \) -form on \( D \) . So, by Theorem 3.3, \( {... | Yes |
Theorem 3.6. Let \( D \subset {\mathbb{C}}^{n} \) be a Runge domain. Then\n\n\[ \n{H}^{r}\left( {D,\mathbb{C}}\right) = 0\;\text{ for }r \geq n.\n\] | Proof. Since, in particular, \( D \) is Stein, by Theorem 3.3 it is enough to show that every holomorphic \( n \) -form \( f \) on \( D \) is the differential \( {du} \) of a holomorphic \( \left( {n - 1}\right) \) -form on \( D \) . Suppose \( f = {hd}{z}_{1} \land \ldots \land d{z}_{n} \in {\Omega }^{n}\left( {\mathb... | Yes |
Theorem 4.1. The ring \( {}_{n}{\mathcal{O}}_{a} \) of germs of holomorphic functions at \( a \in {\mathbb{C}}^{n} \) is an integral domain. | Proof. We must show that if \( {\mathbf{f}}_{a} \cdot {\mathbf{g}}_{a} = {\mathbf{0}}_{a} \) for two germs \( {\mathbf{f}}_{a},{\mathbf{g}}_{a} \in {\mathcal{O}}_{a} \), then at least one of them is zero. Suppose \( {\mathbf{f}}_{a} \neq {\mathbf{0}}_{a} \) . We can then choose representatives \( \left( {f, U}\right) \... | Yes |
Proposition 4.6. \( \mathcal{M}\left( D\right) \) is a field if and only if \( D \) is connected. | The proof is left as an exercise for the reader. | No |
Theorem 4.7. Let \( D \subset {\mathbb{C}}^{n} \) be open and suppose \( m \in \mathcal{M}\left( D\right) \) . Then the singular set \( S\left( m\right) = \left\{ {z \in D : {\mathbf{m}}_{z} \notin {\mathcal{O}}_{z}}\right\} \) is either empty or an analytic set of dimension \( n - 1 \) . | Proof. Notice that the case \( n = 1 \) is trivial. For the general case, let \( a \in D \) and write \( {\mathbf{m}}_{a} = {\mathbf{f}}_{a}/{\mathbf{g}}_{a} \), where, by Theorems 4.3 and 4.4 we can assume that \( f, g \in \mathcal{O}\left( U\right) \), and \( {\mathbf{f}}_{z} \) and \( {\mathbf{g}}_{z} \) are relativ... | Yes |
Theorem 4.9. Let \( D \subset {\mathbb{C}}^{n} \) be Stein (it is enough to assume \( {H}_{\partial }^{1}\left( D\right) = 0 \) ). Then every additive Cousin problem on \( D \) has a solution. | Proof of Theorem 4.9. Suppose the functions \( {m}_{j} \in \mathcal{M}\left( {U}_{j}\right) \) satisfy (4.2). It follows that the functions \( {g}_{ij} = {m}_{j} - {m}_{i} \) satisfy (4.4) in Theorem 4.8. Let \( {g}_{j} \in \mathcal{O}\left( {U}_{j}\right), j \in J \), be the functions given by Theorem 4.8. By (4.5), \... | Yes |
Theorem 5.2. ([Oka], III). Let \( D \) be a Stein domain in \( {\mathbb{C}}^{n} \) (it is enough to assume \( {H}_{\partial }^{1}\left( D\right) = 0 \) ). Then Problem \( Z \) has a holomorphic solution if and only if there is a continuous solution, i.e., there is \( c \in C\left( D\right) \), such that \( c \cdot {f}_... | Proof of Theorem 5.2 (assuming Theorem 5.1). Only one implication is nontrivial. Let \( {f}_{j} \in \mathcal{O}\left( {U}_{j}\right) \) be given, such that \( {g}_{ij} = {f}_{j}{f}_{i}^{-1} \in {\mathcal{O}}^{ * }\left( {{U}_{i} \cap {U}_{j}}\right) \) (cf. (5.1)). If \( c \in C\left( D\right) \) is a continuous soluti... | Yes |
Theorem 5.3. Suppose \( D \subset {\mathbb{C}}^{n} \) has the Oka property and \( {H}_{\bar{\partial }}^{1}\left( D\right) = 0 \) . Then every multiplicative Cousin problem on D has a solution. | Proof. Given an open covering \( \left\{ {{U}_{j}, j \in J}\right\} \) of \( D \) and functions \( {m}_{j} \in {\mathcal{M}}^{ * }\left( {U}_{j}\right) \) with \( {g}_{ij} = {m}_{j}{m}_{i}^{-1} \in {\mathcal{O}}^{ * }\left( {{U}_{i} \cap {U}_{j}}\right) \), the Oka property implies the existence of functions \( {c}_{j}... | Yes |
