more fixes

- Drop the globe icon in the language switcher
- Only show RightArrow on nav menuitem hover
- Fix EU logo colors in light mode

apps/web/app/routes/home.tsx

@@ -43,7 +43,7 @@
    is why every inline edge below is logical (ps-/pe-/start-/end-).
    ========================================================================= */
 
-import { Link } from "react-router";
+import { Link, useRouteLoaderData } from "react-router";
 
 import { ArrowIcon, Button } from "~/components/ui/Button";
 import { Card } from "~/components/ui/Card";
@@ -61,6 +61,7 @@ import {
   buildWebSiteJsonLd,
 } from "~/lib/seo";
 import { safeHref } from "~/lib/safe-href";
+import { DEFAULT_THEME, type AppTheme } from "~/lib/theme";
 
 import type { Route } from "./+types/home";
 import { Accordion } from "./_page-kit/accordion";
@@ -434,6 +435,10 @@ export default function Home({ loaderData }: Route.ComponentProps) {
   const { locale, hero, approach, gallery, partners, news, faqs } = loaderData;
   const t = createTranslator(locale);
   const copy = pageCopy(locale).home;
+  /* The hero badge sits on the page's own surface, so it follows the theme
+     exactly as the footer's badge does: same root cookie, same emblem. */
+  const root = useRouteLoaderData<{ theme?: AppTheme }>("root");
+  const theme: AppTheme = root?.theme ?? DEFAULT_THEME;
   /* The three pillars are the About page's approved copy - one source, so the
      two pages can never drift apart. */
   const pillars = pageCopy(locale).about.pillars;
@@ -495,16 +500,17 @@ export default function Home({ loaderData }: Route.ComponentProps) {
         ]}
         reveal
         /* EU visibility sits at the top of the hero (the draft's badge). The
-           emblem is the dark-surface export because the hero sits on the
-           page's own dark surface. `emblemAlt` is left at its empty default:
-           the statement is real text beside the emblem, so naming the image
-           would read the attribution out twice. */
+           emblem matches the page surface the hero stands on, which the theme
+           cookie decides - dark emblem in dark mode, light emblem in light
+           mode, like the footer's badge. `emblemAlt` is left at its empty
+           default: the statement is real text beside the emblem, so naming the
+           image would read the attribution out twice. */
         funderBadge={
           <EuBadge
             size="sm"
             statement={t("site.euStatement")}
             locale={locale}
-            surface="dark"
+            surface={theme}
             className="self-start"
           />
         }