Intensive Lecture -- Day 1
Complex Analysis in Several Variables
SCVeamer version 1.5 -- English edition

ABOUT THIS SAMPLE

This is the complete English version of the sample lecture, including its
mathematical text, figure labels, interactive examples, prompts and navigation.
It has 47 logical slides and 77 display states, including the Beamer pauses.
The original 46 lecture slides and their timing are retained. A zero-minute
website slide is appended, showing the English Project Holomorphonautica page:
  https://tkoike.com/proj_holomorphonautica/index.html?lang=en
No video files or video demonstration slides are included.
The planned duration remains 135 minutes: 14:55 to 17:10. The lecture slides
take 2-4 minutes each, including discussion and interaction; no separate break
is scheduled.

The final mathematical question asks how strongly 1-complete or strongly 1-convex spaces
differ geometrically from weakly 1-complete spaces. The mathematical content,
examples, timing and blue theme follow the Japanese version 1.3 lecture.
There are no speaker notes or speaker guide for this sample. Branding text is
hidden in the slide footer, while the planned finish time remains visible.
During a circle test in Subharmonic?, the surface stays only slightly transparent.

OPEN THE FINISHED MATERIAL

  index.html     Complete interactive presentation; website viewing is online.
  handout.pdf    The final display state of each logical slide: 47 pages.
  talk.pdf       Every display state, including pauses: 77 pages.

Double-click index.html to open it in a browser. Viewing the finished HTML
requires no Python, TeX installation or local server. The lecture slides and
included interactive figures work offline. The final website slide needs
internet access and has a link to open the specified page in a new tab.
The kit includes this sample; there is no separate lecture ZIP to obtain.
Because this sample has no videos, index.html can be shared as one file.

EDIT AND REBUILD

1. Run scveamer.py from the English kit. On first launch, it automatically
   prepares an editable copy of this sample and opens it as "Sample slides".
2. If the Python dependency has not been installed, click
   "First-time setup (install libraries)".
3. Click "Edit source" to open talk.tex. Use LuaLaTeX in TeXworks.
   The English source uses fontspec and Latin Modern, with no Japanese font
   package requirement. A working TeX distribution is still needed to compile.
4. Save your edits. Keep "Use supplementary data (titles, prompts and speaker
   notes)" enabled and click "Build HTML from TeX". This sample has no speaker
   notes; its supplementary data supplies short titles and Ryokuinzu prompts.
5. After the build succeeds, click "Open HTML". Reload any already open tab.

If another project is open, select "Sample slides" from the project list.
Edit the working copy shown in the editor, rather than the protected bundled
original. To use a copy in another folder, choose "Open my project folder"
and select the folder containing talk.tex and presentation.json.

FILES TO KEEP TOGETHER

talk.tex
  Title page, preamble, mathematical text, editable TikZ sketches and the
  printed explanations used in place of the interactive views in the PDF.
  Edit ordinary frame environments and the observeframe commands.

theme/holomorphonautica.sty and theme/holotalk-games.sty
  The color theme, layout, interactive markers and schedule commands.
  Keep both files with the source.

assets/*.pdf
  Eight static figures for the printed versions of the interactive slides.
  Matplotlib is not needed to rebuild this lecture.

assets/levi-analytic-discs.png
  The supplied analytic-disc illustration, preserved unchanged. It illustrates
  the configuration in Observation III.2.148 and Figure III.4 of the R135 notes.
  Moving talk.tex without its assets will make the figure unavailable.

presentation.json
  Input/output filenames and options, including speaker_notes:false and
  show_brand:false. The TeX engine is lualatex.

lecture-plan.json
  Short slide titles, plus the questions and observation instructions for the
  three Ryokuinzu slides. It contains no speaker notes or planned durations.
  Update its slide numbers if slides are inserted, deleted or reordered.

CREDITS.txt / CREDITS.html
  Research project, joint development, source lecture notes, references and
  software acknowledgments.

START TIME AND PLANNED MINUTES

Set the start time in the preamble:

  \scvstarttime{14:55}

Set the duration near the start of an ordinary frame:

  \begin{frame}{Title}
    \scvminutes{3}
    Your content
  \end{frame}

This lecture uses a local wrapper that also selects the printed explanation:

  \observeframe[3]{sh}{std-b03}{Can a saddle be subharmonic?}{sh-saddle}

The optional [3] means three minutes. The four arguments are the interface,
fixed-example ID, slide title and printed-explanation name. The corresponding
SCVprint@sh-saddle definition is near the top of talk.tex.

The general kit command is:

  \exploreframe[minutes=3,lang=en]{sh}{std-b03}{Title}

If no start time is set, planned finish times are not shown. When a start time
is set, every slide needs a duration. All pauses within one slide share that
slide's time allocation. The displayed time is a cumulative schedule, not an
automatic measurement of elapsed time. Starting at 14:55 with a two-minute
title slide therefore gives a planned finish time of 14:57 for slide 1.

