langchain/docs/modules/document_loaders/examples/online_pdf.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "8d9e1096",
   "metadata": {},
   "source": [
    "# Online PDF\n",
    "\n",
    "This covers how to load online pdfs into a document format that we can use downstream. This can be used for various online pdf sites such as https://open.umn.edu/opentextbooks/textbooks/ and https://arxiv.org/archive/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3dde8f63",
   "metadata": {},
   "outputs": [],
   "source": [
    "from langchain.document_loaders import OnlinePDFLoader"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "04e27651",
   "metadata": {},
   "outputs": [],
   "source": [
    "loader = OnlinePDFLoader(\"https://arxiv.org/pdf/2302.03803.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "85690c73",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "/var/folders/v9/s3kf_mtn2k9f4chcfvsftj380000gq/T/tmp0nnippcn/online_file.pdf\n"
     ]
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[5], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m data \u001b[38;5;241m=\u001b[39m \u001b[43mloader\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mload\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/langchain_db/langchain/document_loaders/online_pdf.py:30\u001b[0m, in \u001b[0;36mOnlinePDFLoader.load\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m     28\u001b[0m file\u001b[38;5;241m.\u001b[39mclose()\n\u001b[1;32m     29\u001b[0m loader \u001b[38;5;241m=\u001b[39m UnstructuredPDFLoader(file_path)\n\u001b[0;32m---> 30\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mloader\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mload\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/langchain_db/langchain/document_loaders/pdf.py:26\u001b[0m, in \u001b[0;36mUnstructuredPDFLoader.load\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m     24\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01munstructured\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mpartition\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mpdf\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m partition_pdf\n\u001b[1;32m     25\u001b[0m elements \u001b[38;5;241m=\u001b[39m partition_pdf(filename\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mfile_path)\n\u001b[0;32m---> 26\u001b[0m text \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;241m.\u001b[39mjoin([\u001b[38;5;28mstr\u001b[39m(el) \u001b[38;5;28;01mfor\u001b[39;00m el \u001b[38;5;129;01min\u001b[39;00m elements])\n\u001b[1;32m     27\u001b[0m metadata \u001b[38;5;241m=\u001b[39m {\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124msource\u001b[39m\u001b[38;5;124m\"\u001b[39m: \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mfile_path}\n\u001b[1;32m     28\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m [Document(page_content\u001b[38;5;241m=\u001b[39mtext, metadata\u001b[38;5;241m=\u001b[39mmetadata)]\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/unstructured/partition/pdf.py:34\u001b[0m, in \u001b[0;36mpartition_pdf\u001b[0;34m(filename, file, url, template, token)\u001b[0m\n\u001b[1;32m     32\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m template \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[1;32m     33\u001b[0m     template \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mlayout/pdf\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[0;32m---> 34\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mpartition_pdf_or_image\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m     35\u001b[0m \u001b[43m    \u001b[49m\u001b[43mfilename\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfilename\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mfile\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfile\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43murl\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtemplate\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtemplate\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtoken\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtoken\u001b[49m\n\u001b[1;32m     36\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/unstructured/partition/pdf.py:73\u001b[0m, in \u001b[0;36mpartition_pdf_or_image\u001b[0;34m(filename, file, url, template, token, is_image)\u001b[0m\n\u001b[1;32m     71\u001b[0m     url \u001b[38;5;241m=\u001b[39m \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;132;01m{\u001b[39;00murl\u001b[38;5;241m.\u001b[39mrstrip(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124m/\u001b[39m\u001b[38;5;124m'\u001b[39m)\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m/\u001b[39m\u001b[38;5;132;01m{\u001b[39;00mtemplate\u001b[38;5;241m.