Archives/langchain
2
0
Fork 1
mirror of https://github.com/hwchase17/langchain synced 2024-10-29 17:07:25 +00:00
Code Issues Packages Projects Releases Wiki Activity
03e5794978
langchain/docs/modules/document_loaders/examples/online_pdf.ipynb
zanderchase c2d1d903fa
Zander/online pdf loader (#984)
2023-02-10 15:42:30 -08:00

131 lines
44 KiB
Plaintext
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

{
"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",
"File \u001b[0;32m~/.pyenv/versions/3.8.1/lib/python3.8/socket.py:669\u001b[0m, in \u001b[0;36mSocketIO.readinto\u001b[0;34m(self, b)\u001b[0m\n\u001b[1;32m 667\u001b[0m \u001b[38;5;28;01mwhile\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m:\n\u001b[1;32m 668\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 669\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_sock\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrecv_into\u001b[49m\u001b[43m(\u001b[49m\u001b[43mb\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 670\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m timeout:\n\u001b[1;32m 671\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_timeout_occurred \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mTrue\u001b[39;00m\n",
"File \u001b[0;32m~/.pyenv/versions/3.8.1/lib/python3.8/ssl.py:1241\u001b[0m, in \u001b[0;36mSSLSocket.recv_into\u001b[0;34m(self, buffer, nbytes, flags)\u001b[0m\n\u001b[1;32m 1237\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m flags \u001b[38;5;241m!=\u001b[39m \u001b[38;5;241m0\u001b[39m:\n\u001b[1;32m 1238\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\n\u001b[1;32m 1239\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mnon-zero flags not allowed in calls to recv_into() on \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m\"\u001b[39m \u001b[38;5;241m%\u001b[39m\n\u001b[1;32m 1240\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m\u001b[38;5;18m__class__\u001b[39m)\n\u001b[0;32m-> 1241\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mread\u001b[49m\u001b[43m(\u001b[49m\u001b[43mnbytes\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mbuffer\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 1242\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 1243\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28msuper\u001b[39m()\u001b[38;5;241m.\u001b[39mrecv_into(buffer, nbytes, flags)\n",
"File \u001b[0;32m~/.pyenv/versions/3.8.1/lib/python3.8/ssl.py:1099\u001b[0m, in \u001b[0;36mSSLSocket.read\u001b[0;34m(self, len, buffer)\u001b[0m\n\u001b[1;32m 1097\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m 1098\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m buffer \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[0;32m-> 1099\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_sslobj\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mread\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mlen\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mbuffer\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 1100\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 1101\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_sslobj\u001b[38;5;241m.\u001b[39mread(\u001b[38;5;28mlen\u001b[39m)\n",
"\u001b[0;31mKeyboardInterrupt\u001b[0m: "
]
}
],
"source": [
"data = loader.load()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "2d48610e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[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. Pub-', lookup_str='', metadata={'source': '/var/folders/v9/s3kf_mtn2k9f4chcfvsftj380000gq/T/tmpp13fkggm/online_file.pdf'}, lookup_index=0)]\n"
]
}
],
"source": [
"print(data)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d3258869",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.1"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
Reference in New Issue View Git Blame Copy Permalink