Are there any temples for Matsya, Kurma, and Varaha avatars?

Question

Are there any temples for the following three avatars of Vishnu?

1. Matsya
2. Kurma
3. Varaha

If yes, please share some information about some popular temples for them.

Answer 1

Yes, there are temples for all three.

1. Matsya

   Most temples to Matsya are in South India. First of all, there is the Vedanarayana Swami Temple in Nagalapuram, a village in Andhra Pradesh. Matsya is called Vedanarayana because he rescued the Vedas after the demon Hayagrivasura stole them from Brahma. You can see Matsya on the exterior of the temple (taken from here):

   

   Then there is the very ancient Rajagopala Swami Temple in Mannarkovil, a village in Tamil Nadu which is most famous because of the poems of the Vaishnava saint Kulashekhara Alwar. Ravana's brother Vibhishana is said to have worshipped Matsya at this location while he was heading back to Lanka after Rama's coronation. (This would have been right after he lost the Ranganatha statue at Sri Rangam, since Sri Rangam is north of Mannarkovil.) Here is a depiction of Matsya in the temple:

2. Kurma

   Here is the idol of Kurma from the Kurma Varadaraja Swami Temple, a Pallava-era temple in the village of Kurmai (named after Kurma) in Andhra Pradesh:

   

   That idol doesn't look much like a turtle, but the famous Swayambhu (divinely created) idol at the Sri Kurmam Temple, also in Andhra Pradesh, clearly looks like a turtle:

   

3. Varaha

   The question of Varaha temples is the most interesting, because like Vishnu's incarnation Parashurama, Varaha is still alive on the earth today. Most people only know about how Varaha defeated the demon Hiranyaksha and retrieved the Earth from the water. But afterwards he came down and started living on the Earth for the benefit of his devotees, as described in the Venkatachala Mahatmya of the Skanda Purana (page 12 of this PDF):

   After eulogizing the Lord thus Goddess Earth [Vishnu's wife Bhumidevi] saluted him (falling on his feet. On seeing her saluting the eyes of The Lord beamed with delight. He lifted up Goddess Earth with her arms and Embraced her. Her sniffed the face of [Bhumidevi] and placed her on his left lap. He then rode on Lord Garuda and went to Vrishabhachala. The Lord of the Earth was Eulogized by the leading sages beginning with Narada. On the western bank of Swamipushkarini, honored by all the worlds, remains seated The Lord having the face of Boar. He is worshipped there by the leading sages, Vaikhanasas ("anchorites") of great magnificence, noble souls on a par with Brahma.

   This area near the Swamipushkarini lake came to be known as Adi Varaha Kshetra (the place of the first Varaha), and it has been called the "oldest place on Earth", because Varaha has been living there since the beginning of the present Kalpa. (That's why the present Kalpa is called Shwetavaraha Kalpa - the Kalpa of white Varaha.)

   But this place became even more famous when Lakshmi abandoned Vishnu after being kicked by Bhrighu. Lakshmi came down to the Earth and started living in Adi Varaha Kshetra, and then Vishnu himself came down there to find her. He requested permission to stay in that place from Vahara (Vishnu's own incarnation!). Varaha agreed, but on the condition that anyone who worships Vishnu in Adi Varaha Kshetra must first pay his respects to Varaha. Not to keep you in suspense, Adi Varaha Kshetra is now called Tirupati, and the story I'm alluding to is the famous story of Venkateshwara.

   And to this day, devotees to Vishnu's temple in Tirupati, the world-renowned Tirumala Venkateshwara Temple located on the southern bank of Swamipushkarini, are supposed to first go and worship Varaha. Since Varaha lives on the western bank of Pushkarini, a temple was created there for him by Ramunjacharya, as described in this excerpt from the Venkatachala Itihasamala. It is the great Varaha Swami Temple:

   

   So the next time you go to Tirupati, make sure you go to that temple first. As the Skanda Purana describes, you might even see a glimpse of Varaha himself running around as a white boar!