OUTLINE AND TIMING

Slides  Topic                                                  Minutes
 1-13   One-variable review, subharmonicity and distributions       37
14-19   Holomorphicity, domains of holomorphy, Hartogs, Ryokuinzu    17
20-22   Interactive ball, bidisc and Hartogs figure                  9
23-26   Tangent bundles, complex tangents and boundary Levi form    13
27-31   Boundary signs, analytic discs, estimates and extension     15
32      Levi pseudoconvexity                                        2
33-38   Psh definition, examples, complex Hessian and currents       18
39-41   Holomorphic convexity, exhaustions and Oka's theorem         10
42-46   Complex manifolds, 1-convexity/1-completeness and review      14
47      Project Holomorphonautica website                            0

Planned checkpoints:
  Slide 13  15:32       Slide 19  15:49       Slide 22  15:58
  Slide 26  16:11       Slide 31  16:26       Slide 32  16:28
  Slide 38  16:46       Slide 41  16:56       Slide 46  17:10
  Slide 47  17:10 (website slide; no additional planned minutes)

BOUNDARY LEVI FORM AND EXTENSION

Five slides connect boundary geometry to the behavior of holomorphic functions:

27  Boundary normal form in holomorphic coordinates, stated as a fact.  3 min
28  Positive Levi form: the local holomorphic function 1/w.             3 min
29  Negative Levi direction: analytic discs and the supplied figure.   3 min
30  Maximum principle: an interior compact set bounds the centers.     2 min
31  Local extension by a Cauchy integral and small parameter changes.  4 min

Observation III.2.148 directly gives a uniform estimate at the disc centers,
and the resulting failure of holomorphic convexity. Holomorphic extension is
explained separately using a Cauchy integral with varying parameters. The
estimate does not say that a holomorphic function is constant on each disc.
The function 1/w in the positive-Levi case is local; no global function is
claimed at that step. The normal-form proof is not included in the lecture.

PRESENTATION CONTROLS

Right arrow / Space / PageDown: next display state.
Left arrow / PageUp: previous display state.
Click an interactive figure to manipulate it. F2 switches between interacting
with the figure and controlling the slides. "Show answer and explanation"
and "Back to the question" keep the same fixed example.
Use the fullscreen button, or the browser's F11 shortcut, for fullscreen.
Compare the planned finish time at the lower right with the current time to
judge your pace. Speaker notes and the notes button are disabled in this sample.
The previous/next controls stay in fixed positions when changing slides at the
same window size. Slide 47 displays the website while keeping those controls.
The PDF and handout show its title and clickable URL instead of a live page.

NOTATION AND STATEMENT ENDINGS

The sign convention follows the R135 notes:
  Delta = -(d/dx)^2 - (d/dy)^2; subharmonicity means Delta u <= 0.
The "Laplacian measure" in Subharmonic? corresponds to -Delta u here.
The "Levi matrix" in Plurisubharmonic? is the complex Hessian H(u) of a
function. The boundary Levi form is the restriction to the complex tangent
space of the form obtained from a defining function.

Subharmonic and plurisubharmonic functions are defined as upper semicontinuous,
with -infinity allowed; the continuous plots are examples, not extra hypotheses.
Definitions, theorems, propositions, examples and questions end automatically
with a right-aligned square. Proofs use Beamer's standard QED symbol; do not
add a second square manually.

CHANGING AN INTERACTIVE EXAMPLE

The example ID and the printed figure/explanation are separate choices.
Changing an ID does not automatically change the static PDF figure. Update
the matching SCVprint@... definition and figure asset as well.
The Ryokuinzu questions and instructions come from prompt / observation in
lecture-plan.json. The SH and PSH interface text is not replaced through
those two fields. The complete 194-example catalog is at the end of the kit
manual, together with insertion instructions.

THE EIGHT FIXED EXAMPLES USED HERE

Slide  Interface / ID                  Meaning
 5     sh / std-b03                    Re(z^2)=x^2-y^2: a harmonic saddle.
 6     sh / std-b02                    -|z|^2: its circle average lies below
                                      its center value, so it is not subharmonic.
10     sh / std-i02                    max{0,Re z}: subharmonic with a crease.
20     ryokuinzu / lecture-ball        |z|^2+|w|^2<1: shrinking disc fibers.
21     ryokuinzu / lecture-polydisc    |z|<1, |w|<1: constant unit-disc fibers.
22     ryokuinzu / n18                 Hartogs figure: discs change to annuli.
34     psh / N01                      |z|^2+|w|^2: H(u)=I.
35     psh / A03                      |z|^2+|w|^2-4 Re(z conjugate(w)):
                                      positive in coordinate directions,
                                      negative in a diagonal direction.

REFERENCES AND PROJECT

The sample follows the supplied R135 lecture notes and annotated lecture
slides. Its mathematical reference map is summarized in CREDITS.txt.
Nishino's account of Oka and Ryokuinzu remains cited on the relevant slide.
The source notes and annotated PDFs are not redistributed in this kit.

Created as part of Project Holomorphonautica, the project on visualization and
geometric intuition in complex analytic geometry and several complex variables.
Research project: Developing geometric intuition in several complex variables
through AI-assisted visualization.
Program: MEXT's Supporting Pioneering Research through AI for 1,000 Discovery
challenges (SPReAD).
Award period: June 2026-January 2027.
Principal investigator: Takayuki Koike, Institute of Pure and Applied Sciences,
University of Tsukuba.

SCVeamer is jointly developed by Takayuki Koike with:
  Kyoka Marui -- Tohoku University
  Masakazu Takakura -- Tokyo Metropolitan University
  Ryo Uchimoto -- Kyushu University
This credit covers the three interactive interfaces, editor, conversion engine
and presentation player. The games' existing credits are retained.
ChatGPT (Chapiokun) assisted with coding and documentation.
Project website: https://tkoike.com/proj_holomorphonautica/

See CREDITS.txt / CREDITS.html for the complete acknowledgments and references.