Theorem 5.4. Let \( {D}_{v},1 \leq v \leq n \), be open subsets of \( {\mathbb{C}}^{1} \) and suppose that all but one of them are simply connected. Then \( D = {D}_{1} \times {D}_{2} \times \cdots \times {D}_{n} \subset {\mathbb{C}}^{n} \) has the Oka property. In particular, every region \( D \subset {\mathbb{C}}^{1}... | For the proof of Theorem 5.4 we fix an open covering \( \left\{ {{U}_{j}, j \in J}\right\} \) of \( D \) and functions \( {c}_{ij} \in {C}^{ * }\left( {{U}_{i} \cap {U}_{j}}\right) \) which satisfy (5.15). We say that a set \( K \subset D \) has property \( S \) (with respect to \( \left\{ {c}_{ij}\right\} \) ) if ther... | Yes |
Lemma 5.6. Let \( K = {K}_{1} \times \cdots \times {K}_{n} \subset D \), where \( {K}_{1} \subset \mathbb{C} \) is a compact set whose boundary consists of finitely many compact intervals parallel to the coordinate axis, and \( {K}_{v} \subset \mathbb{C} \) is compact and simply connected for \( v \geq 2 \) . Let \( \l... | Proof of Lemma 5.6. Let \( {W}_{\mu },\mu = 1,2 \), be neighborhoods of \( {K}^{\prime } \) and \( {K}^{\prime \prime } \) on which we have solutions \( {c}_{j}^{\left( \mu \right) } \in {C}^{ * }\left( {{W}_{\mu } \cap {U}_{j}}\right) \) of (5.16). Since \( {K}^{\prime } \cap {K}^{\prime \prime } = \left\{ {{z}_{1} \i... | Yes |
Lemma 5.7. Let \( K \) be as in Lemma 5.6, and let \( h \) denote either one of the functions \( \operatorname{Re}{z}_{1},\operatorname{Im}{z}_{1} \). Suppose \( K\left( \lambda \right) = \{ z \in K : h\left( z\right) = \lambda \} \) has property \( S \) for each \( \lambda \in \mathbb{R} \). Then \( K \) has property ... | Proof. Since the geometric assumptions on \( {K}_{1} \) in Lemma 5.6 are not changed if \( {K}_{1} \) is multiplied by \( \sqrt{-1} \), it is enough to prove the case \( h\left( z\right) = \operatorname{Re}{z}_{1} \). Let \( m \) be the supremum of the set of those \( \lambda \in \mathbb{R} \) for which \( \left\{ {z \... | Yes |
Lemma 5.8. Let \( K = {K}_{1} \times \cdots \times {K}_{n} \subset D \), where each \( {K}_{v} \subset \mathbb{C} \) is a compact set whose boundary consists of finitely many compact intervals parallel to the coordinate axis. Suppose that all but one of the \( {K}_{v} \) ’s are simply connected. Then \( K \) has proper... | Proof. Without loss of generality assume that \( {K}_{v} \) is simply connected for \( 1 \leq v \leq n - 1 \) . We will prove by induction on \( l \) that for \( a \in K \) the set\n\n\[ \n{L}_{l}\left( a\right) = {K}_{1} \times \cdots \times {K}_{l} \times \left\{ {a}_{l + 1}\right\} \times \cdots \times \left\{ {a}_{... | Yes |
Lemma 5.9. Let \( D \subset {\mathbb{C}}^{n} \) . Every divisor \( \delta \) on \( D \) is the quotient \( \delta = {\delta }^{ + }/{\delta }^{ - } \) of two positive divisors \( {\delta }^{ + },{\delta }^{ - } \) on \( D \), such that \( {\delta }^{ + }\left( z\right) \) and \( {\delta }^{ - }\left( z\right) \) are re... | Proof. We will need some of the algebraic properties of \( {}_{n}{\mathcal{O}}_{a} \) stated in \( §{4.1} \) . Let \( \delta \in \mathcal{D}\left( D\right) \) . By (5.23), for each \( a \in D \) there are a neighborhood \( {U}_{a} \) and holomorphic functions \( {f}^{\left( a\right) },{g}^{\left( a\right) } \in \mathca... | Yes |