Answer 2

Sri Matsya Narayana Temple

The Matsya Narayana Temple at Omkar hills in south Bangalore is a temple dedicated to the Matsya Avatar of Vishnu. From the official page of the temple:

Sri Matsyanarayana Temple in Omkara Ashrama is the only temple of its kind in Karnataka of Bhagavan Sri Matsyanarayana Swamy.Matsya Avatar was the first Avatar of Lord Vishnu, among the Dashavatara of Mahavishnu. Matsya means ‘fish’ in Sanskrit and Matsya Avatar is the incarnation of Lord Vishnu in the form of Fish.

One photo of the main deity is available here:

If you want to pay a visit, the address is Omkar Hills, Srinivaspura, Bengaluru-60, India.

---

Gavi Ranganatha Temple

The Kurmai and Kurmam temples mentioned in Keshav's answer are undoubtedly the most famous Kurmavatar temples. Located in Gavirangapura, the Gavi Ranganatha temple is another place where the main deity is Kurma (despite it's name). There aren't many resources available about this particular place, online. I found a small slideshow of a few pictures from that place. There is another YouTube video from inside the temple.

---

Shweta Varaha Swamy Temple

The Shweta Varahaswamy temple is perhaps the most famous temple in the complex of the Mysore Palace. From it's wikipedia page

According to the Archaeological Survey of India, the Shveta Varahaswamy temple was constructed in the Hoysala style during the rule of Chikkadevararaja Wodeyar (r.1673-1704).

Varaha avatar has a special place because it was the family deity of the erstwhile Mysore Wodeyar family. You can read more about the temple in this blog. There are no photographs of the deity available as photography is banned inside the temple.

Bhoo Varaha Swamy Temple

The Bhoo Varahaswamy temple is present on the banks of Hemavathi river in Mandya district of the State of Karnataka, India. The legend of this temple is detailed in this blog, and the history behind the renovation of the temple is mentioned in this news report

The photo of the main deity is available here

Answer 3

Sreevaraham Lakshmivaraha Temple Thiruvananthapuram

Sreevaraham Lakshmi Varaha Temple is situated at Sreevaraham near Sree Padmanabhaswami Temple in Thiruvananthapuram City. This a west facing Mahakshtera with golden Dwaja Prathishta. The main deity in sreevaraham temple is Lord Vishnu in Varaha avatar along with Goddess Mahalakshmi on his left lap.

Varaha Teertham

The pond infront of the temple is famous as Varaha Theertham. This is one among the Theerthas mentioned in Syanandoora Purana Samuchaya - a Purana describing the glory of Padmanabha Swami Temple. The pond is the biggest temple pond in Kerala (about 8 Acres). A mandapam is located at the centre of the pond. Kunnandan Mahaganapathy temple (facing Sree Varaha Moorthy) is situated at the west side of the pond. Varahtheertham was the bathing place for the Brahmins/Nampoothiris who participated in Murajapam ritual conducted once in six years at Padmanabhaswamy Temple. In ancient days the Karamana river was conneced with Padmatheertham of Padmanabha Swamy Temple and it is connected with Varahatheertham through a canal for maintaining the water quality. The pond was cleaned in 2012 after several years with the help of Thiruvananthapuram Corporation, Travancore Devaswom Board and local people. The pond is big enough to conduct even Vallamkali (boat race).

Answer 4

"Information on Varaha Temple Only"

On the South Bank of the Pushkarini tank in Tirumala, next to the main temple is a temple dedicated to Varaha Swamy. The Sthala Purana has it that the temple was orginally dedicated to Varaha Swamy who resides in a divine vimana over the present temple.

Skanda Purana in 'Venkatesha Mahatmiyam' describes Bhoodevi approaching Varahaswamy at his residence in Mount Meru. Varaha Swamy questions Bhoodevi as to what she now requires, since she is securely placed on one of the hoods of Sesha Naga. Bhoodevi requests Varaha Swamy to stay on Earth till the end of Shweta Varaha Kalpa and the Lord obliges.

Back to Sthala Purana - When Srinivasa needed a place to stay since His marriage to Padmavati had been solmenised, He met Varaha Swamy. Varaha Swamy willingly agreed to move out of the temple, but wanted to dwell near the Pushkarini Teertha. So Srinivasa requested Him to construct a dwelling for Himself. Being obliged to Varaha Swamy, Srinivasa declared that whoever wanted to have His darshan would first have to visit and Have the Darshan of Lord Varaha before coming to Him.

This is an important Temple of Varaha Swamy.