\u001b[39mlstrip(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124m/\u001b[39m\u001b[38;5;124m'\u001b[39m)\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m     72\u001b[0m     \u001b[38;5;66;03m# NOTE(alan): Remove \"data=data\" after different models are handled by routing\u001b[39;00m\n\u001b[0;32m---> 73\u001b[0m     layout_elements \u001b[38;5;241m=\u001b[39m \u001b[43m_partition_via_api\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m     74\u001b[0m \u001b[43m        \u001b[49m\u001b[43mfilename\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfilename\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mfile\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfile\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43murl\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtoken\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtoken\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mdata\u001b[49m\n\u001b[1;32m     75\u001b[0m \u001b[43m    \u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     77\u001b[0m elements: List[Element] \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mlist\u001b[39m()\n\u001b[1;32m     78\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m layout_element \u001b[38;5;129;01min\u001b[39;00m layout_elements:\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/unstructured/partition/__init__.py:34\u001b[0m, in \u001b[0;36m_partition_via_api\u001b[0;34m(filename, file, url, token, data)\u001b[0m\n\u001b[1;32m     26\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mendpoint api healthcheck has failed!\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n\u001b[1;32m     28\u001b[0m file_: Mapping[\u001b[38;5;28mstr\u001b[39m, Tuple[\u001b[38;5;28mstr\u001b[39m, Union[BinaryIO, \u001b[38;5;28mbytes\u001b[39m]]] \u001b[38;5;241m=\u001b[39m {\n\u001b[1;32m     29\u001b[0m     \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mfile\u001b[39m\u001b[38;5;124m\"\u001b[39m: (\n\u001b[1;32m     30\u001b[0m         filename,\n\u001b[1;32m     31\u001b[0m         file \u001b[38;5;28;01mif\u001b[39;00m file \u001b[38;5;28;01melse\u001b[39;00m \u001b[38;5;28mopen\u001b[39m(filename, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mrb\u001b[39m\u001b[38;5;124m\"\u001b[39m),\n\u001b[1;32m     32\u001b[0m     )\n\u001b[1;32m     33\u001b[0m }\n\u001b[0;32m---> 34\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[43mrequests\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mpost\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m     35\u001b[0m \u001b[43m    \u001b[49m\u001b[43murl\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m     36\u001b[0m \u001b[43m    \u001b[49m\u001b[43mheaders\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43m{\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mAuthorization\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m:\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;124;43mf\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mBearer \u001b[39;49m\u001b[38;5;132;43;01m{\u001b[39;49;00m\u001b[43mtoken\u001b[49m\u001b[38;5;132;43;01m}\u001b[39;49;00m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mif\u001b[39;49;00m\u001b[43m \u001b[49m\u001b[43mtoken\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01melse\u001b[39;49;00m\u001b[43m \u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m}\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m     37\u001b[0m \u001b[43m    \u001b[49m\u001b[43mfiles\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfile_\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m     38\u001b[0m \u001b[43m    \u001b[49m\u001b[43mdata\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mdata\u001b[49m\u001b[43m,\u001b[49m\u001b[43m  \u001b[49m\u001b[38;5;66;43;03m# NOTE(alan): Remove after unstructured API is using routing\u001b[39;49;00m\n\u001b[1;32m     39\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     41\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m response\u001b[38;5;241m.\u001b[39mstatus_code \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m200\u001b[39m:\n\u001b[1;32m     42\u001b[0m     pages \u001b[38;5;241m=\u001b[39m response\u001b[38;5;241m.\u001b[39mjson()[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mpages\u001b[39m\u001b[38;5;124m\"\u001b[39m]\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/requests/api.py:115\u001b[0m, in \u001b[0;36mpost\u001b[0;34m(url, data, json, **kwargs)\u001b[0m\n\u001b[1;32m    103\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mpost\u001b[39m(url, data\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mNone\u001b[39;00m, json\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mNone\u001b[39;00m, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs):\n\u001b[1;32m    104\u001b[0m \u001b[38;5;250m    \u001b[39m\u001b[38;5;124mr\u001b[39m\u001b[38;5;124;03m\"\"\"Sends a POST request.