SCVeamer version 1.3 -- English sample lecture

ENGLISH EDITION

The full lecture text, theorem/definition labels, figure annotations, contents
titles, Ryokuinzu questions, editing instructions and references are in English.
The HTML embeds the English interactive interfaces and English player controls.
The TeX source uses LuaLaTeX with fontspec and Latin Modern. Japanese font
packages are not required for this English source.

The sample preserves the mathematical content, eight fixed-example IDs,
blue color theme, 46 logical slides, 76 display states and 135-minute schedule
of the Japanese version 1.3 lecture. The start time is 14:55; the final slide
is scheduled to finish at 17:10. Speaker notes and footer branding remain off.
The final question compares the geometry of strongly 1-complete / strongly
1-convex spaces with weakly 1-complete spaces.

VERSION 1.3 LECTURE REVISION

- Replaced the final slide's forward-looking question with the short comparison
  above, preserving the rest of the lecture content and timing.
- Included the lecture directly in the kit, so first launch prepares its editable
  copy automatically. A separate lecture ZIP is unnecessary.
- Retained the version 1.2 conversion and presentation behavior; the English
  edition localizes the text and language defaults.

BOUNDARY LEVI FORM AND HOLOMORPHIC EXTENSION REVISION

Five slides (15 minutes) were inserted between the definition of the boundary
Levi form and Levi pseudoconvexity:
  27  Boundary normal form, used as a fact without proof.                 3 min
  28  Positive Levi form and the local holomorphic function 1/w.          3 min
  29  Negative Levi direction and the supplied analytic-disc figure.     3 min
  30  Uniform bound at the disc centers by the maximum principle.        2 min
  31  Local holomorphic extension by a Cauchy integral.                   4 min

The direct conclusion of R135 Observation III.2.148 is a uniform bound using
one interior compact set for every holomorphic function; this shows that its
holomorphic hull is not compact. The local-extension argument additionally
uses a Cauchy integral and parameter variation. These are distinct steps.
In the positive-Levi case, 1/w is constructed locally, not globally.
The supplied image remains unchanged at assets/levi-analytic-discs.png.

SETTINGS RETAINED FROM VERSION 1.2

- The Subharmonic? surface remains 90% opaque during circle tests.
- The lecture footer omits system name, lecture title and subtitle.
- This sample has no speaker notes, operation notes or generated speaker guide.
  The kit's optional notes feature remains available for other projects.
- Start time and per-slide minutes are written in TeX. Planned finish time is
  always visible at the lower right, without opening the notes panel.
- The final square marks the end of theorem-like environments automatically.

ALIGNMENT WITH THE R135 NOTES

1. Define Delta_r(a) as a disc explicitly. In Cauchy's formula, assume f is
   holomorphic on Delta_R(a), and use a contour with 0<r<R.
2. Explain that holomorphic functions have local convergent power series,
   and therefore are analytic and C-infinity.
3. Define subharmonic and psh functions using upper semicontinuity, allowing
   the identically -infinity function. Require the closed disc to lie inside
   the open set when applying the circle-average inequality.
4. Use the negative-sign Laplacian Delta, so subharmonicity means Delta u<=0.
5. Define complex derivatives, partial and dbar, the decomposition d=partial+dbar,
   square-zero identities and anticommutation.
6. State the maximum principle before the C2 test. Its hypothesis is attainment
   of the supremum at an interior point.
7. Use roman Re/Im, sqrt(-1), H(phi), Psh(Omega) and Lebesgue measure d lambda.
8. Give distributions in an example environment and explain the topology on
   test functions, Frechet spaces and the strict countable LF-space.
9. Characterize real L1_loc equivalence classes by positivity, stating the
   existence and uniqueness of a subharmonic or psh representative.
10. Define multi-indices and explain the nontrivial part of Hartogs' separate
    holomorphicity theorem: no joint continuity or local boundedness is assumed.
11. Use f(z,w)=z to show why an accumulating zero set does not suffice for the
    several-variable identity theorem.
12. Use the quantifiers over U and every component V in the definition of a
    domain of holomorphy, with a sketch of the overlapping domains.
13. Start with the real tangent bundle, then complexify and decompose by type.
    Complex tangent spaces consist of tangent vectors, not just coefficient tuples.
14. Distinguish positivity of a function from the existence of an exhaustion
    with the required positivity on a manifold.
15. For strong 1-convexity, strict positivity is required outside a compact set;
    for weak 1-completeness, semipositivity is required everywhere.
    The example P1 x C distinguishes exhaustion from strict positivity.
16. Keep source references separate from the presentation text. The short
    Nishino quotation retains its visible bibliographical attribution.

EDITABLE SKETCHES AND FIXED EXAMPLES

The source retains editable TikZ sketches of a center and circle, overlapping
domains, v and Jv, a ball and shell, complex directions, nested sublevel sets,
the unbounded sublevel sets of |w|^2, an exceptional compact set and compact
fibers of P1 x C. The positive-Levi local supporting picture is also retained.
The eight static PDFs and supplied analytic-disc PNG need no image generator
to rebuild. SH IDs: std-b03, std-b02, std-i02. Ryokuinzu IDs: lecture-ball,
lecture-polydisc, n18. PSH IDs: N01, A03.

REFERENCES

R135 Observation III.2.148, printed p. 492 / PDF p. 510;
Figure III.4, printed p. 493 / PDF p. 511.
Local-extension supplement: Jiri Lebl, Tasty Bits of Several Complex Variables,
v4.4, Theorem 2.3.11, printed pp. 75-76: https://jirka.org/scv/scv.pdf
Further references and complete project credits are in CREDITS.txt / CREDITS.html.
