diff --git a/swh/deposit/api/checks.py b/swh/deposit/api/checks.py
--- a/swh/deposit/api/checks.py
+++ b/swh/deposit/api/checks.py
@@ -14,7 +14,6 @@
 from typing import Dict, Optional, Tuple
 
 MANDATORY_FIELDS_MISSING = "Mandatory fields are missing"
-ALTERNATE_FIELDS_MISSING = "Mandatory alternate fields are missing"
 
 
 def check_metadata(metadata: Dict) -> Tuple[bool, Optional[Dict]]:
@@ -28,36 +27,22 @@
           ok (False, <detailed-error>) otherwise.
 
     """
-    # following fields are mandatory
-    required_fields = {
-        "atom:author": False,
-    }
     # at least one value per couple below is mandatory
     alternate_fields = {
-        ("atom:name", "atom:title"): False,
+        ("atom:name", "atom:title", "codemeta:name"): False,
+        ("atom:author", "codemeta:author"): False,
     }
 
     for field, value in metadata.items():
-        for name in required_fields:
-            if name in field:
-                required_fields[name] = True
-
         for possible_names in alternate_fields:
             for possible_name in possible_names:
                 if possible_name in field:
                     alternate_fields[possible_names] = True
                     continue
 
-    mandatory_result = [k for k, v in required_fields.items() if not v]
-    optional_result = [" or ".join(k) for k, v in alternate_fields.items() if not v]
+    mandatory_result = [" or ".join(k) for k, v in alternate_fields.items() if not v]
 
-    if mandatory_result == [] and optional_result == []:
+    if mandatory_result == []:
         return True, None
-    detail = []
-    if mandatory_result != []:
-        detail.append({"summary": MANDATORY_FIELDS_MISSING, "fields": mandatory_result})
-    if optional_result != []:
-        detail.append(
-            {"summary": ALTERNATE_FIELDS_MISSING, "fields": optional_result,}
-        )
+    detail = [{"summary": MANDATORY_FIELDS_MISSING, "fields": mandatory_result}]
     return False, {"metadata": detail}
diff --git a/swh/deposit/tests/api/test_checks.py b/swh/deposit/tests/api/test_checks.py
--- a/swh/deposit/tests/api/test_checks.py
+++ b/swh/deposit/tests/api/test_checks.py
@@ -23,11 +23,12 @@
             "atom:title": "bar",
             "atom:author": "no one",
         },
+        {"atom:url": "some url", "codemeta:name": "bar", "codemeta:author": "no one",},
     ],
 )
 def test_api_checks_check_metadata_ok(metadata_ok, swh_checks_deposit):
     actual_check, detail = check_metadata(metadata_ok)
-    assert actual_check is True
+    assert actual_check is True, detail
     assert detail is None
 
 
@@ -41,8 +42,8 @@
                 "atom:author": "someone",
             },
             {
-                "summary": "Mandatory alternate fields are missing",
-                "fields": ["atom:name or atom:title"],
+                "summary": "Mandatory fields are missing",
+                "fields": ["atom:name or atom:title or codemeta:name"],
             },
         ),
         (
@@ -51,7 +52,34 @@
                 "atom:external_identifier": "something-else",
                 "atom:title": "foobar",
             },
-            {"summary": "Mandatory fields are missing", "fields": ["atom:author"],},
+            {
+                "summary": "Mandatory fields are missing",
+                "fields": ["atom:author or codemeta:author"],
+            },
+        ),
+        (
+            {
+                "atom:url": "something",
+                "atom:external_identifier": "something-else",
+                "codemeta:title": "bar",
+                "atom:author": "someone",
+            },
+            {
+                "summary": "Mandatory fields are missing",
+                "fields": ["atom:name or atom:title or codemeta:name"],
+            },
+        ),
+        (
+            {
+                "atom:url": "something",
+                "atom:external_identifier": "something-else",
+                "atom:title": "foobar",
+                "author": "foo",
+            },
+            {
+                "summary": "Mandatory fields are missing",
+                "fields": ["atom:author or codemeta:author"],
+            },
         ),
     ],
 )
diff --git a/swh/deposit/tests/api/test_deposit_private_check.py b/swh/deposit/tests/api/test_deposit_private_check.py
--- a/swh/deposit/tests/api/test_deposit_private_check.py
+++ b/swh/deposit/tests/api/test_deposit_private_check.py
@@ -7,7 +7,7 @@
 import pytest
 from rest_framework import status
 
-from swh.deposit.api.checks import ALTERNATE_FIELDS_MISSING, MANDATORY_FIELDS_MISSING
+from swh.deposit.api.checks import MANDATORY_FIELDS_MISSING
 from swh.deposit.api.private.deposit_check import (
     MANDATORY_ARCHIVE_INVALID,
     MANDATORY_ARCHIVE_MISSING,
@@ -130,13 +130,15 @@
         assert len(details["archive"]) == 1
         assert details["archive"][0]["summary"] == MANDATORY_ARCHIVE_UNSUPPORTED
         # metadata check failure
-        assert len(details["metadata"]) == 2
+        assert len(details["metadata"]) == 1
         mandatory = details["metadata"][0]
         assert mandatory["summary"] == MANDATORY_FIELDS_MISSING
-        assert set(mandatory["fields"]) == set(["atom:author"])
-        alternate = details["metadata"][1]
-        assert alternate["summary"] == ALTERNATE_FIELDS_MISSING
-        assert alternate["fields"] == ["atom:name or atom:title"]
+        assert set(mandatory["fields"]) == set(
+            [
+                "atom:author or codemeta:author",
+                "atom:name or atom:title or codemeta:name",
+            ]
+        )
 
         deposit = Deposit.objects.get(pk=deposit.id)
         assert deposit.status == DEPOSIT_STATUS_REJECTED
diff --git a/swh/deposit/tests/api/test_deposit_update.py b/swh/deposit/tests/api/test_deposit_update.py
--- a/swh/deposit/tests/api/test_deposit_update.py
+++ b/swh/deposit/tests/api/test_deposit_update.py
@@ -796,11 +796,12 @@
     assert response.status_code == status.HTTP_400_BAD_REQUEST
     assert b"Functional metadata checks failure" in response.content
     # detail on the errors
-    assert b"- Mandatory fields are missing (atom:author)" in response.content
-    assert (
-        b"- Mandatory alternate fields are missing (atom:name or atom:title)"
-        in response.content
+    msg = (
+        b"- Mandatory fields are missing ("
+        b"atom:name or atom:title or codemeta:name, "
+        b"atom:author or codemeta:author)"
     )
+    assert msg in response.content
 
 
 def test_put_atom_with_create_origin_and_external_identifier(