\u001b[39;00m\n\u001b[1;32m    105\u001b[0m \n\u001b[1;32m    106\u001b[0m \u001b[38;5;124;03m    :param url: URL for the new :class:`Request` object.\u001b[39;00m\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m    112\u001b[0m \u001b[38;5;124;03m    :rtype: requests.Response\u001b[39;00m\n\u001b[1;32m    113\u001b[0m \u001b[38;5;124;03m    \"\"\"\u001b[39;00m\n\u001b[0;32m--> 115\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mrequest\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mpost\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mdata\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mjson\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mjson\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/requests/api.py:59\u001b[0m, in \u001b[0;36mrequest\u001b[0;34m(method, url, **kwargs)\u001b[0m\n\u001b[1;32m     55\u001b[0m \u001b[38;5;66;03m# By using the 'with' statement we are sure the session is closed, thus we\u001b[39;00m\n\u001b[1;32m     56\u001b[0m \u001b[38;5;66;03m# avoid leaving sockets open which can trigger a ResourceWarning in some\u001b[39;00m\n\u001b[1;32m     57\u001b[0m \u001b[38;5;66;03m# cases, and look like a memory leak in others.\u001b[39;00m\n\u001b[1;32m     58\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m sessions\u001b[38;5;241m.\u001b[39mSession() \u001b[38;5;28;01mas\u001b[39;00m session:\n\u001b[0;32m---> 59\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43msession\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrequest\u001b[49m\u001b[43m(\u001b[49m\u001b[43mmethod\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mmethod\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43murl\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/requests/sessions.py:587\u001b[0m, in \u001b[0;36mSession.request\u001b[0;34m(self, method, url, params, data, headers, cookies, files, auth, timeout, allow_redirects, proxies, hooks, stream, verify, cert, json)\u001b[0m\n\u001b[1;32m    582\u001b[0m send_kwargs \u001b[38;5;241m=\u001b[39m {\n\u001b[1;32m    583\u001b[0m     \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mtimeout\u001b[39m\u001b[38;5;124m\"\u001b[39m: timeout,\n\u001b[1;32m    584\u001b[0m     \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mallow_redirects\u001b[39m\u001b[38;5;124m\"\u001b[39m: allow_redirects,\n\u001b[1;32m    585\u001b[0m }\n\u001b[1;32m    586\u001b[0m send_kwargs\u001b[38;5;241m.\u001b[39mupdate(settings)\n\u001b[0;32m--> 587\u001b[0m resp \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43msend\u001b[49m\u001b[43m(\u001b[49m\u001b[43mprep\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43msend_kwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    589\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m resp\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/requests/sessions.py:701\u001b[0m, in \u001b[0;36mSession.send\u001b[0;34m(self, request, **kwargs)\u001b[0m\n\u001b[1;32m    698\u001b[0m start \u001b[38;5;241m=\u001b[39m preferred_clock()\n\u001b[1;32m    700\u001b[0m \u001b[38;5;66;03m# Send the request\u001b[39;00m\n\u001b[0;32m--> 701\u001b[0m r \u001b[38;5;241m=\u001b[39m \u001b[43madapter\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43msend\u001b[49m\u001b[43m(\u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    703\u001b[0m \u001b[38;5;66;03m# Total elapsed time of the request (approximately)\u001b[39;00m\n\u001b[1;32m    704\u001b[0m elapsed \u001b[38;5;241m=\u001b[39m preferred_clock() \u001b[38;5;241m-\u001b[39m start\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/requests/adapters.py:489\u001b[0m, in \u001b[0;36mHTTPAdapter.send\u001b[0;34m(self, request, stream, timeout, verify, cert, proxies)\u001b[0m\n\u001b[1;32m    487\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m    488\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m chunked:\n\u001b[0;32m--> 489\u001b[0m         resp \u001b[38;5;241m=\u001b[39m \u001b[43mconn\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43murlopen\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    490\u001b[0m \u001b[43m            \u001b[49m\u001b[43mmethod\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mrequest\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmethod\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    491\u001b[0m \u001b[43m            \u001b[49m\u001b[43murl\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    492\u001b[0m \u001b[43m            \u001b[49m\u001b[43mbody\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mrequest\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mbody\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    493\u001b[0m \u001b[43m            \u001b[49m\u001b[43mheaders\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mrequest\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mheaders\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    494\u001b[0m \u001b[43m            \u001b[49m\u001b[43mredirect\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43;01mFalse\u001b[39;49;00m\u001b[43m,\u001b[49m\n\u001b[1;32m    495\u001b[0m \u001b[43m            \u001b[49m\u001b[43massert_same_host\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43;01mFalse\u001b[39;49;00m\u001b[43m,\u001b[49m\n\u001b[1;32m    496\u001b[0m \u001b[43m            \u001b[49m\u001b[43mpreload_content\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43;01mFalse\u001b[39;49;00m\u001b[43m,\u001b[49m\n\u001b[1;32m    497\u001b[0m \u001b[43m            \u001b[49m\u001b[43mdecode_content\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43;01mFalse\u001b[39;49;00m\u001b[43m,\u001b[49m\n\u001b[1;32m    498\u001b[0m \u001b[43m            \u001b[49m\u001b[43mretries\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmax_retries\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    499\u001b[0m \u001b[43m            \u001b[49m\u001b[43mtimeout\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtimeout\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    500\u001b[0m \u001b[43m        \u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    502\u001b[0m     \u001b[38;5;66;03m# Send the request.\u001b[39;00m\n\u001b[1;32m    503\u001b[0m     \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m    504\u001b[0m         \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28mhasattr\u001b[39m(conn, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mproxy_pool\u001b[39m\u001b[38;5;124m\"\u001b[39m):\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/urllib3/connectionpool.py:703\u001b[0m, in \u001b[0;36mHTTPConnectionPool.urlopen\u001b[0;34m(self, method, url, body, headers, retries, redirect, assert_same_host, timeout, pool_timeout, release_conn, chunked, body_pos, **response_kw)\u001b[0m\n\u001b[1;32m    700\u001b[0m     \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_prepare_proxy(conn)\n\u001b[1;32m    702\u001b[0m \u001b[38;5;66;03m# Make the request on the httplib connection object.\u001b[39;00m\n\u001b[0;32m--> 703\u001b[0m httplib_response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_make_request\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m    704\u001b[0m \u001b[43m    \u001b[49m\u001b[43mconn\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    705\u001b[0m \u001b[43m    \u001b[49m\u001b[43mmethod\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    706\u001b[0m \u001b[43m    \u001b[49m\u001b[43murl\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    707\u001b[0m \u001b[43m    \u001b[49m\u001b[43mtimeout\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtimeout_obj\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    708\u001b[0m \u001b[43m    \u001b[49m\u001b[43mbody\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mbody\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    709\u001b[0m \u001b[43m    \u001b[49m\u001b[43mheaders\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mheaders\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    710\u001b[0m \u001b[43m    \u001b[49m\u001b[43mchunked\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mchunked\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m    711\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    713\u001b[0m \u001b[38;5;66;03m# If we're going to release the connection in ``finally:``, then\u001b[39;00m\n\u001b[1;32m    714\u001b[0m \u001b[38;5;66;03m# the response doesn't need to know about the connection. Otherwise\u001b[39;00m\n\u001b[1;32m    715\u001b[0m \u001b[38;5;66;03m# it will also try to release it and we'll have a double-release\u001b[39;00m\n\u001b[1;32m    716\u001b[0m \u001b[38;5;66;03m# mess.\u001b[39;00m\n\u001b[1;32m    717\u001b[0m response_conn \u001b[38;5;241m=\u001b[39m conn \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m release_conn \u001b[38;5;28;01melse\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/urllib3/connectionpool.py:449\u001b[0m, in \u001b[0;36mHTTPConnectionPool._make_request\u001b[0;34m(self, conn, method, url, timeout, chunked, **httplib_request_kw)\u001b[0m\n\u001b[1;32m    444\u001b[0m             httplib_response \u001b[38;5;241m=\u001b[39m conn\u001b[38;5;241m.