Theorem 5.10. Suppose \( D \subset {\mathbb{C}}^{n} \) has the Oka property and \( {H}_{\bar{\partial }}^{1}\left( D\right) = 0 \) . If \( m \in {\mathcal{M}}^{ * }\left( D\right) \), then there are \( F, G \in \mathcal{O}\left( D\right) \) such that \( m = F/G \) and \( {\mathbf{F}}_{z} \) and \( {\mathbf{G}}_{z} \) a... | Proof. By Lemma 5.9 we can write \( \operatorname{div}m = {\delta }^{ + }/{\delta }^{ - } \), where \( {\delta }^{ + } \) and \( {\delta }^{ - } \) are positive relatively prime divisors. By Theorem 5.3’, there exist \( f, G \in \mathcal{O}\left( D\right) \), such that \( {\delta }^{ + } = \operatorname{div}f \) and \(... | Yes |
On the domain \( D \subset {\mathbb{C}}^{2} \) of the Oka example discussed in \( §{5.2} \), we introduce the following Cousin I distribution (the notation is as in \( §{5.2} \)): \[ {m}_{1} = 1/{f}_{1} \in \mathcal{M}\left( {U}_{1}\right) ,\;{m}_{2} = 1 \in \mathcal{M}\left( {U}_{2}\right) . \] Notice that \( {m}_{1} ... | Since \( D \) is Stein, by Theorem 4.9 there is \( m \in \mathcal{M}\left( D\right) \), such that \( m - {m}_{i} \in \mathcal{O}\left( {U}_{i}\right), i = 1,2 \) . Clearly \( m ≢ 0 \) on \( D \), so \( m \in {\mathcal{M}}^{ * }\left( D\right) \) . Suppose \( m = F/G \), with \( F, G \in \mathcal{O}\left( D\right) \), a... | Yes |
Lemma 6.1. Let \( D \subset {\mathbb{C}}^{n} \) be open. Given \( f \in \mathcal{O}\left( D\right) \), define \( {s}_{f} : D \rightarrow {\mathcal{O}}_{{\mathbb{C}}^{n}} \) by \( {s}_{f}\left( a\right) = {\mathbf{f}}_{a} \) for \( a \in D \) . Then \( {s}_{f} \) is continuous, and hence a section of \( \mathcal{O} \) .... | The proof is an elementary consequence of the definitions and is left to the reader. | No |
Corollary 6.3. If the sheaf \( \mathcal{S} \) is a Hausdorff space, then two sections \( {s}_{1} \) and \( {s}_{2} \) of \( \mathcal{S} \) over \( Y \) which agree at one point \( {x}_{0} \in Y \) agree on the connected component of \( Y \) which contains \( {x}_{0} \) . | Proof. The set \( \Omega = \left\{ {x \in Y : {s}_{1}\left( x\right) = {s}_{2}\left( x\right) }\right\} \) contains \( {x}_{0} \) and is open, by Lemma 6.2; if \( \mathcal{S} \) is Hausdorff, \( \Omega \) is also closed. | Yes |
Lemma 6.4. For any open cover \( \mathcal{U} \) of \( X \) and sheaf \( \mathcal{S} \) of Abelian groups one has\n\n(i)\n\[ {H}^{0}\left( {X,\mathcal{S}}\right) = {H}^{0}\left( {\mathcal{U},\mathcal{S}}\right) = {Z}^{0}\left( {\mathcal{U},\mathcal{S}}\right) = \Gamma \left( {X,\mathcal{S}}\right) \]\n\nand\n\n(ii) if \... | Notice that (i) is an immediate consequence of the definitions. The proof of (ii) is an abstract reformulation of the second part of the proof of Theorem 5.1. | No |
If in the exact sequence (6.7) one has \( {H}^{1}\left( {X,{\mathcal{S}}^{\prime }}\right) = 0 \), then\n\n\[ \n{\psi }^{0} : \Gamma \left( {X,\mathcal{S}}\right) \rightarrow \Gamma \left( {X,{\mathcal{S}}^{\prime \prime }}\right)\n\]\n\nis surjective. | Proof. The relevant portion of the cohomology sequence reads\n\n\[ \n\cdots \rightarrow {H}^{0}\left( {X,\mathcal{S}}\right) \overset{{\psi }^{0}}{ \rightarrow }{H}^{0}\left( {X,{\mathcal{S}}^{\prime \prime }}\right) \overset{{\delta }^{0}}{ \rightarrow }0 \rightarrow \cdots ;\n\]\n\nexactness at \( {H}^{0}\left( {X,{\... | Yes |
Corollary 6.10. If \( D \) is Stein, then\n\n\[ \n{H}^{q}\left( {D,\mathcal{O}}\right) = 0\;\text{ for }q \geq 1.\n\] | Proof. Use Theorem 6.9 and Theorem 1.4. | No |
