Why can't I use a variable for the homepage in Flutter?

I'm trying to bypass the Login page by using a variable for the homepage widget. I check FirebaseAuth to determine if the user is signed in already. Then I set the homepage based on yes or no. But it gives me a blank white screen. Why doesn't this work?

    @override
      Widget build(BuildContext context) {
    
        Widget? home;

    //Check to see if a user is already logged in.
    FirebaseAuth.instance
      .authStateChanges()
      .listen((User? user) {
        if (user == null) {
          home = Signup();
        } else {
          home = const HomePage();
        }
      });

    return MaterialApp(
      navigatorKey: NavigationService.navigatorKey,
      debugShowCheckedModeBanner: false, // turn off debug banner
      title: 'My App',
     
      home: home,
    );
   }
 }

Since you have not provided much code to understand the issue.Here is full code to achieve your desired way.

import 'package:flutter/material.dart';
import 'package:firebase_core/firebase_core.dart';
import 'package:firebase_auth/firebase_auth.dart';

Future<void> main() async {
  WidgetsFlutterBinding.ensureInitialized();
  await Firebase.initializeApp();
  runApp(MyApp());
}

class MyApp extends StatefulWidget {
const MyApp({super.key});

@override
State<MyApp> createState() => _MyAppState();
}

class _MyAppState extends State<MyApp> {
 // Navigator key for managing navigation globally
 final navigatorKey = GlobalKey<NavigatorState>();

 Widget? home;

 void checkLogin() {
   // Check to see if a user is already logged in.
  FirebaseAuth.instance.authStateChanges().listen((User? user) {
    if (user == null) {
      home = Signup();
    } else {
     home = const HomePage();
   }
  setState(() {});
  });
}

@override
void initState() {
  super.initState();
 checkLogin();
}

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      navigatorKey: navigatorKey,
      debugShowCheckedModeBanner: false, // Turn off debug banner
      title: 'My App',
      home: home ?? LoadingScreen(),
    );
  }
}

class NavigationService {
  static final navigatorKey = GlobalKey<NavigatorState>();
}

// Placeholder for Signup screen
class Signup extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: const Text('Signup'),
      ),
      body: Center(
        child: Text('Signup Screen'),
      ),
    );
  }
}

// Placeholder for Home screen
class HomePage extends StatelessWidget {
  const HomePage({super.key});

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: const Text('Home'),
      ),
      body: Center(
        child: Text('Welcome to the Home Page!'),
      ),
    );
  }
}

// Placeholder for Loading screen
class LoadingScreen extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(
        child: CircularProgressIndicator(),
      ),
    );
  }
}