\u001b[39mgetresponse()\n\u001b[1;32m    445\u001b[0m         \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mBaseException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m    446\u001b[0m             \u001b[38;5;66;03m# Remove the TypeError from the exception chain in\u001b[39;00m\n\u001b[1;32m    447\u001b[0m             \u001b[38;5;66;03m# Python 3 (including for exceptions like SystemExit).\u001b[39;00m\n\u001b[1;32m    448\u001b[0m             \u001b[38;5;66;03m# Otherwise it looks like a bug in the code.\u001b[39;00m\n\u001b[0;32m--> 449\u001b[0m             \u001b[43msix\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mraise_from\u001b[49m\u001b[43m(\u001b[49m\u001b[43me\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mNone\u001b[39;49;00m\u001b[43m)\u001b[49m\n\u001b[1;32m    450\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m (SocketTimeout, BaseSSLError, SocketError) \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m    451\u001b[0m     \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_raise_timeout(err\u001b[38;5;241m=\u001b[39me, url\u001b[38;5;241m=\u001b[39murl, timeout_value\u001b[38;5;241m=\u001b[39mread_timeout)\n",
      "File \u001b[0;32m<string>:3\u001b[0m, in \u001b[0;36mraise_from\u001b[0;34m(value, from_value)\u001b[0m\n",
      "File \u001b[0;32m~/langchain_db/.venv/lib/python3.8/site-packages/urllib3/connectionpool.py:444\u001b[0m, in \u001b[0;36mHTTPConnectionPool._make_request\u001b[0;34m(self, conn, method, url, timeout, chunked, **httplib_request_kw)\u001b[0m\n\u001b[1;32m    441\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mTypeError\u001b[39;00m:\n\u001b[1;32m    442\u001b[0m     \u001b[38;5;66;03m# Python 3\u001b[39;00m\n\u001b[1;32m    443\u001b[0m     \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 444\u001b[0m         httplib_response \u001b[38;5;241m=\u001b[39m \u001b[43mconn\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mgetresponse\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    445\u001b[0m     \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mBaseException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m    446\u001b[0m         \u001b[38;5;66;03m# Remove the TypeError from the exception chain in\u001b[39;00m\n\u001b[1;32m    447\u001b[0m         \u001b[38;5;66;03m# Python 3 (including for exceptions like SystemExit).\u001b[39;00m\n\u001b[1;32m    448\u001b[0m         \u001b[38;5;66;03m# Otherwise it looks like a bug in the code.\u001b[39;00m\n\u001b[1;32m    449\u001b[0m         six\u001b[38;5;241m.\u001b[39mraise_from(e, \u001b[38;5;28;01mNone\u001b[39;00m)\n",
      "File \u001b[0;32m~/.pyenv/versions/3.8.1/lib/python3.8/http/client.py:1322\u001b[0m, in \u001b[0;36mHTTPConnection.getresponse\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1320\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m   1321\u001b[0m     \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m-> 1322\u001b[0m         \u001b[43mresponse\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mbegin\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m   1323\u001b[0m     \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mConnectionError\u001b[39;00m:\n\u001b[1;32m   1324\u001b[0m         \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mclose()\n",
      "File \u001b[0;32m~/.pyenv/versions/3.8.1/lib/python3.8/http/client.py:303\u001b[0m, in \u001b[0;36mHTTPResponse.begin\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    301\u001b[0m \u001b[38;5;66;03m# read until we get a non-100 response\u001b[39;00m\n\u001b[1;32m    302\u001b[0m \u001b[38;5;28;01mwhile\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m:\n\u001b[0;32m--> 303\u001b[0m     version, status, reason \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_read_status\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    304\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m status \u001b[38;5;241m!=\u001b[39m CONTINUE:\n\u001b[1;32m    305\u001b[0m         \u001b[38;5;28;01mbreak\u001b[39;00m\n",
      "File \u001b[0;32m~/.pyenv/versions/3.8.1/lib/python3.8/http/client.py:264\u001b[0m, in \u001b[0;36mHTTPResponse._read_status\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    263\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m_read_status\u001b[39m(\u001b[38;5;28mself\u001b[39m):\n\u001b[0;32m--> 264\u001b[0m     line \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mstr\u001b[39m(\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mfp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mreadline\u001b[49m\u001b[43m(\u001b[49m\u001b[43m_MAXLINE\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m+\u001b[39;49m\u001b[43m \u001b[49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124miso-8859-1\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n\u001b[1;32m    265\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28mlen\u001b[39m(line) \u001b[38;5;241m>\u001b[39m _MAXLINE:\n\u001b[1;32m    266\u001b[0m         \u001b[38;5;28;01mraise\u001b[39;00m LineTooLong(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mstatus line\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n",