Corollary 6.11. If \( \mathcal{U} \) is a covering of \( D \) by Stein domains, then\n\n\[ \n{H}^{q}\left( {\mathcal{U},\mathcal{O}}\right) = {H}^{q}\left( {D,\mathcal{O}}\right) = {H}_{\bar{\partial }}^{q}\left( D\right) \;\text{ for }q \geq 0.\n\] | Proof. Since finite intersections of Stein domains are Stein, Corollary 6.10 implies that \( {H}^{q}\left( {{U}_{{i}_{0}} \cap \ldots \cap {U}_{{i}_{l}},\mathcal{O}}\right) = 0 \) for \( q \geq 1 \) and \( l \geq 0 \) . Thus \( \mathcal{U} \) is acyclic for \( \mathcal{O} \), and the desired result follows from Leray’s... | Yes |
Lemma 6.14. If \( X \) is paracompact, then \( {H}^{q}\left( {X,\mathcal{C}}\right) = 0 \) for \( q \geq 1 \) . | The proof of Lemma 6.14 uses a construction based on the existence of partitions of unity analogous to the proof of Step 1 in Theorem 4.8, which, in the present language, states that \( {H}^{1}\left( {D,{\mathcal{C}}^{\infty }}\right) = 0 \) . The same method of proof gives the following for any \( k = 1,2,\ldots ,\inf... | No |
If \( D \) is Stein, the inclusion \( {\mathcal{O}}^{ * } \hookrightarrow {\mathcal{C}}^{ * } \) induces an isomorphism\n\n\[ \n{H}^{1}\left( {D,{\mathcal{O}}^{ * }}\right) \simeq {H}^{1}\left( {D,{\mathcal{C}}^{ * }}\right) \n\] | The fact that the two groups are isomorphic is clear from Lemmas 6.12 and 6.15. To see that the isomorphism is induced by the inclusion map requires some \ | No |
Theorem 6.17. If \( D \) is Stein, then a divisor on \( D \) is principal if and only if its Chern class is 0. | At this point the reader should turn back to the beginning of \( §{5.3} \) where, essentially, we had explicitly constructed the Chern class of the divisor \( d \) associated to the Cousin II distribution \( \left\{ {{m}_{j} \in {\mathcal{M}}^{ * }\left( {U}_{j}\right) }\right\} \) . In fact, the collection \( \left\{ ... | No |
Does every holomorphic function \( f \) on \( A \) have an extension to a holomorphic function \( F \) on \( D \), i.e., is there \( F \in \mathcal{O}\left( D\right) \) with \( F\left( z\right) = f\left( z\right) \) for \( z \in A \) ? | For Problem 2, notice that after natural identifications a holomorphic function on \( A \) is precisely a section over \( A \) in the sheaf \( \mathcal{O}/\mathcal{I}\left( A\right) \), and that each section \( f \in \Gamma \left( {A,\mathcal{O}/\mathcal{I}\left( A\right) }\right) \) has a unique extension \( \widehat{... | Yes |
Theorem 6.18. Suppose \( D \) is Stein and that \( A = Z\left( f\right) = \{ z \in D : f\left( z\right) = 0\} \), where \( f \in \mathcal{O}\left( D\right) \) satisfies \( {df} \neq 0 \) on \( A \) . Then \( {H}^{1}\left( {D,\mathcal{I}\left( A\right) }\right) = 0 \) . | Proof. The hypotheses imply that \( A \) is a complex submanifold of dimension \( n - 1 \) and that \( \mathcal{I}\left( A\right) = f{\mathcal{O}}_{D} \) . Since \( {\mathcal{O}}_{a} \) is an integral domain, multiplication by \( f \) defines a sheaf isomorphism \( {\varphi }_{f} : {\mathcal{O}}_{D} \simeq f{\mathcal{O... | Yes |
Lemma 6.21. Suppose \( \mathcal{A} \) is a coherent analytic sheaf over \( D \) . Then for every point \( a \in D \) there are a neighborhood \( U \) of \( a \), positive integers \( {l}_{1},{l}_{2} \), and a homomorphism \( \psi : {\mathcal{O}}_{U}^{{l}_{1}} \rightarrow {\mathcal{O}}_{U}^{{l}_{2}} \) so that\n\n\[{\ma... | Proof. Since \( \mathcal{A} \) is of finite type, given \( a \in D \) there is a surjective homomorphism \( \varphi : {\mathcal{O}}_{U}^{{l}_{2}} \rightarrow {\mathcal{A}}_{U} \) for some neighborhood \( U \) of \( a \) . By the definition of coherence, \( {\mathcal{K}}_{ei}\varphi \) is of finite type, so, after shrin... | Yes |
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