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      "[Document(page_content='A WEAK ( k, k ) -LEFSCHETZ THEOREM FOR PROJECTIVE TORIC ORBIFOLDS\\n\\nWilliam D. Montoya\\n\\nInstituto de Matem´atica, Estat´ıstica e Computa¸c˜ao Cient´ıfica,\\n\\nFirstly we show a generalization of the ( 1 , 1 ) -Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2 k -dimensional quasi-smooth hyper- surfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational ( k, k ) -cohomology class is algebraic, i.e., the Hodge conjecture holds\\n\\nIn [3] we proved that, under suitable conditions, on a very general codimension s quasi- smooth intersection subvariety X in a projective toric orbifold P d Σ with d + s = 2 ( k + 1 ) the Hodge conjecture holds, that is, every ( p, p ) -cohomology class, under the Poincar´e duality is a rational linear combination of fundamental classes of algebraic subvarieties of X . The proof of the above-mentioned result relies, for p ≠ d + 1 − s , on a Lefschetz\\n\\nKeywords: (1,1)- Lefschetz theorem, Hodge conjecture, toric varieties, complete intersection Email: wmontoya@ime.unicamp.br\\n\\ntheorem ([7]) and the Hard Lefschetz theorem for projective orbifolds ([11]). When p = d + 1 − s the proof relies on the Cayley trick, a trick which associates to X a quasi-smooth hypersurface Y in a projective vector bundle, and the Cayley Proposition (4.3) which gives an isomorphism of some primitive cohomologies (4.2) of X and Y . The Cayley trick, following the philosophy of Mavlyutov in [7], reduces results known for quasi-smooth hypersurfaces to quasi-smooth intersection subvarieties. The idea in this paper goes the other way around, we translate some results for quasi-smooth intersection subvarieties to quasi-smooth hypersurfaces, mainly the ( 1 , 1 ) -Lefschetz theorem.\\n\\nAcknowledgement. I thank Prof. Ugo Bruzzo and Tiago Fonseca for useful discus- sions. I also acknowledge support from FAPESP postdoctoral grant No. 2019/23499-7.\\n\\nPreliminaries and Notation\\n\\nLet M be a free abelian group of rank d , let N = Hom ( M, Z ) , and N R = N ⊗ Z R\\n\\nif there exist k linearly independent primitive elements e\\n\\n, . . . , e k ∈ N such that σ = { µ\\n\\ne\\n\\n+ ⋯ + µ k e k } . • The generators e i are integral if for every i and any nonnegative rational number µ the product µe i is in N only if µ is an integer. • Given two rational simplicial cones σ , σ ′ one says that σ ′ is a face of σ ( σ ′ < σ ) if the set of integral generators of σ ′ is a subset of the set of integral generators of σ . • A finite set Σ = { σ\\n\\n, . . . , σ t } of rational simplicial cones is called a rational simplicial complete d -dimensional fan if:\\n\\nall faces of cones in Σ are in Σ ;\\n\\nif σ, σ ′ ∈ Σ then σ ∩ σ ′ < σ and σ ∩ σ ′ < σ ′ ;\\n\\nN R = σ\\n\\n∪ ⋅ ⋅ ⋅ ∪ σ t .\\n\\nA rational simplicial complete d -dimensional fan Σ defines a d -dimensional toric variety P d Σ having only orbifold singularities which we assume to be projective. Moreover, T ∶ = N ⊗ Z C ∗ ≃ ( C ∗ ) d is the torus action on P d Σ . We denote by Σ ( i ) the i -dimensional cones\\n\\nFor a cone σ ∈ Σ, ˆ σ is the set of 1-dimensional cone in Σ that are not contained in σ\\n\\nand x ˆ σ ∶ = ∏ ρ ∈ ˆ σ x ρ is the associated monomial in S .\\n\\nDefinition 2.2. The irrelevant ideal of P d Σ is the monomial ideal B Σ ∶ =< x ˆ σ ∣ σ ∈ Σ > and the zero locus Z ( Σ ) ∶ = V ( B Σ ) in the affine space A d ∶ = Spec ( S ) is the irrelevant locus.\\n\\nProposition 2.3 (Theorem 5.1.11 [5]) . The toric variety P d Σ is a categorical quotient A d ∖ Z ( Σ ) by the group Hom ( Cl ( Σ ) , C ∗ ) and the group action is induced by the Cl ( Σ ) - grading of S .\\n\\nNow we give a brief introduction to complex orbifolds and we mention the needed theorems for the next section. Namely: de Rham theorem and Dolbeault theorem for complex orbifolds.\\n\\nDefinition 2.4. A complex orbifold of complex dimension d is a singular complex space whose singularities are locally isomorphic to quotient singularities C d / G , for finite sub- groups G ⊂ Gl ( d, C ) .\\n\\nDefinition 2.5. A differential form on a complex orbifold Z is defined locally at z ∈ Z as a G -invariant differential form on C d where G ⊂ Gl ( d, C ) and Z is locally isomorphic to d\\n\\nRoughly speaking the local geometry of orbifolds reduces to local G -invariant geometry.\\n\\nWe have a complex of differential forms ( A ● ( Z ) , d ) and a double complex ( A ● , ● ( Z ) , ∂, ¯ ∂ ) of bigraded differential forms which define the de Rham and the Dolbeault cohomology groups (for a fixed p ∈ N ) respectively:\\n\\n(1,1)-Lefschetz theorem for projective toric orbifolds\\n\\nDefinition 3.1. A subvariety X ⊂ P d Σ is quasi-smooth if V ( I X ) ⊂ A #Σ ( 1 ) is smooth outside\\n\\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub-\\n\\nExample 3.2 . Quasi-smooth hypersurfaces or more generally quasi-smooth intersection sub- varieties are quasi-smooth subvarieties (see [2] or [7] for more details).\\n\\nRemark 3.3 . Quasi-smooth subvarieties are suborbifolds of P d Σ in the sense of Satake in [8]. Intuitively speaking they are subvarieties whose only singularities come from the ambient\\n\\nProof. From the exponential short exact sequence\\n\\nwe have a long exact sequence in cohomology\\n\\nH 1 (O ∗ X ) → H 2 ( X, Z ) → H 2 (O X ) ≃ H 0 , 2 ( X )\\n\\nwhere the last isomorphisms is due to Steenbrink in [9]. Now,\\n\\nH 2 ( X, Z ) / / (cid:15) (cid:15) H 2 ( X, O X ) ≃ Dolbeault (cid:15) (cid:15) H 2 ( X, C ) deRham ≃ (cid:15) (cid:15) H 2 dR ( X, C ) / / H 0 , 2 ¯ ∂ ( X )\\n\\nof the proof follows as the ( 1 , 1 ) -Lefschetz theorem in [6].\\n\\nRemark 3.5 . For k = 1 and P d Σ as the projective space, we recover the classical ( 1 , 1 ) - Lefschetz theorem.\\n\\nBy the Hard Lefschetz Theorem for projective orbifolds (see [11] for details) we\\n\\nBy the Hard Lefschetz Theorem for projective orbifolds (see [11] for details) we get an\\n\\ngiven by the Lefschetz morphism and since it is a morphism of Hodge structures, we have:\\n\\nH 1 , 1 ( X, Q ) ≃ H dim X − 1 , dim X − 1 ( X, Q )\\n\\nCorollary 3.6. If the dimension of X is 1 , 2 or 3 . The Hodge conjecture holds on X\\n\\nProof. If the dim C X = 1 the result is clear by the Hard Lefschetz theorem for projective orbifolds. The dimension 2 and 3 cases are covered by Theorem 3.5 and the Hard Lefschetz.\\n\\nCayley trick and Cayley proposition\\n\\nThe Cayley trick is a way to associate to a quasi-smooth intersection subvariety a quasi- smooth hypersurface. Let L 1 , . . . , L s be line bundles on P d Σ and let π ∶ P ( E ) → P d Σ be the projective space bundle associated to the vector bundle E = L 1 ⊕ ⋯ ⊕ L s . It is known that P ( E ) is a ( d + s − 1 ) -dimensional simplicial toric variety whose fan depends on the degrees of the line bundles and the fan Σ. Furthermore, if the Cox ring, without considering the grading, of P d Σ is C [ x 1 , . . . , x m ] then the Cox ring of P ( E ) is\\n\\nMoreover for X a quasi-smooth intersection subvariety cut off by f 1 , . . . , f s with deg ( f i ) = [ L i ] we relate the hypersurface Y cut off by F = y 1 f 1 + ⋅ ⋅ ⋅ + y s f s which turns out to be quasi-smooth. For more details see Section 2 in [7].\\n\\nWe will denote P ( E ) as P d + s − 1 Σ ,X to keep track of its relation with X and P d Σ .\\n\\nThe following is a key remark.\\n\\nRemark 4.1 . There is a morphism ι ∶ X → Y ⊂ P d + s − 1 Σ ,X . Moreover every point z ∶ = ( x, y ) ∈ Y with y ≠ 0 has a preimage. Hence for any subvariety W = V ( I W ) ⊂ X ⊂ P d Σ there exists W ′ ⊂ Y ⊂ P d + s − 1 Σ ,X such that π ( W ′ ) = W , i.e., W ′ = { z = ( x, y ) ∣ x ∈ W } .\\n\\nFor X ⊂ P d Σ a quasi-smooth intersection variety the morphism in cohomology induced by the inclusion i ∗ ∶ H d − s ( P d Σ , C ) → H d − s ( X, C ) is injective by Proposition 1.4 in [7].\\n\\nDefinition 4.2. The primitive cohomology of H d − s prim ( X ) is the quotient H d − s ( X, C )/ i ∗ ( H d − s ( P d Σ , C )) and H d − s prim ( X, Q ) with rational coefficients.\\n\\nH d − s ( P d Σ , C ) and H d − s ( X, C ) have pure Hodge structures, and the morphism i ∗ is com- patible with them, so that H d − s prim ( X ) gets a pure Hodge structure.\\n\\nThe next Proposition is the Cayley proposition.\\n\\nProposition 4.3. [Proposition 2.3 in [3] ] Let X = X 1 ∩⋅ ⋅ ⋅∩ X s be a quasi-smooth intersec- tion subvariety in P d Σ cut off by homogeneous polynomials f 1 . . . f s . Then for p ≠ d + s − 1 2 , d + s − 3 2\\n\\nRemark 4.5 . The above isomorphisms are also true with rational coefficients since H ● ( X, C ) = H ● ( X, Q ) ⊗ Q C . See the beginning of Section 7.1 in [10] for more details.\\n\\nTheorem 5.1. Let Y = { F = y 1 f 1 + ⋯ + y k f k = 0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to the quasi-smooth intersection surface X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f k ⊂ P k + 2 Σ . Then on Y the Hodge conjecture holds.\\n\\nthe Hodge conjecture holds.\\n\\nProof. If H k,k prim ( X, Q ) = 0 we are done. So let us assume H k,k prim ( X, Q ) ≠ 0. By the Cayley proposition H k,k prim ( Y, Q ) ≃ H 1 , 1 prim ( X, Q ) and by the ( 1 , 1 ) -Lefschetz theorem for projective\\n\\ntoric orbifolds there is a non-zero algebraic basis λ C 1 , . . . , λ C n with rational coefficients of H 1 , 1 prim ( X, Q ) , that is, there are n ∶ = h 1 , 1 prim ( X, Q ) algebraic curves C 1 , . . . , C n in X such that under the Poincar´e duality the class in homology [ C i ] goes to λ C i , [ C i ] ↦ λ C i . Recall that the Cox ring of P k + 2 is contained in the Cox ring of P 2 k + 1 Σ ,X without considering the grading. Considering the grading we have that if α ∈ Cl ( P k + 2 Σ ) then ( α, 0 ) ∈ Cl ( P 2 k + 1 Σ ,X ) . So the polynomials defining C i ⊂ P k + 2 Σ can be interpreted in P 2 k + 1 X, Σ but with different degree. Moreover, by Remark 4.1 each C i is contained in Y = { F = y 1 f 1 + ⋯ + y k f k = 0 } and\\n\\nfurthermore it has codimension k .\\n\\nClaim: { C i } ni = 1 is a basis of prim ( ) . It is enough to prove that λ C i is different from zero in H k,k prim ( Y, Q ) or equivalently that the cohomology classes { λ C i } ni = 1 do not come from the ambient space. By contradiction, let us assume that there exists a j and C ⊂ P 2 k + 1 Σ ,X such that λ C ∈ H k,k ( P 2 k + 1 Σ ,X , Q ) with i ∗ ( λ C ) = λ C j or in terms of homology there exists a ( k + 2 ) -dimensional algebraic subvariety V ⊂ P 2 k + 1 Σ ,X such that V ∩ Y = C j so they are equal as a homology class of P 2 k + 1 Σ ,X ,i.e., [ V ∩ Y ] = [ C j ] . It is easy to check that π ( V ) ∩ X = C j as a subvariety of P k + 2 Σ where π ∶ ( x, y ) ↦ x . Hence [ π ( V ) ∩ X ] = [ C j ] which is equivalent to say that λ C j comes from P k + 2 Σ which contradicts the choice of [ C j ] .\\n\\nRemark 5.2 . Into the proof of the previous theorem, the key fact was that on X the Hodge conjecture holds and we translate it to Y by contradiction. So, using an analogous argument we have:\\n\\nargument we have:\\n\\nProposition 5.3. Let Y = { F = y 1 f s +⋯+ y s f s = 0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to a quasi-smooth intersection subvariety X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f s ⊂ P d Σ such that d + s = 2 ( k + 1 ) . If the Hodge conjecture holds on X then it holds as well on Y .\\n\\nCorollary 5.4. If the dimension of Y is 2 s − 1 , 2 s or 2 s + 1 then the Hodge conjecture holds on Y .\\n\\nProof. By Proposition 5.3 and Corollary 3.6.\\n\\n[\\n\\n] Angella, D. Cohomologies of certain orbifolds. Journal of Geometry and Physics\\n\\n(\\n\\n),\\n\\n–\\n\\n[\\n\\n] Batyrev, V. V., and Cox, D. A. On the Hodge structure of projective hypersur- faces in toric varieties. Duke Mathematical Journal\\n\\n,\\n\\n(Aug\\n\\n). [\\n\\n] Bruzzo, U., and Montoya, W. On the Hodge conjecture for quasi-smooth in- tersections in toric varieties. S˜ao Paulo J. Math. Sci. Special Section: Geometry in Algebra and Algebra in Geometry (\\n\\n). [\\n\\n] Caramello Jr, F. C. Introduction to orbifolds. a\\n\\niv:\\n\\nv\\n\\n(\\n\\n). [\\n\\n] Cox, D., Little, J., and Schenck, H. Toric varieties, vol.\\n\\nAmerican Math- ematical Soc.,\\n\\n[\\n\\n] Griffiths, P., and Harris, J. Principles of Algebraic Geometry. John Wiley & Sons, Ltd,\\n\\n[\\n\\n] Mavlyutov, A. R. Cohomology of complete intersections in toric varieties. Pub- lished in Pacific J. of Math.\\n\\nNo.\\n\\n(\\n\\n),\\n\\n–\\n\\n[\\n\\n] Satake, I. On a Generalization of the Notion of Manifold. Proceedings of the National Academy of Sciences of the United States of America\\n\\n,\\n\\n(\\n\\n),\\n\\n–\\n\\n[\\n\\n] Steenbrink, J. H. M. Intersection form for quasi-homogeneous singularities. Com- positio Mathematica\\n\\n,\\n\\n(\\n\\n),\\n\\n–\\n\\n[\\n\\n] Voisin, C. Hodge Theory and Complex Algebraic Geometry I, vol.\\n\\nof Cambridge Studies in Advanced Mathematics . Cambridge University Press,\\n\\n[\\n\\n] Wang, Z. Z., and Zaffran, D. A remark on the Hard Lefschetz theorem for K¨ahler orbifolds. Proceedings of the American Mathematical Society\\n\\n,\\n\\n(Aug\\n\\n).\\n\\n[2] Batyrev, V. V., and Cox, D. A. On the Hodge structure of projective hypersur- faces in toric varieties. Duke Mathematical Journal 75, 2 (Aug 1994).\\n\\n[\\n\\n] Bruzzo, U., and Montoya, W. On the Hodge conjecture for quasi-smooth in- tersections in toric varieties. S˜ao Paulo J. Math. Sci. Special Section: Geometry in Algebra and Algebra in Geometry (\\n\\n).\\n\\n[3] Bruzzo, U., and Montoya, W. On the Hodge conjecture for quasi-smooth in- tersections in toric varieties. S˜ao Paulo J. Math. Sci. Special Section: Geometry in Algebra and Algebra in Geometry (2021).\\n\\nCaramello Jr, F. C. Introduction to orbifolds. arXiv:1909.08699v6 (2019).\\n\\nA. R. Cohomology of complete intersections in toric varieties. 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